{ "course": "MaThematical_Modeling", "course_id": "CO2011", "schema_version": "material.v1", "slides": [ { "page_index": 0, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_001.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_001.png", "page_index": 0, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:03+07:00" }, "raw_text": "Propositional Logic Review I Nguyen An Khuong. Chapter 1a Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Propositional Logic Review I BK TP.HCM Mathematical Modeling (CO2011) Contents Introduction Declarative Sentences Natural Deduction Materials drawn from Chapter 1 in: Sequents Rules for natural deduction \"Michael Huth and Mark Ryan. Logic in Computer Science: Modelling and Basic and Derived Rules Reasoning about Systems, 2nd Ed., Cambridge University Press, 2006.\" Intuitionistic Logic Formal Language Nguyen An Khuong. Semantics Meaning of Logical Tran Van Hoai, Connectives Huynh Tuong Nguyen, Preview: Soundness and Completeness Le H`ng Trang Normal Form Faculty of Computer Science and Engineering Homeworks and Next Week Plan? University of Technology, VNU-HCM 1a.1" }, { "page_index": 1, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_001.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_001.png", "page_index": 1, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:07+07:00" }, "raw_text": "Propositional Logic Review Il Nguyen An Khuong. Chapter 1b Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai Propositional Logic Review Il BK (SAT Solving and Application) TP.HCM Mathematics Modeling Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P Materials drawn from Chapter 1 in: An example 'Michael Huth and Mark Ryan. Logic in Computer Science: Modelling and UNSAT Graphical View of 2-SAT Reasoning about Systems, 2nd Ed., Cambridge University Press, 2006' SAT Solvers and some other sources) WalkSAT: Idea Nguyen An Khuong, DPLL: Idea A Linear Solver Le Hong Trang, A Cubic Solver Huynh Tuong Nguyen, Homeworks and Next Week Plan? Tran Van Hoai Faculty of Computer Science and Engineering University of Technology, VNU-HCM 1b.1" }, { "page_index": 2, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_001.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_001.png", "page_index": 2, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:11+07:00" }, "raw_text": "Advanced Predicate Logic Nguyen An Khuong. Chapter 1c Huynh Tuong Nguyen Advanced Predicate Logic BK TP.HCM Discrete Mathematics /l Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Materials drawn from Chapter 2 in: Proof Theory of Predicate Logic \"Michael Huth and Mark Ryan. Logic in Computer Science: Modelling and Quantifier Equivalences Reasoning about Systems, 2nd Ed., Cambridge University Press, 2006.\") Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Compactness of Predicate Calculus Faculty of Computer Science and Engineering Homeworks and Next University of Technology, VNU-HCM Week Plan? 1c.1" }, { "page_index": 3, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_001.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_001.png", "page_index": 3, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:14+07:00" }, "raw_text": "Examples on Using Proposition and Predicate Logic Chapter 1d Nguyen An Khuong. Huynh Tuong Nguyen Examples on Using Proposition and BK Predicate Logic TP.HCM Contents Discrete Mathematics il Natural Deduction in Propositional Logic: Electing Puzzle Expressing specifications by Predicate Logic Protocol Requirements (Materials drawn from Chapter 2 in: \"Michael Huth and Mark Ryan. Logic in Computer Science: Modelling and Reasoning about Systems, 2nd Ed., Cambridge University Press, 2006.\") Nguyen An Khuong, Huynh Tuong Nguyen Faculty of Computer Science and Engineering University of Technology, VNU-HCM 1d.1" }, { "page_index": 4, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_001.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_001.png", "page_index": 4, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:17+07:00" }, "raw_text": "Predicate Logic and Program Verification Nguyen An Khuong. Chapter 1e Huynh Tuong Nguyen Predicate Logic and Program Verification BK TP.HCM Mathematical Modeling (CO2011) Contents Warm-up questions Program Verification Homeworks Materials drawn from \"Michael Huth and Mark Ryan. Logic in Computer Science: Modelling and Reasoning about Systems, 2nd Ed., Cambridge University Press, 2006.\") Nguyen An Khuong, Huynh Tuong Nguyen Faculty of Computer Science and Engineering University of Technology, VNU-HCM 1e.1" }, { "page_index": 5, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_001.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_001.png", "page_index": 5, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:19+07:00" }, "raw_text": "Program Verification Nguyen An Khuong Chapter 1f BK Program Verification TP.HCM Mathematical Modeling (CO2011) Contents Core Programming Language Hoare Triples;Partial and Total Correctness Proof Calculus for Partial Correctness Materials drawn from: Practical Aspects of Correctness Proofs \"Michael Huth and Mark Ryan. Logic in Computer Science: Modelling and Correctness of the Reasoning about Systems, 2nd Ed., Cambridge University Press, 2006.\") Factorial Function Proof Calculus for Total Correctness Homeworks Nguyen An Khuong Faculty of Computer Science and Engineering University of Technology, VNU-HCM 1f.1" }, { "page_index": 6, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_002.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_002.png", "page_index": 6, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:23+07:00" }, "raw_text": "Contents Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Propositional Calculus: Declarative Sentences Propositional Calculus: Natural Deduction BK TP.HCM Sequents Rules for natural deduction Contents Basic and Derived Rules Excursion: Intuitionistic Logic Introduction Declarative Sentences Natural Deduction Propositional Logic as a Formal Language Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Semantics of Propositional Logic Formal Language Meaning of Logical Connectives Semantics Preview: Soundness and Completeness Meaning of Logical Connectives Preview: Soundness and Completeness 5) Conjunctive Normal Form Normal Form Homeworks and Next Week Plan? 1a.2" }, { "page_index": 7, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_002.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_002.png", "page_index": 7, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:27+07:00" }, "raw_text": "Contents Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. 1 Introduction Tran Van Hoai Quick review Boolean Satisfiability (SAT) BK Intermezzo: Classification of problems according to their TP.HCM difficulty Contents 2 2-SAT is in P Introduction Quick review An example Boolean Satisfiability SAT) An Efficient Algorithm based on Unit Clause Propogation P and NP 2-SAT is in P Graphical View of 2-SAT An example UNSAT Graphical View of 2SAT SAT Solvers SAT Solvers WalkSAT: ldea WalkSAT: Idea DPLL: Idea DPLL: Idea A Linear Solver A Cubic Solver A Linear Solver Homeworks and Next A Cubic Solver Week Plan? 1b.2" }, { "page_index": 8, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_002.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_002.png", "page_index": 8, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:30+07:00" }, "raw_text": "Contents Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen 1 Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language BK TP.HCM Predicate Logic as Formal Language Proof Theory of Predicate Logic Contents Quantifier Equivalences Predicate Logic Motivation, Syntax, Proof Theory Semantics of Predicate Logic Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Soundness and Completeness of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Undecidability of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Compactness of Predicate Calculus Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.2" }, { "page_index": 9, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_002.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_002.png", "page_index": 9, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:32+07:00" }, "raw_text": "Contents Examples on Using Proposition and Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Natural Deduction in Propositional Logic: Electing Puzzle Contents Natural Deduction in Propositional Logic: Electing Puzzle Expressing specifications by Expressing specifications by Predicate Logic: Protocol Predicate Logic: Protocol Requirements Requirements 1d.2" }, { "page_index": 10, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_002.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_002.png", "page_index": 10, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:34+07:00" }, "raw_text": "Contents Predicate Logic and Program Verification Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Warm-up questions Contents Warm-up questions Program Verification Homeworks Program Verification 3 Homeworks 1e.2" }, { "page_index": 11, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_002.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_002.png", "page_index": 11, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:37+07:00" }, "raw_text": "Contents Program Verification Nguyen An Khuong Core Programming Language BK TP.HCM Hoare Triples; Partial and Total Correctness Contents Core Programming Language 3) Proof Calculus for Partial Correctness Hoare Triples; Partial and Total Correctness Proof Calculus for Practical Aspects of Correctness Proofs Partial Correctness Practical Aspects of Correctness Proofs Correctness of the Factorial Function Correctness of the Factorial Function Proof Calculus for Total Correctness 6 Proof Calculus for Total Correctness Homeworks Homeworks 1f.2" }, { "page_index": 12, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_003.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_003.png", "page_index": 12, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:40+07:00" }, "raw_text": "Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang Propositional Calculus: Declarative Sentences BK 2 Propositional Calculus: Natural Deduction TP.HCM 3 Propositional Logic as a Formal Language Contents Introduction Declarative Sentences 4 Semantics of Propositional Logic Natural Deduction Sequents Rules for natural deduction 5 Conjunctive Normal Form Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.3" }, { "page_index": 13, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_003.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_003.png", "page_index": 13, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:43+07:00" }, "raw_text": "Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai 1 lntroduction Quick review BK Boolean Satisfiability (SAT) TP.HCM Intermezzo: Classification of problems according to their difficulty Contents Introduction Quick review 2 2-SAT is in P Boolean Satisfiability (SAT) P and NP 2-SAT is in P 3 SAT Solvers An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.3" }, { "page_index": 14, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_003.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_003.png", "page_index": 14, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:48+07:00" }, "raw_text": "Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language BK TP.HCM Proof Theory of Predicate Logic Quantifier Equivalences Contents Predicate Logic: 2 Semantics of Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal 3 Soundness and Completeness of Predicate Logic Language Proof Theory of Predicate Logic Quantifier Equivalences 4 Undecidability of Predicate Logic Semantics of Predicate Logic Soundness and Completeness of 5 Compactness of Predicate Calculus Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.3" }, { "page_index": 15, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_003.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_003.png", "page_index": 15, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:50+07:00" }, "raw_text": "Electing Puzzle Examples on Using Proposition and Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK Four men and four women are nominated for two positions. TP.HCM Exactly one man and one woman are elected Contents The men are A,B,C.D and the women are E,F,G,H. We Natural Deduction in know: Propositional Logic Electing Puzzle . if neither A nor E won, then G won Expressing specifications by . if neither A nor F won, then B won Predicate Logic: Protocol Requirements . if neither B nor G won, then C won . if neither C nor F won, then E won. Who were the two people elected? 1d.3" }, { "page_index": 16, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_003.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_003.png", "page_index": 16, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:54+07:00" }, "raw_text": "Warm-up questions Predicate Logic and Program Verification Nguyen An Khuong, Huynh Tuong Nguyen a) Are there expressions in Predicate Logic that do not evaluate to TRUE or FALSE? If so, give an example. Ans.: Terms, unlike predicates and formulas, do not evaluate to the BK TP.HCM distinguished symbols true or false. Examples of terms include: a, a constant (or 0-ary function); x, a variable; f(t): a unary function f applied to a term t. Contents Warm-up questions b) ls p(a) -> 3x.p(x) a valid formula? Program Verification Ans.: Yes Homeworks c) How do you represent a propositional variable (as used in Propositional Logic) in a Predicate Logic formula? Ans.: As a 0-ary predicate. d) Fermat's Last Theorem is the name of the statement in number theory that: It is impossible to separate any power higher than the second into two like powers. Or, more precisely: If an integer n is greater than 2, then the equation x\" + y\" = z\" has no solutions in positive integers x, y, and z. Formulate the above statement in Predicate Logic with Equality? 1e.3" }, { "page_index": 17, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_003.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_003.png", "page_index": 17, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:24:58+07:00" }, "raw_text": "Motivation Program Verification Nguyen An Khuong BK TP.HCM One way of checking the correctness of programs is to explore Contents the possible states that a computation system can reach Core Programming Language during the execution of the program. Hoare Triples; Partial . Problems with this mode/ checking approach: and Total Correctness Proof Calculus for - Models become infinite. Partial Correctness - Satisfaction/validity becomes undecidable Practical Aspects of Correctness Proofs In this lecture, we cover a proof-based framework for program Correctness of the Factorial Function verification. Proof Calculus fo Total Correctness Homeworks 1f.3" }, { "page_index": 18, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_004.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_004.png", "page_index": 18, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:02+07:00" }, "raw_text": "Propositional Calculus Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Study of atomic propositions Propositions are built from sentences whose internal structure is BK not of concern. TP.HCM Building propositions Contents Boolean operators are used to construct propositions out of Introduction simpler propositions Declarative Sentences Natural Deduction Sequents Example for Propositional Calculus Rules for natural deduction Basic and Derived Rules Intuitionistic Logic : Atomic proposition: One plus one equals two Formal Language Atomic proposition: The earth revolves around the sun. Semantics Meaning of Logical Combined proposition: One plus one equals two and the Connectives Preview: Soundness and earth revolves around the sun. Completeness Normal Form Homeworks and Next Week Plan? 1a.4" }, { "page_index": 19, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_004.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_004.png", "page_index": 19, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:04+07:00" }, "raw_text": "Motivated Example - A Logic Puzzle Propositional Logic Review II Nguyen An Khuong Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.4" }, { "page_index": 20, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_004.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_004.png", "page_index": 20, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:07+07:00" }, "raw_text": "More Declarative Sentences Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Propositional logic can easily handle simple declarative BK statements such as: TP.HCM Example Contents Student Hung enrolled in DMll Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.4" }, { "page_index": 21, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_004.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_004.png", "page_index": 21, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:11+07:00" }, "raw_text": "Huth and Ryan [2], Exercises 2.1.5: Protocol Requirements Examples on Using Proposition and Predicate Logic : The following sentences are taken from the RFC3157 Nguyen An Khuong. Internet Task-force Document 'Securely Available Huynh Tuong Nguyen Credentials - Requirements.' Specify it in predicate logic, defining predicate symbols as BK appropriate: TP.HCM a. An attacker can persuade a server that a successful login has occurred, even if it hasn't. Contents b. An attacker can overwrite someone else's credentials on the Natural Deduction in server. Propositional Logic: Electing Puzzle c. All users enter passwords instead of names. Expressing d. Credential transfer both to and from a device MUST be specificationsby Predicate Logic supported. Protocol Requirements e. Credentials MUST NOT be forced by the protocol to be present in cleartext at any device other than the end user's f. The protocol MUST support a range of cryptographic algorithms, including symmetric and asymmetric algorithms hash algorithms, and MAC algorithms. g. Credentials MUST only be downloadable following user authentication or else only downloadable in a format that requires completion of user authentication for deciphering. h. Different end user devices MAY be used to download, upload or manage the same set of credentials. 1d.4" }, { "page_index": 22, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_004.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_004.png", "page_index": 22, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:13+07:00" }, "raw_text": "Warm-up questions (cont'd): An answer to Fermat's Last Predicate Logic and Program Verification Theorem Formulation Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Contents Warm-up questions Vn.integer(n) n > 2 -> Vx,y,z.integer(x) integer(y) Program Verificatior integer(z) x >0y >0Az > 0->xn +y\" # zn. Homeworks 1e.4" }, { "page_index": 23, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_004.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_004.png", "page_index": 23, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:16+07:00" }, "raw_text": "Characteristics of the Approach Program Verification Nguyen An Khuong BK TP.HCM Contents Proof-based instead of model checking Core Programming Language Semi-automatic instead of automatic Hoare TriplesPartial and Total Correctness Property-oriented not using full specification Proof Calculus for Partial Correctness Application domain fixed to sequential programs using integers Practical Aspects of Interleaved with development rather than a-posteriori verification Correctness Proofs Correctness of the Factorial Function Proof Calculus fol Total Correctness Homeworks 1f.4" }, { "page_index": 24, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_005.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_005.png", "page_index": 24, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:19+07:00" }, "raw_text": "Goals and Main Result of Propositional Calculus Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Meaning of formula BK Associate meaning to a set of formulas by assigning a value true or TP.HCM fa/se to every formula in the set. Contents Proofs Introduction Declarative Sentences Symbol sequence that formally establishes whether a formula is Natural Deduction always true. Sequents Rules for natural deduction Basic and Derived Rules Soundness and completeness Intuitionistic Logic Formal Language The set of provable formulas is the same as the set of formulas Semantics which are always true. Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.5" }, { "page_index": 25, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_005.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_005.png", "page_index": 25, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:21+07:00" }, "raw_text": "Motivated Example - A Logic Puzzle Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai . If the unicorn is mythical, then it is immortal; BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.4" }, { "page_index": 26, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_005.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_005.png", "page_index": 26, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:25+07:00" }, "raw_text": "More Declarative Sentences Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Propositional logic can easily handle simple declarative BK statements such as: TP.HCM Example Contents Student Hung enrolled in DMll. Predicate Logic Motivation, Syntax, Proof Theory Propositional logic can also handle combinations of such Need for Richer Language statements such as: Predicate Logic as Formal Language Proof Theory of Predicate Example Logic Quantifier Equivalences Student Hung enrolled in Tutorial 1, and student Cuong is enrolled Semantics of Predicate Logic in Tutorial 2. Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.4" }, { "page_index": 27, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_005.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_d/slide_005.png", "page_index": 27, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:27+07:00" }, "raw_text": "Huth and Ryan [2], Exercises 2.1.5: Solutions Examples on Using Proposition and Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM a. An attacker can persuade a server that a successful login has Contents occurred, even if it hasn't: Natural Deduction in Propositional Logic: :=3as((-loggedIn(a,s)) ->(canPersuade(a,s))) Electing Puzzle Expressing b. An attacker can overwrite someone else's credentials on the specifications by server: $ := 3ucs3d((-ownsCredentials(u,c) - Predicate Logic Protocol Requirements canWrite(u,c,s,d)) 1d.5" }, { "page_index": 28, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_005.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_005.png", "page_index": 28, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:30+07:00" }, "raw_text": "Program Verification Predicate Logic and Program Verification Nguyen An Khuong. Huynh Tuong Nguyen . Below is a function written in an imperative programming language to perform binary search, by returning TRUE iff the BK array a contains the value e and FALSE otherwise, under the TP.HCM assumption that the input range is sorted. bool binarySearch ( int [] a, int l, int u, int e) Contents if (l > u) return false ; Warm-up questions else { Program Verification int m = (l + u) div 2; Homeworks if (a[m] == e) return true ; else if (a[m] < e) return binarySearch (a, m + 1, u, e); else return binarySearch (a, l, m - 1, e); } } As a first step towards determining whether an implementation (such as that in the function above) fulfills its specification, the specification has to be formalized. We do so in terms of preconditions and postconditions 1e.5" }, { "page_index": 29, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_005.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_005.png", "page_index": 29, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:33+07:00" }, "raw_text": "Reasons for Program Verification Program Verification Nguyen An Khuong BK TP.HCM Documentation. Program properties formulated as theorems can Contents Core Programming serve as concise documentation Language Time-to-market. Verification prevents/catches bugs and can Hoare Triples Partial and Total Correctness reduce development time Proof Calculus for Partial Correctness Reuse. Clear specification provides basis for reuse Practical Aspects of Certification. Verification is reguired in safety-critical domains Correctness Proofs Correctness of the such as nuclear power stations and aircraft cockpits Factorial Function Proof Calculus for Total Correctness Homeworks 1f.5" }, { "page_index": 30, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_006.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_006.png", "page_index": 30, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:37+07:00" }, "raw_text": "Uses of Propositional Calculus Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Hardware design BK The production of logic circuits uses propositional calculus at all TP.HCM phases; specification, design, testing. Contents Verification Introduction Declarative Sentences Verification of hardware and software makes extensive use of Natural Deduction propositional calculus Sequents Rules for natural deduction Basic and Derived Rules Problem solving Intuitionistic Logic Formal Language Decision problems (scheduling, timetabling, etc) can be expressed Semantics as satisfiability problems in propositional calculus. Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.6" }, { "page_index": 31, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_006.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_006.png", "page_index": 31, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:40+07:00" }, "raw_text": "Motivated Example - A Logic Puzzle Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai If the unicorn is mythical, then it is immortal;and BK TP.HCM If the unicorn is not mythical, then it is a mortal mammal; Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.4" }, { "page_index": 32, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_006.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_006.png", "page_index": 32, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:44+07:00" }, "raw_text": "More Declarative Sentences Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Propositional logic can easily handle simple declarative BK statements such as: TP.HCM Example Contents Student Hung enrolled in DMll. Predicate Logic Motivation, Syntax, Proof Theory Propositional logic can also handle combinations of such Need for Richer Language statements such as: Predicate Logic as Formal Language Proof Theory of Predicate Example Logic Quantifier Equivalences Student Hung enrolled in Tutorial 1, and student Cuong is enrolled Semantics of Predicate Logic in Tutorial 2. Soundness and Completeness of . But: How about statements with \"there exists...\" or \"every... Predicate Logic or \"among...\"? Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.4" }, { "page_index": 33, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_006.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_006.png", "page_index": 33, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:48+07:00" }, "raw_text": "Program Verification (cont'd) Predicate Logic and Program Verification Nguyen An Khuong, Huynh Tuong Nguyen BK A precondition specifies what should be true upon entering TP.HCM the function (i.e., under what inputs the function is expected to work). Contents Warm-up questions The postcondition is a formula G whose free variables include Program Verification only the formal parameters and the special variable ru Homeworks representing the return value of the function. The postcondition relates the function's output (the return value ru) to its input (the parameters) Prob: Formulate in Predicate Logic the precondition and the postcondition for binarySearch. 1e.6" }, { "page_index": 34, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_006.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_006.png", "page_index": 34, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:51+07:00" }, "raw_text": "Framework for Software Verification Program Verification Nguyen An Khuong BK TP.HCM Contents Convert informal description R of requirements for an Core Programming Language application domain into formula $R: Hoare TriplesPartial and Total Correctness Write program P that meets oR Proof Calculus for Prove that P satisfies oR. Partial Correctness Practical Aspects of Correctness Proofs Each step provides risks and opportunities Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.6" }, { "page_index": 35, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_007.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_007.png", "page_index": 35, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:54+07:00" }, "raw_text": "Predicate Calculus: Central ideas Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai Huynh Tuong Nguyen Le Höng Trang Richer language BK Instead of dealing with atomic propositions, predicate calculus TP.HCM provides the formulation of statements involving sets, functions and relations on these sets Contents Introduction Quantifiers Declarative Sentences Predicate calculus provides statements that all or some elements Natural Deduction Sequents of a set have specified properties Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Compositionality Formal Language Similar to propositional calculus, formulas can be built from Semantics Meaning of Logical composites using logical connectives. Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.7" }, { "page_index": 36, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_007.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_007.png", "page_index": 36, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:25:58+07:00" }, "raw_text": "Motivated Example - A Logic Puzzle Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK If the unicorn is mythical, then it is immortal;and TP.HCM If the unicorn is not mythical, then it is a mortal mammal;and Contents If the unicorn is either immortal or a mammal, then it is Introduction horned; Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.4" }, { "page_index": 37, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_007.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_007.png", "page_index": 37, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:01+07:00" }, "raw_text": "What is needed? Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK Example TP.HCM Every student is younger than some instructor. Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.5" }, { "page_index": 38, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_007.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_007.png", "page_index": 38, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:03+07:00" }, "raw_text": "Program Verification (cont'd): Answer Predicate Logic and Program Verification Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Contents First precondition: 0 < l u 3i.l < i < u a= e 1e.7" }, { "page_index": 39, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_007.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_007.png", "page_index": 39, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:06+07:00" }, "raw_text": "Program Verification Nguyen An Khuong Core Programming Language BK TP.HCM 2 Hoare Triples Partial and Total Correctness Contents Proof Calculus for Partial Correctness Core Programming Language Hoare Triples; Partial 4 Practical Aspects of Correctness Proofs and Total Correctness Proof Calculus for Partial Correctness 5 Correctness of the Factorial Function Practical Aspects of Correctness Proofs Correctness of the Factorial Function 6 Proof Calculus for Total Correctness Proof Calculus for Total Correctness Homeworks 7 Homeworks 1f.7" }, { "page_index": 40, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_008.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_008.png", "page_index": 40, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:10+07:00" }, "raw_text": "The uses of Predicate Calculus Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Progamming Language Semantics BK The meaning of programs such as TP.HCM ifx >= 0theny := sqrt(x)elsey := abs(x) can be captured with formulas of predicate calculus: Contents Introduction VxVy(x'=x (x 0-y'=Vx) (-(x 0)-y'=x1)) Declarative Sentences Natural Deduction Sequents Other Uses of Predicate Calculus Rules for natural deduction Basic and Derived Rules Specification: Formally specify the purpose of a program in order to Intuitionistic Logic serve as input for software design, Formal Language Verification: Prove the correctness of a program with respect to its Semantics Meaning of Logical specification. Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.8" }, { "page_index": 41, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_008.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_008.png", "page_index": 41, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:13+07:00" }, "raw_text": "Motivated Example - A Logic Puzzle Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai If the unicorn is mythical, then it is immortal;and BK TP.HCM If the unicorn is not mythical, then it is a mortal mammal;and Contents If the unicorn is either immortal or a mammal, then it is Introduction Quick review horned;and Boolean Satisfiability (SAT) P and NP The unicorn is magical if it is horned 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.4" }, { "page_index": 42, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_008.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_008.png", "page_index": 42, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:16+07:00" }, "raw_text": "What is needed? Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK Example TP.HCM Every student is younger than some instructor. Contents What is this statement about? Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.5" }, { "page_index": 43, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_008.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_e/slide_008.png", "page_index": 43, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:19+07:00" }, "raw_text": "HW Predicate Logic and Program Verification Nguyen An Khuong. Huynh Tuong Nguyen 1. Do all HWs which have not been done in previous lectures. 2. Try to understand deeply the following notations/terms BK TP.HCM arity, expression, term, formula, atomic formula, sentence, clause, Backus Naur form (BNF), parse tree, precondition, Contents postcondition, binding priorities, provability, witness, scope, Warm-up questions bound, verification, model checking, Hoare triple, and their Program Verification other related notation/terms. Homeworks 3. Do exercise 1.5.14 on page 89 in [2] 4. Consider the following program temp := x x := y y := temp What does this tinny program do? Find preconditions, postconditions and verify its correctness? 1e.8" }, { "page_index": 44, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_008.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_008.png", "page_index": 44, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:22+07:00" }, "raw_text": "Motivation of Core Language Program Verification Nguyen An Khuong BK TP.HCM Real-world languages are quite large; many features and Contents Core Programming constructs Language Verification framework would exceed time we have in CS5209 Hoare Triples;Partial and Total Correctness Theoretical constructions such as Turing machines or lambda Proof Calculus for Partial Correctness calculus are too far from actual applications; too low-level Practical Aspects of Idea: use subset of Pascal/C/C++/Java Correctness Proofs Correctness of the Benefit: we can study useful \"realistic\" examples Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.8" }, { "page_index": 45, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_009.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_009.png", "page_index": 45, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:25+07:00" }, "raw_text": "An Example for Specification Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM Let P be a program of the form while a <> b do Contents if a > b then a := a - b else a:= b - a; Introduction Declarative Sentences The specification of the program is given by the formula Natural Deduction Sequents {a 0Ab 0} P{a=gcd(a,b)} Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.9" }, { "page_index": 46, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_009.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_009.png", "page_index": 46, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:29+07:00" }, "raw_text": "Motivated Example - A Logic Puzzle Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai If the unicorn is mythical, then it is immortal;and BK TP.HCM If the unicorn is not mythical, then it is a mortal mammal;and Contents If the unicorn is either immortal or a mammal, then it is Introduction horned;and Quick review Boolean Satisfiability (SAT) The unicorn is magical if it is horned. P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers Q: Is the unicorn mythical? Is it magical? Is it horned? WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.4" }, { "page_index": 47, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_009.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_009.png", "page_index": 47, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:31+07:00" }, "raw_text": "What is needed? Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK Example TP.HCM Every student is younger than some instructor. Contents What is this statement about? Predicate Logic Motivation, Syntax, Proof Theory Being a student Need for Richer Language Predicate Logic as Formal Being an instructor Language Proof Theory of Predicate Being younger than somebody else Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.5" }, { "page_index": 48, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_009.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_009.png", "page_index": 48, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:26:34+07:00" }, "raw_text": "Expressions in Core Language Program Verification Nguyen An Khuong BK TP.HCM Expressions come as arithmetic expressions E Contents Core Programming E:=nx(-E)(E+E)(E-E)(E*E Language Hoare Triples Partial and Total Correctness and boolean expressions B: Proof Calculus for Partial Correctness B::=truefalse(!B)(B&B)(BB)(E 2 is the sum of two prime numbers. Rules for natural deduction Basic and Derived Rules 4 All Martians like pepperoni on their pizza. Intuitionistic Logic Formal Language Semantics Not Examples Meaning of Logical Connectives Preview: Soundness and . Could you please pass me the salt? Completeness Normal Form . Ready, steady, go! Homeworks and Next Week Plan? . May fortune come your way. 1a.12" }, { "page_index": 58, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_012.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_012.png", "page_index": 58, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:07+07:00" }, "raw_text": "CNF Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai Boolean formula is defined over a set of propositional variables P1,..., Pn, using the standard propositional connectives -,, V,- ?,< ?, and parenthesis BK . The domain of propositional variables is {0,1} TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.5" }, { "page_index": 59, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_012.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_012.png", "page_index": 59, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:10+07:00" }, "raw_text": "Predicates Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Example Contents Every student is younger than some instructor. Predicate Logic Motivation, Syntax. Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.6" }, { "page_index": 60, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_012.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_012.png", "page_index": 60, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:12+07:00" }, "raw_text": "Program Verification Nguyen An Khuong 1 Core Programming Language BK TP.HCM Hoare Triples; Partial and Total Correctness Contents Proof Calculus for Partial Correctness Core Programming Language Hoare Triples Partial 4 Practical Aspects of Correctness Proofs and Total Correctness Proof Calculus for Partial Correctness 5 Correctness of the Factorial Function Practical Aspects of Correctness Proofs Correctness of the Factorial Function 6 Proof Calculus for Total Correctness Proof Calculus for Total Correctness Homeworks 7 Homeworks 1f.12" }, { "page_index": 61, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_013.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_013.png", "page_index": 61, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:16+07:00" }, "raw_text": "Putting Propositions Together Propositional Logic Review I Nguyen An Khuong. Example 1.1 Tran Van Hoai Huynh Tuong Nguyen If the train arrives late and Le Höng Trang there are no taxis at the station then John is late for his meeting BK TP.HCM John is not late for his meeting The train did arrive late. Contents Introduction Therefore, there were taxis at the station Declarative Sentences Natural Deduction Sequents Example 1.2 Rules for natural deduction Basic and Derived Rules If it is raining and Intuitionistic Logic Jane does not have her umbrella with her then Formal Language she will get wet. Semantics Meaning of Logical Connectives Jane is not wet. Preview: Soundness and Completeness Normal Form It is raining Homeworks and Next Week Plan? Therefore, Jane has her umbrella with her 1a.13" }, { "page_index": 62, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_013.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_013.png", "page_index": 62, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:19+07:00" }, "raw_text": "CNF Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai Boolean formula is defined over a set of propositional variables P1,..., Pn, using the standard propositional connectives -,A, V,- ?,< , and parenthesis BK The domain of propositional variables is {0,1} TP.HCM Example: $(p1,P2,p3) =((-p1 p2) V p3) (-p2 V p3) Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.5" }, { "page_index": 63, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_013.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_013.png", "page_index": 63, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:22+07:00" }, "raw_text": "Predicates Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Example Contents Every student is younger than some instructor. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language S(An) could denote that An is a student Predicate Logic as Formal Language I(Binh) could denote that Binh is an instructor. Proof Theory of Predicate Logic Y(An, Binh) could denote that An is younger than Binh. Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.6" }, { "page_index": 64, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_013.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_013.png", "page_index": 64, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:25+07:00" }, "raw_text": "Example Program Verification Nguyen An Khuong BK TP.HCM Contents Core Programming Language 1; Hoare Triples.Partial z = 0; and Total Correctness while (z != x) Proof Calculus for Partial Correctness Practical Aspects of Correctness Proofs Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.13" }, { "page_index": 65, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_014.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_014.png", "page_index": 65, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:29+07:00" }, "raw_text": "Focus on Structure Propositional Logic Review 1 Nguyen An Khuong. We are primarily concerned about the structure of arguments in Tran Van Hoai, this class, not the validity of statements in a particular domain. Huynh Tuong Nguyen Le Höng Trang We therefore simply abbreviate sentences by letters such as p, q, r, P1, P2 etc. BK TP.HCM From Concrete Propositions to Letters - Example 1.1 If the train arrives late and Contents there are no taxis at the station then Introduction John is late for his meeting Declarative Sentences Natural Deduction John is not late for his meeting Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic The train did arrive late. Formal Language Semantics Therefore, there were taxis at the station Meaning of Logical Connectives Preview: Soundness and becomes Completeness Normal Form Letter version Homeworks and Next Week Plan? If p and not q, then r. Not r. p. Therefore, q. 1a.14" }, { "page_index": 66, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_014.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_014.png", "page_index": 66, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:32+07:00" }, "raw_text": "CNF Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai Boolean formula o is defined over a set of propositional variables P1, ..., Pn, using the standard propositional connectives -,, V,-,, and parenthesis BK The domain of propositional variables is {0,1} TP.HCM Example: $(p1,P2,p3) =((-p1 p2) V p3) (-p2 V p3) A formula in conjunctive normal form (CNF) is a Contents conjunction of disjunctions (clauses) of literals, where a literal Introduction Quick review is a variable or its complement. Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.5" }, { "page_index": 67, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_014.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_014.png", "page_index": 67, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:35+07:00" }, "raw_text": "The Need for Variables Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Example Contents Every student is younger than some instructor. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.7" }, { "page_index": 68, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_014.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_014.png", "page_index": 68, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:38+07:00" }, "raw_text": "Example Program Verification Nguyen An Khuong BK TP.HCM Contents Core Programming y = 1; Language z = 0; Hoare Triples.Partial while z != x) and Total Correctness = y * z; } Proof Calculus for Partial Correctness We need to be able to say that at the end, y is x! Practical Aspects of Correctness Proofs Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.14" }, { "page_index": 69, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_015.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_015.png", "page_index": 69, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:42+07:00" }, "raw_text": "Focus on Structure Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai Huynh Tuong Nguyen. Le Höng Trang From Concrete Propositions to Letters - Example 1.2 If it is raining and BK Jane does not have her umbrella with her then TP.HCM she will get wet. Contents Jane is not wet. Introduction Declarative Sentences It is raining Natural Deduction Sequents Rules for natural deduction Therefore, Jane has her umbrella with her Basic and Derived Rules Intuitionistic Logic Formal Language has Semantics the same letter version Meaning of Logical Connectives If p and not q, then r. Not r. p. Therefore, q. Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.15" }, { "page_index": 70, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_015.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_015.png", "page_index": 70, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:45+07:00" }, "raw_text": "CNF Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai Boolean formula δ is defined over a set of propositional variables P1, ..., Pn, using the standard propositional connectives -,, V,-,, and parenthesis BK . The domain of propositional variables is {0,1} TP.HCM Example: $(p1,P2,p3) =((-p1 p2) V p3) (-p2 V p3) A formula in conjunctive normal form (CNF) is a Contents conjunction of disjunctions (clauses) of literals, where a literal Introduction Quick review is a variable or its complement. Boolean Satisfiability (SAT) Example: $(p1,P2,p3) =(-p1 V p2) (-p2 V p3). P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.5" }, { "page_index": 71, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_015.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_015.png", "page_index": 71, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:48+07:00" }, "raw_text": "The Need for Variables Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Example Contents Every student is younger than some instructor. Predicate Logic Motivation, Syntax, Proof Theory We use the predicate S to denote student-hood Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.7" }, { "page_index": 72, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_015.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_015.png", "page_index": 72, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:51+07:00" }, "raw_text": "Example Program Verification Nguyen An Khuong BK TP.HCM Contents y = 1; Core Programming 0; Language z= while (z != x) { z = z + 1; HoareTriples.Partial y = y * z; } and Total Correctness Proof Calculus for We need to be able to say that at the end, y is x! Partial Correctness Practical Aspects of That means we require a post-condition y = x! Correctness Proofs Correctness of the Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.15" }, { "page_index": 73, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_016.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_016.png", "page_index": 73, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:56+07:00" }, "raw_text": "Logical Connectives Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Notations/Symbols Sentences like \"If p and not q, then r.\" occur frequently. Instead BK TP.HCM of English words such as \"if...then\", \"and\", \"not\", it is more convenient to use symbols such as -?, A, -. Contents -: negation of p is denoted by -p Introduction Declarative Sentences V: disjunction of p and r is denoted by p V r, meaning at least Natural Deduction one of the two statements is true. Sequents Rules for natural deduction A: conjunction of p and r is denoted by p r, meaning both are Basic and Derived Rules Intuitionistic Logic true. Formal Language >: implication between p and r is denoted by p - r, meaning Semantics Meaning of Logical that r is a logical consequence of p. p is called the Connectives Preview: Soundness and antecedent, and r the consequent. Completeness Normal Form Homeworks and Next Week Plan? 1a.16" }, { "page_index": 74, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_016.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_016.png", "page_index": 74, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:27:59+07:00" }, "raw_text": "CNF Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai Boolean formula δ is defined over a set of propositional variables P1, ..., Pn, using the standard propositional connectives -,, V,-?,<, and parenthesis BK . The domain of propositional variables is {0,1} TP.HCM Example: $(p1,P2,p3) =((-p1 p2) V p3) (-p2 V p3) . A formula δ in conjunctive normal form (CNF) is a Contents conjunction of disjunctions (clauses) of literals, where a literal Introduction Quick review is a variable or its complement. Boolean Satisfiability (SAT) : Example: $(p1,p2,p3) =(-p1 V p2) (-p2 V p3). P and NP 2-SAT is in P An example Proposition (see [2, Subsection 1.5.1]) UNSAT Graphical View of 2-SAT There is an algorithm to translate any Boolean formula into CNF SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.5" }, { "page_index": 75, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_016.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_016.png", "page_index": 75, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:02+07:00" }, "raw_text": "The Need for Variables Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Example Contents Every student is younger than some instructor. Predicate Logic Motivation, Syntax, Proof Theory We use the predicate S to denote student-hood Need for Richer Language How do we express \"every student\"? Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.7" }, { "page_index": 76, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_016.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_016.png", "page_index": 76, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:04+07:00" }, "raw_text": "Example Program Verification Nguyen An Khuong BK TP.HCM Contents Core Programming y = 1; Language z= 0; Hoare Triples.Partial while (z != x) and Total Correctness 7 + y y * Z Proof Calculus for Partial Correctness Do we need pre-conditions, too? Practical Aspects of Correctness Proofs Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.16" }, { "page_index": 77, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_017.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_017.png", "page_index": 77, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:08+07:00" }, "raw_text": "Example 1.1 Revisited Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang From Example 1.1 If the train arrives late and BK there are no taxis at the station then TP.HCM John is late for his meeting Contents Symbolic Propositions Introduction Declarative Sentences We replaced \"the train arrives late\"' by p, etc. Natural Deduction Sequents The statement becomes: If p and not q, then r Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Symbolic Connectives Formal Language Semantics With symbolic connectives, the statement becomes: Meaning of Logical Connectives Preview: Soundness and p-qr Completeness Normal Form Homeworks and Next Week Plan? 1a.17" }, { "page_index": 78, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_017.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_017.png", "page_index": 78, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:12+07:00" }, "raw_text": "CNF Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai Boolean formula δ is defined over a set of propositional variables P1, ..., Pn, using the standard propositional connectives -,, V,-?,<, and parenthesis BK . The domain of propositional variables is {0,1} TP.HCM Example: $(p1,P2,P3) =((-p1 p2) V p3) (-p2 V p3) A formula in conjunctive normal form (CNF) is a Contents conjunction of disjunctions (clauses) of literals, where a literal Introduction Quick review is a variable or its complement. Boolean Satisfiability (SAT) : Example: $(p1,p2,p3) =(-p1 V p2) (-p2 V p3). P and NP 2-SAT is in P An example Proposition (see [2, Subsection 1.5.1]) UNSAT Graphical View of 2-SAT There is an algorithm to translate any Boolean formula into CNF SAT Solvers WalkSAT: Idea DPLL: Idea Proposition 1.45, p. 57 A Linear Solver A Cubic Solver -satisfiable iff -o-not tautology Homeworks and Next Week Plan? 1b.5" }, { "page_index": 79, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_017.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_017.png", "page_index": 79, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:15+07:00" }, "raw_text": "The Need for Variables Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Example Contents Every student is younger than some instructor. Predicate Logic Motivation, Syntax, Proof Theory We use the predicate S to denote student-hood Need for Richer Language How do we express \"every student\"? Predicate Logic as Formal Language Proof Theory of Predicate Logic We need variab/es that can stand for constant values, and a Quantifier Equivalences Semantics of Predicate quantifier symbol that denotes \"every\" Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.7" }, { "page_index": 80, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_017.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_017.png", "page_index": 80, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:18+07:00" }, "raw_text": "Example Program Verification Nguyen An Khuong BK TP.HCM Contents y = 1; z = 0; Core Programming Language while z I = Z. HoareTriples.Partial andTotal Correctness Do we need pre-conditions, too? Proof Calculus for Partial Correctness Yes, they specify what needs to be the case before Practical Aspects of execution. Correctness Proofs Example: x > 0 Correctness of the Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.17" }, { "page_index": 81, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_018.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_018.png", "page_index": 81, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:21+07:00" }, "raw_text": "Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. 1 Propositional Calculus: Declarative Sentences Le Höng Trang Propositional Calculus: Natural Deduction BK Sequents TP.HCM Rules for natural deduction Basic and Derived Rules Contents Excursion: Intuitionistic Logic Introduction Declarative Sentences Natural Deduction 3 Propositional Logic as a Formal Language Sequents Rules for natural deduction Basic and Derived Rules Semantics of Propositional Logic Intuitionistic Logic Formal Language Semantics 5 Conjunctive Normal Form Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.18" }, { "page_index": 82, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_018.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_018.png", "page_index": 82, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:23+07:00" }, "raw_text": "SAT Propositional Logic Review II Nguyen An Khuong Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai BK TP.HCM Contents Introduction Quick review Booiean SatisfiabilitySAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.6" }, { "page_index": 83, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_018.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_018.png", "page_index": 83, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:26+07:00" }, "raw_text": "The Need for Variables Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Example Contents Every student is younger than some instructor. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.8" }, { "page_index": 84, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_018.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_018.png", "page_index": 84, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:29+07:00" }, "raw_text": "Example Program Verification Nguyen An Khuong BK TP.HCM y = 1; Contents z = 0; Core Programming while (z != x) y Z: Language HoareTriples.Partial and Total Correctness Do we need pre-conditions, too? Proof Calculus for Yes, they specify what needs to be the case before Partial Correctness execution. Practical Aspects of Correctness Proofs Example: x > 0 Correctness of the Factorial Function Do we have to prove the postcondition in one go? Proof Calculus fo Total Correctness Homeworks 1f.18" }, { "page_index": 85, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_019.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_019.png", "page_index": 85, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:32+07:00" }, "raw_text": "Introduction Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai Huynh Tuong Nguyen. Le Höng Trang Objective BK We would like to develop a ca/cu/us for reasoning about TP.HCM propositions, so that we can establish the validity of statements such as Example 1.1 Contents Introduction Idea Declarative Sentences We introduce proof rules that allow us to derive a formula from Natural Deduction Sequents a number of other formulas 1, 2, : . . n Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Notation Formal Language We write a sequent 1,$2,...,$n F Semantics Meaning of Logical to denote that we can derive from 1,2,: : Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.19" }, { "page_index": 86, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_019.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_019.png", "page_index": 86, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:34+07:00" }, "raw_text": "SAT Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai Problem BK TP.HCM Contents Introduction Quick review Booiean SatisfiabilitySAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.6" }, { "page_index": 87, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_019.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_019.png", "page_index": 87, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:37+07:00" }, "raw_text": "The Need for Variables Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Example Contents Every student is younger than some instructor. Predicate Logic Motivation, Syntax, Using variables and quantifiers, we can write: Proof Theory Need for Richer Language Predicate Logic as Formal Vx(S(x) ->(Ey(I(y) Y(x,y)))) Language Proof Theory of Predicate Logic Quantifier Equivalences Literally: For every ε, if x is a student, then there is some y such Semantics of Predicate that y is an instructor and x is younger than y Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.8" }, { "page_index": 88, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_019.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_019.png", "page_index": 88, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:41+07:00" }, "raw_text": "Example Program Verification Nguyen An Khuong BK TP.HCM z= 0; Contents while z 1 = Core Programming Language Do we need pre-conditions, too? Hoare Triples.Partial and Total Correctness Yes, they specify what needs to be the case before Proof Calculus for execution. Partial Correctness Example: x > 0 Practical Aspects of Correctness Proofs Do we have to prove the postcondition in one go? Correctness of the Factorial Function No, the postcondition of one line can be the Proof Calculus for pre-condition of the next! Total Correctness Homeworks 1f.19" }, { "page_index": 89, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_020.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_020.png", "page_index": 89, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:44+07:00" }, "raw_text": "Example 1.1 Revisited Propositional Logic Review I Nguyen An Khuong. English Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang If the train arrives late and there are no taxis at the station then John is late for his meeting. BK TP.HCM John is not late for his meeting Contents The train did arrive late. Introduction Declarative Sentences Therefore, there were taxis at the station Natural Deduction Sequents Rules for natural deduction Sequent Basic and Derived Rules Intuitionistic Logic Formal Language pA-q-r,-r,pFq Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Remaining task Normal Form Homeworks and Next Develop a set of proof rules that allows us to establish such Week Plan? sequents. 1a.20" }, { "page_index": 90, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_020.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_020.png", "page_index": 90, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:47+07:00" }, "raw_text": "SAT Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai Problem BK Find an assignment to the variables P1, ...,Pn such that TP.HCM β(p1, ...,Pn) = 1, or prove that no such assignment exists Contents Introduction Quick review Booiean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.6" }, { "page_index": 91, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_020.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_020.png", "page_index": 91, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:49+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen English BK TP.HCM Not all birds can fly Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.9" }, { "page_index": 92, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_020.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_020.png", "page_index": 92, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:51+07:00" }, "raw_text": "Assertions on Programs Program Verification Nguyen An Khuong BK TP.HCM Shape of assertions Contents Core Programming Language 9$D P Q4D HoareTriples.Partial and Total Correctness Proof Calculus for Partial Correctness Informal meaning Practical Aspects of Correctness Proofs If the program P is run in a state that satisfies , then the state Correctness of the resulting from P's execution will satisfy W Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.20" }, { "page_index": 93, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_021.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_021.png", "page_index": 93, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:54+07:00" }, "raw_text": "Rules for Conjunction Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang Introduction of Conjunction BK TP.HCM [i] Contents Introduction Declarative Sentences Natural Deduction Elimination of Conjunction Sequents Ruies for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics e1 [e2] Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.21" }, { "page_index": 94, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_021.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_021.png", "page_index": 94, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:57+07:00" }, "raw_text": "SAT Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai Problem BK Find an assignment to the variables P1, ...,Pn such that TP.HCM β(p1, ...,Pn) = 1, or prove that no such assignment exists Contents Facts: SAT is an NP-complete decision problem [Cook'71 Introduction Quick review Booiean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.6" }, { "page_index": 95, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_021.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_021.png", "page_index": 95, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:28:59+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen English BK TP.HCM Not all birds can fly Contents Predicates Predicate Logic Motivation, Syntax, Proof Theory B(x): x is a bird Need for Richer Language F(x): x can fly Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.9" }, { "page_index": 96, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_021.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_021.png", "page_index": 96, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:02+07:00" }, "raw_text": "Slightly Trivial) Example Program Verification Nguyen An Khuong Informal specification BK Given a positive number x, the program P calculates a number y TP.HCM whose square is less than :c. Contents Assertion Core Programming Language Hoare Triples.Partial and Total Correctness (x>OD P(yyOD y = O(yyOD P (y`y 0, a concrete function fM : Ar - A; Correctness of the Factorial Function Q for each P e P with arity n > 0,a set PM C Ar Proof Calculus for Total Correctness Homeworks 1f.23" }, { "page_index": 105, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_024.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_024.png", "page_index": 105, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:29+07:00" }, "raw_text": "Where are we heading with this? Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang We would like to prove sequents of the form BK 01, 02.. ...n F V TP.HCM We introduce rules that allow us to form \"legal'\" proofs Then any proof of any formula using the premises Contents Introduction 1, 2,. .., n is considered \"correct\". Declarative Sentences Can we say that sequents with a correct proof are somehow Natural Deduction \"valid\", or \"meaningful\"? Sequents Ruies for natural deduction What does it mean to be meaningful? Basic and Derived Rules Intuitionistic Logic Can we say that any meaningful sequent has a valid proof? Formal Language Semantics ...but first back to the proof rules... Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.24" }, { "page_index": 106, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_024.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_024.png", "page_index": 106, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:32+07:00" }, "raw_text": "SAT Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai Problem BK Find an assignment to the variables P1, ..., Pn such that TP.HCM β(p1, ...,Pn) = 1, or prove that no such assignment exists Contents Facts: SAT is an NP-complete decision problem [Cook'71] Introduction Quick review Booiean Satisfiability(SAT) SAT was the first problem to be shown NP-complete P and NP 2-SAT is in P There are no known polynomial time algorithms for SAT. An example UNSAT More-than-35-year old conjecture: Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.6" }, { "page_index": 107, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_024.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_024.png", "page_index": 107, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:35+07:00" }, "raw_text": "A Third Example Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen English BK Every girl is younger than her mother. TP.HCM Predicates Contents Predicate Logic Motivation, Syntax, G(x): x is a girl Proof Theory Need for Richer Language M(x,y): y is x's mother Predicate Logic as Formal Language Y(x,y): x is younger than y Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.10" }, { "page_index": 108, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_024.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_024.png", "page_index": 108, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:38+07:00" }, "raw_text": "Recall: Satisfaction Relation Program Verification Nguyen An Khuong BK TP.HCM The model M satisfies with respect to environment l, written M F o: Contents Core Programming in case is of the form P(t1,t2, . ..,tn), if the result Language (a1, a2, . . ., an) of evaluating t1,t2, . . .,tn with respect to l is Hoare Triples.Partial and Total Correctness in pM. Proof Calculus for Partial Correctness in case $ has the form Vx,if the M Fl[x-a] holds for all Practical Aspects of a E A; Correctness Proofs in case $ has the form 3x,if the M Fl[xHa] holds for Correctness of the Factorial Function some a E A; Proof Calculus for Total Correctness Homeworks 1f.24" }, { "page_index": 109, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_025.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_025.png", "page_index": 109, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:40+07:00" }, "raw_text": "Rules of Double Negation and Eliminating Implication Propositional Logic Review I Nguyen An Khuong. Double Negation Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang BK TP.HCM Contents Introduction Eliminating Implication Declarative Sentences Natural Deduction Sequents Ruies for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Example Normal Form Homeworks and Next p:= \"It rained,\" and p - q:= \"If it rained, then the street is wet.\" Week Plan? We can conclude from these two that the street is indeed wet. 1a.25" }, { "page_index": 110, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_025.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_025.png", "page_index": 110, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:43+07:00" }, "raw_text": "SAT Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai Problem BK Find an assignment to the variables P1, ..., Pn such that TP.HCM $(p1, ..., Pn) = 1, or prove that no such assignment exists Contents Facts: SAT is an NP-complete decision problem [Cook'71] Introduction Quick review Booiean Satisfiability(SAT) SAT was the first problem to be shown NP-complete P and NP 2-SAT is in P There are no known polynomial time algorithms for SAT. An example UNSAT More-than-35-year old conjecture: Graphical View of 2SAT \"Any algorithm that solves SAT is exponential in the number SAT Solvers of variables, in the worst-case.' WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.6" }, { "page_index": 111, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_025.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_025.png", "page_index": 111, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:47+07:00" }, "raw_text": "A Third Example Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen English BK Every girl is younger than her mother. TP.HCM Predicates Contents Predicate Logic Motivation, Syntax, G(x): x is a girl Proof Theory Need for Richer Language M(x,y): y is x's mother Predicate Logic as Formal Language Y(x,y): x is younger than y Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate The sentence in predicate logic Logic Soundness and Completeness of Predicate Logic VxVy(G(x) M(x,y) -Y(x,y)) Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.10" }, { "page_index": 112, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_025.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_025.png", "page_index": 112, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:49+07:00" }, "raw_text": "Recall: Satisfaction Relation (continued) Program Verification Nguyen An Khuong BK TP.HCM in case o has the form -, if M W does not hold; Contents Core Programming in case o has the form 1 V V2, if M l 1 holds or Language M Fi V2 holds; HoareTriples.Partial and Total Correctness in case has the form 1 W2, if M 1 holds and Proof Calculus for Partial Correctness M Fi w2 holds; and Practical Aspects of in case o has the form V1 -> 2, if M i 1 holds whenever Correctness Proofs Correctness of the M Fi w2 holds. Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.25" }, { "page_index": 113, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_026.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_026.png", "page_index": 113, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:52+07:00" }, "raw_text": "Modus Ponens Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai Huynh Tuong Nguyen. Le Höng Trang The rule BK TP.HCM Contents Introduction is often called \"Modus Ponens\" (or MP) Declarative Sentences Natural Deduction Origin of term Sequents Ruies for natural deduction \"Modus ponens\" is an abbreviation of the Latin \"modus ponendo Basic and Derived Rules Intuitionistic Logic ponens\" which means in English \"mode that affirms by affirming\" Formal Language More precisely, we could say \"mode that affirms the antecedent of Semantics Meaning of Logical an implication\". Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.26" }, { "page_index": 114, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_026.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_026.png", "page_index": 114, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:54+07:00" }, "raw_text": "Polynomial time reductions and NP-Completeness Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Denote Huynh Tuong Nguyen, Tran Van Hoai BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.7" }, { "page_index": 115, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_026.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_026.png", "page_index": 115, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:29:58+07:00" }, "raw_text": "A \"Mother\" Function Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen The sentence in predicate logic BK TP.HCM VxVy(G(x) M(x,y)-Y(x,y) Contents Predicate Logic Note that y is only introduced to denote the mother of x Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.11" }, { "page_index": 116, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_026.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_026.png", "page_index": 116, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:00+07:00" }, "raw_text": "Hoare Triples Program Verification Nguyen An Khuong BK TP.HCM Definition Contents An assertion of the form (@) P (D is called a Hoare triple Core Programming o is called the precondition, is called the postcondition Language HoareTriples.Partial . A state of a Core program P is a function l that assigns each and Total Correctness variable x in P to an integer l(c). Proof Calculus for Partial Correctness . A state l satisfies if M i $, where M contains integers Practical Aspects of Correctness Proofs and gives the usual meaning to the arithmetic operations Correctness of the : Quantifiers in $ and bind only variables that do not occur Factorial Function Proof Calculus fo in the program P. Total Correctness Homeworks 1f.26" }, { "page_index": 117, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_027.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_027.png", "page_index": 117, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:03+07:00" }, "raw_text": "Modus Tollens Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Hong Trang A similar rule of \"Modus Ponens\" BK TP.HCM MT Contents Introduction Declarative Sentences is called \"Modus Tollens\" (or MT) Natural Deduction Sequents Ruies for natural deduction Origin of term Basic and Derived Rules Intuitionistic Logic \"Modus tollens\" is an abbreviation of the Latin \"modus tollendo Formal Language tollens\" which means in English \"mode that denies by denying\" Semantics More precisely, we could say \"mode that denies the consequent of Meaning of Logical Connectives an implication\". Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.27" }, { "page_index": 118, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_027.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_027.png", "page_index": 118, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:06+07:00" }, "raw_text": "Polynomial time reductions and NP-Completeness Propositional Logic Review II Nguyen An Khuong. Le Hong Trang . Denote Huynh Tuong Nguyen. Tran Van Hoai EXP = {Decision problems solvable in exponential time} BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.7" }, { "page_index": 119, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_027.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_027.png", "page_index": 119, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:09+07:00" }, "raw_text": "A \"Mother\" Function Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen The sentence in predicate logic BK TP.HCM VxVy(G(x)M(x,y) -Y(x,y) Contents Predicate Logic Note that y is only introduced to denote the mother of x Motivation, Syntax, Proof Theory Need for Richer Language If everyone has exactly one mother, the predicate M(ε,y) is a Predicate Logic as Formal Language function, when read from right to left. Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.11" }, { "page_index": 120, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_027.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_027.png", "page_index": 120, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:11+07:00" }, "raw_text": "Example Program Verification Nguyen An Khuong BK TP.HCM Contents Core Programming Let l(x) = -2,l(y) = 5 and l(z) = -1. We have Language . l F -(x + y< z Hoare Triples.Partial and Total Correctness . l Y y=x.z< z Proof Calculus for Partial Correctness .lAVu(y< u-yz< uz Practical Aspects of Correctness Proofs Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.27" }, { "page_index": 121, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_028.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_028.png", "page_index": 121, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:15+07:00" }, "raw_text": "Example Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM p-(q->r),p,-rF-q Contents 1 p-(q-r) premise Introduction 2 p premise Declarative Sentences 3 -r premise Natural Deduction Sequents 4 q-r -e 1,2 Ruies for natural deduction 5 MT 4,3 Basic and Derived Rules -q Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.28" }, { "page_index": 122, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_028.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_028.png", "page_index": 122, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:18+07:00" }, "raw_text": "Polynomial time reductions and NP-Completeness Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, . Denote Huynh Tuong Nguyen. Tran Van Hoai : EXP = {Decision problems solvable in exponential time} P = {Decision problems solvable in polynomial time} BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.7" }, { "page_index": 123, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_028.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_028.png", "page_index": 123, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:21+07:00" }, "raw_text": "A \"Mother\" Function Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen The sentence in predicate logic BK TP.HCM VxVy(G(x) M(x,y) -Y(x,y) Contents Predicate Logic Note that y is only introduced to denote the mother of x Motivation, Syntax, Proof Theory Need for Richer Language If everyone has exactly one mother, the predicate M(ε,y) is a Predicate Logic as Formal Language function, when read from right to left. Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate We introduce a function symbol m that can be applied to Logic variables and constants as in Soundness and Completeness of Predicate Logic Vx(G(x)-Y(x,m(x) Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.11" }, { "page_index": 124, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_028.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_028.png", "page_index": 124, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:23+07:00" }, "raw_text": "Partial Correctness Program Verification Nguyen An Khuong BK TP.HCM Definition Contents Core Programming We say that the triple (oD P ( is satisfied under partial Language correctness if, for all states which satisfy o, the state resulting HoareTriples.Partial and Total Correctness from P's execution satisfies W, provided that P terminates Proof Calculus for Partial Correctness Practical Aspects of Notation Correctness Proofs We write Fpar 1$D P 1D Correctness of the Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.28" }, { "page_index": 125, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_029.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_029.png", "page_index": 125, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:27+07:00" }, "raw_text": "How to introduce implication? Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang BK Compare the sequent (MT) TP.HCM p->q,-qF-p Contents Introduction with the sequent Declarative Sentences p->qF-q--p Natural Deduction Sequents The second sequent should be provable, but we don't have a rule Ruies for natural deduction Basic and Derived Rules to introduce implication yet! Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.29" }, { "page_index": 126, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_029.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_029.png", "page_index": 126, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:29+07:00" }, "raw_text": "Polynomial time reductions and NP-Completeness Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Denote Huynh Tuong Nguyen Tran Van Hoai EXP = {Decision problems solvable in exponential time} P = {Decision problems solvable in polynomial time} NP = {Decision problems where Yes solution can verified in BK polynomial time} TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.7" }, { "page_index": 127, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_029.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_029.png", "page_index": 127, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:32+07:00" }, "raw_text": "A Drastic Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen English An and Binh have the same maternal grandmother. BK TP.HCM Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.12" }, { "page_index": 128, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_029.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_029.png", "page_index": 128, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:33+07:00" }, "raw_text": "Extreme Example Program Verification Nguyen An Khuong BK TP.HCM Contents Core Programming Language while true { x = O; } (y Hoare Triples.Partial and Total Correctness holds for all o and Proof Calculus for Partial Correctness Practical Aspects of Correctness Proofs Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.29" }, { "page_index": 129, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_030.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_030.png", "page_index": 129, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:37+07:00" }, "raw_text": "A Proof We Would Like To Have Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Hong Trang BK p->qF-q->-p TP.HCM 1 p->q premise 2 q assumption Contents 3 MT 1,2 Introduction -p Declarative Sentences 4 -q->-p >i 2-3 Natural Deduction Sequents Ruies for natural deduction We can start a box with an assumption, and use previously proven Basic and Derived Rules Intuitionistic Logic propositions (including premises) from the outside in the box. Formal Language We cannot use assumptions from inside the box in rules outside Semantics the box. Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.30" }, { "page_index": 130, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_030.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_030.png", "page_index": 130, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:40+07:00" }, "raw_text": "Polynomial time reductions and NP-Completeness Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Denote Huynh Tuong Nguyen Tran Van Hoai EXP = {Decision problems solvable in exponential time} P = {Decision problems solvable in polynomial time} NP = {Decision problems where Yes solution can verified in BK polynomial time} TP.HCM A major open question in theoretical computer science is if P = NP or not. Contents Introduction Quick review Boolean Satisfiability (SAT P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.7" }, { "page_index": 131, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_030.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_030.png", "page_index": 131, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:42+07:00" }, "raw_text": "A Drastic Example Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen English An and Binh have the same maternal grandmother BK TP.HCM The sentence in predicate logic without functions Contents Predicate Logic Motivation, Syntax, Proof Theory VxVyVuVv(M(y,x) M(An,y) Need for Richer Language Predicate Logic as Formal M(v,u) M(Binh,v)->x=u) Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.12" }, { "page_index": 132, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_030.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_030.png", "page_index": 132, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:45+07:00" }, "raw_text": "Total Correctness Program Verification Nguyen An Khuong BK TP.HCM Definition Contents Core Programming We say that the triple (oD P (WD is satisfied under total Language correctness if, for all states which satisfy o, P is guaranteed to HoareTriples.Partial and Total Correctness terminate and the resulting state satisfies V. Proof Calculus for Partial Correctness Practical Aspects of Notation Correctness Proofs We write Ftot 1oD P 0yD Correctness of the Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.30" }, { "page_index": 133, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_031.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_031.png", "page_index": 133, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:47+07:00" }, "raw_text": "Rule for Introduction of Implication Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang Introduction of Implication BK TP.HCM Contents Introduction Declarative Sentences Natural Deductior Sequents Ruies for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.31" }, { "page_index": 134, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_031.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_031.png", "page_index": 134, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:51+07:00" }, "raw_text": "Polynomial time reductions and NP-Completeness Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Denote Huynh Tuong Nguyen. Tran Van Hoai EXP = {Decision problems solvable in exponential time} P = {Decision problems solvable in polynomial time} . NP = {Decision problems where Yes solution can verified in BK polynomial time} TP.HCM A major open question in theoretical computer science is if P = NP or not. Contents Introduction Introduce the notion of polynomial time reductions Quick review X

x=u Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate The same sentence in predicate logic with functions Logic Soundness and Completeness of Predicate Logic Undecidability of m(m(An)) = m(m(Binh) Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.12" }, { "page_index": 136, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_031.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_031.png", "page_index": 136, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:56+07:00" }, "raw_text": "Back to Factorial Program Verification Nguyen An Khuong BK TP.HCM Contents Core Programming Consider Fac1: Language Hoare Triples.Partial y = 1; and Total Correctness z = 0; Proof Calculus for while (z != x Z y Partial Correctness y Practical Aspects of Correctness Proofs Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.31" }, { "page_index": 137, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_032.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_032.png", "page_index": 137, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:30:59+07:00" }, "raw_text": "Rule for Disjunction Propositional Logic Review I Nguyen An Khuong. Introduction of Disjunction Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang [Vi1] BK V22 TP.HCM o V V o V 4 Contents Introduction Elimination of Disjunction Declarative Sentences Natural Deduction Sequents Ruies for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical x x Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next x Week Plan? 1a.32" }, { "page_index": 138, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_032.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_032.png", "page_index": 138, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:31:02+07:00" }, "raw_text": "Polynomial time reductions and NP-Completeness Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Denote Huynh Tuong Nguyen Tran Van Hoai EXP = {Decision problems solvable in exponential time} P = {Decision problems solvable in polynomial time} NP = {Decision problems where Yes solution can verified in BK polynomial time} TP.HCM A major open question in theoretical computer science is if P = NP or not. Contents Introduction Introduce the notion of polynomial time reductions Quick review X

p,p V p Contents Introduction : We denote the proposition that is always true (tautology)) Declarative Sentences using the symbol T. Natural Deduction Sequents Ruies for natural deduction Another Special Proposition Basic and Derived Rules Intuitionistic Logic Similarly, we denote the proposition that is always false Formal Language Semantics (contradiction) using the symbol L. Meaning of Logical Connectives Example: p -p Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.34" }, { "page_index": 146, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_034.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_034.png", "page_index": 146, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:31:29+07:00" }, "raw_text": "Polynomial time reductions and NP-Completeness Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Denote Huynh Tuong Nguyen Tran Van Hoai EXP = {Decision problems solvable in exponential time} P = {Decision problems solvable in polynomial time} NP = {Decision problems where Yes solution can verified in BK polynomial time} TP.HCM A major open question in theoretical computer science is if P = NP or not. Contents Introduction Introduce the notion of polynomial time reductions Quick review X

OD P qx>OD Homeworks 1f.40" }, { "page_index": 173, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_041.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_041.png", "page_index": 173, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:32:43+07:00" }, "raw_text": "Some Derived Rules: Introduction of Double Negation Propositional Logic Review 1 Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Hóng Trang BK TP.HCM Contents Introduction Declarative Sentences Natural Deduction Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.41" }, { "page_index": 174, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_041.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_041.png", "page_index": 174, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:32:47+07:00" }, "raw_text": "P=NP question Propositional Logic Review Il Nguyen An Khuong. The problems in VPC are the hardest problems in NP and Le Hong Trang, Huynh Tuong Nguyen the key to resolving the P = VP question. Tran Van Hoai If one problem Y E NPC is in P then P = NP. If one problem Y E NP is not in P then NPC N P = 0. BK TP.HCM By now a lot of problems have been proved VP-complete We think the world looks like this-but we really do not know: Contents NP-EXP Introduction Quick review Boolean Satisfiability (SAT) P and NP NPC 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL. Idea If someone found a polynomial time solution to a problem in A Linear Solver NPC our world would \"collapse\" and a lot of smart people A Cubic Solver Homeworks and Next have tried really hard to solve VPC problems efficiently Week Plan? We regard Y E NPC a strong evidence for Y being hard! 1b.8" }, { "page_index": 175, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_041.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_041.png", "page_index": 175, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:32:50+07:00" }, "raw_text": "Arity of Functions and Predicates Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Every function symbol in F and predicate symbol in P comes with Contents a fixed arity, denoting the number of arguments the symbol can Predicate Logic Motivation, Syntax, take. Proof Theory Need for Richer Language Special case Predicate Logic as Formal Language Proof Theory of Predicate Function symbols with arity 0 are called constants Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.16" }, { "page_index": 176, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_041.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_041.png", "page_index": 176, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:32:53+07:00" }, "raw_text": "More Examples Program Verification Nguyen An Khuong BK TP.HCM Let P be the program x = x + 1. Using Contents Core Programming Language Assignment HoareTriplesPartial and Total Correctness 8[x-E]vD x=E(vD Proof Calculus for Partial Correctness Practical Aspects of we can prove: Correctness Proofs 0x+1=2D P qx=2D Correctness of the Factorial Function (x+1=yD P (x=yD Proof Calculus for Total Correctness Homeworks 1f.41" }, { "page_index": 177, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_042.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_042.png", "page_index": 177, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:32:57+07:00" }, "raw_text": "Example: Deriving [-- i] from [- i] and [- e] Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM 1 premise Contents 2 assumption Introduction 3 1 -e 1,2 Declarative Sentences Natural Deduction 4 -i 2-3 Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.42" }, { "page_index": 178, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_042.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_042.png", "page_index": 178, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:00+07:00" }, "raw_text": "VP-Complete Problems Propositional Logic Review II Nguyen An Khuong. The following lemma helps us to prove a problem Le Hong Trang Huynh Tuong Nguyen VP-complete using another VP-complete problem Tran Van Hoai BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.9" }, { "page_index": 179, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_042.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_042.png", "page_index": 179, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:02+07:00" }, "raw_text": "Terms Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM t::= xcf(t,...,t Contents Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.17" }, { "page_index": 180, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_042.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_042.png", "page_index": 180, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:04+07:00" }, "raw_text": "Rules for Partial Correctness (continued) Program Verification Nguyen An Khuong BK TP.HCM Q$ABD C1 04D 0&A-BD C2 0wD Contents If-statement Core Programming Language (gD if B{ C1 } else { Cz }(4D Hoare Triples Partial and Total Correctness Proof Calculus for Partial Correctness Practical Aspects of Correctness Proofs (ABD C 14D Correctness of the Factorial Function Partial-while Proof Calculus for Total Correctness 04D while B { C } (VA-BD Homeworks 1f.42" }, { "page_index": 181, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_043.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_043.png", "page_index": 181, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:07+07:00" }, "raw_text": "Some Derived Rules: Modus Tollens Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM Contents Introduction MT Declarative Sentences Natural Deduction Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.43" }, { "page_index": 182, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_043.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_043.png", "page_index": 182, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:10+07:00" }, "raw_text": "V P-Complete Problems Propositional Logic Review II Nguyen An Khuong. The following lemma helps us to prove a problem Le Hong Trang, Huynh Tuong Nguyen. NP-complete using another NP-complete problem. Tran Van Hoai Lemma:lf Y E NP and X

$ 0$D C 04D FAR -w' Hoare Triples; Partial and Total Correctness Implied] Proof Calculus for Partial Correctness 1$'D C 0'D Practical Aspects of Correctness Proofs Correctness of the Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.43" }, { "page_index": 185, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_044.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_044.png", "page_index": 185, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:18+07:00" }, "raw_text": "Some Derived Rules: Proof By Contradiction Propositional Logic Review 1 Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM Contents .. Introduction Declarative Sentences Natural Deduction Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.44" }, { "page_index": 186, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_044.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_044.png", "page_index": 186, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:21+07:00" }, "raw_text": "VP-Complete Problems Propositional Logic Review II Nguyen An Khuong. The following lemma helps us to prove a problem Le Hong Trang, Huynh Tuong Nguyen. VP-complete using another VP-complete problem Tran Van Hoai Lemma:lf Y E NP and X

0. Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.17" }, { "page_index": 192, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_045.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_045.png", "page_index": 192, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:38+07:00" }, "raw_text": "Interleave Formulas with Code Program Verification Nguyen An Khuong BK 0sD Ci (nD QnD C2 0xD TP.HCM [Composition ($D C1;C2 (wD Contents Core Programming Language Shape of rule suggests format for proof of C1; C2; . . . ; Cn: Hoare Triples; Partial and Total Correctness C1; Proof Calculus for Partial Correctness 181 D justification Practical Aspects of C2, Correctness Proofs Correctness of the Factorial Function Proof Calculus for ($n-1D justification Total Correctness Cn; Homeworks 18nD justification 1f.45" }, { "page_index": 193, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_046.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_046.png", "page_index": 193, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:42+07:00" }, "raw_text": "Motivation Propositional Logic Review I Nguyen An Khuong. Consider the following theorem Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Theorem There exist irrational numbers a and b such that ab is rational BK Let us call this theorem X. We give a Proof Outline for X TP.HCM Let p be the following proposition. Proposition p Contents Introduction is rational. Declarative Sentences Natural Deduction Then the proof of x goes like this. Sequents Rules for natural deduction Basic and Derived Rules p rp Intuitionistic Logic Formal Language LEM .. Semantics p V-p x x Meaning of Logical Connectives Preview: Soundness and Completeness Ve Normal Form Homeworks and Next x Week Plan? 1a.46" }, { "page_index": 194, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_046.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_046.png", "page_index": 194, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:33:45+07:00" }, "raw_text": "VP-Complete Problems Propositional Logic Review Il Nguyen An Khuong. The following lemma helps us to prove a problem Le Hong Trang, Huynh Tuong Nguyen NP-complete using another NP-complete problem. Tran Van Hoai Lemma:If Y E NP and X

1 Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.20" }, { "page_index": 212, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_050.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_050.png", "page_index": 212, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:34:40+07:00" }, "raw_text": "Example Program Verification Nguyen An Khuong BK TP.HCM Consider this implementation of Succ: Contents Core Programming a = x + 1; Language if (a - == 0 { Hoare Triples; Partial y = 1; and Total Correctness } else t Proof Calculus for Partial Correctness y = a; 1 Practical Aspects of Correctness Proofs Can we prove (TD Succ (y= x +1D ? Correctness of the Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.50" }, { "page_index": 213, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_051.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_051.png", "page_index": 213, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:34:43+07:00" }, "raw_text": "Intuitionistic Logic Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK Intuitionistic logic does not accept the derived rule LEM. TP.HCM The underlying argument for LEM is elimination of double negation. Contents Introduction Declarative Sentences Natural Deduction Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.51" }, { "page_index": 214, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_051.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_051.png", "page_index": 214, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:34:48+07:00" }, "raw_text": "VP-Complete Problems Propositional Logic Review Il Nguyen An Khuong. The following lemma helps us to prove a problem Le Hong Trang, Huynh Tuong Nguyen NP-complete using another NP-complete problem. Tran Van Hoai Lemma: lf Y E NP and X

1, Logic Quantifier Equivalences : t; are terms over F and Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.20" }, { "page_index": 216, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_051.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_051.png", "page_index": 216, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:34:53+07:00" }, "raw_text": "Another Example Program Verification Nguyen An Khuong BK TP.HCM if (a-1==0 ) { Contents g1=x+1D If-Statement 2 Core Programming Language y = 1; Hoare Triples Partial (y=x+1D Assignment and Total Correctness } else { Proof Calculus for Partial Correctness 0a=x+1D If-Statement 2 Practical Aspects of Correctness Proofs y = a; Correctness of the (y=x +1D Assignment Factorial Function } Proof Calculus for Total Correctness =x+1D If-Statement 2 Homeworks 1f.51" }, { "page_index": 217, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_052.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_052.png", "page_index": 217, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:34:57+07:00" }, "raw_text": "Deriving LEM using Basic Rules Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang 1 -(v-) assumption BK TP.HCM 2 assumption 3 v- Vi1 2 Contents 4 - e 3,1 Introduction 5 - i 2-4 Declarative Sentences 6 8 v -$ V i2 5 Natural Deduction Sequents 7 1 - e 6,1 Rules for natural deduction Basic and Derived Rules 8 -(V-) -i 1-7 Intuitionistic Logic 9 ov-o Formal Language nne Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.52" }, { "page_index": 218, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_052.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_052.png", "page_index": 218, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:01+07:00" }, "raw_text": "VP-Complete Problems Propositional Logic Review Il Nguyen An Khuong. The following lemma helps us to prove a problem Le Hong Trang, Huynh Tuong Nguyen NP-complete using another NP-complete problem. Tran Van Hoai Lemma: If Y E NP and X

)1(Vxo)1(Exo) Motivation, Syntax, Proof Theory Need for Richer Language where Predicate Logic as Formal Language Proof Theory of Predicate P E P is a predicate symbol of arity n > 1, Logic Quantifier Equivalences : t; are terms over F and Semantics of Predicate Logic x is a variable. Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.20" }, { "page_index": 220, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_052.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_052.png", "page_index": 220, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:07+07:00" }, "raw_text": "Another Example Program Verification Nguyen An Khuong OTD BK TP.HCM x+1-1=0-1=x+1 (-(x+1-1=0) -x+1=x+1)D Implied Contents a= x + 1; Core Programming ((a-1=0-1=x+1) Language -(a-1=0) -a=x+1)b Assignment Hoare Triples; Partial and Total Correctness if (a-1==0 ) { Proof Calculus for g1=x+1D If-Statement 2 Partial Correctness Practical Aspects of y = 1; Correctness Proofs (y=x+1D Assignment Correctness of the Factorial Function } else { Proof Calculus fo 0a=x+1D If-Statement 2 Total Correctness y = a; Homeworks 0y=x+1D Assignment 1f.52" }, { "page_index": 221, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_053.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_053.png", "page_index": 221, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:11+07:00" }, "raw_text": "Intuitionistic Logic Propositional Logic Review I Intuitionistic logic is obtained from natural deduction by removing Nguyen An Khuong. Tran Van Hoai. the rule --e. Huynh Tuong Nguyen Le Höng Trang History of Intuitionistic Logic : Late 19th century: Gottlob Frege proposes to reduce BK mathematics to set theory. TP.HCM Russell destroys this programme via paradox Contents In response, L.E.J. Brouwer proposes intuitionistic Introduction mathematics, with intuitionistic /ogic as its formal foundation Declarative Sentences . An alternative response is Hilbert's formalistic position Natural Deduction Sequents Rules for natural deduction Basic and Derived Rules Applications of Intuitionistic Logic Intuitionistic Logic Formal Language : Intuitionistic logic has a strong connection to computability Semantics Meaning of Logical . For example, if we have an intuitionistic proof of Connectives Preview: Soundness and Completeness Theorem Normal Form There exist irrational numbers a and b such that ab is rational. Homeworks and Next Week Plan? then we would know irrational a and b such that ab is rational. 1a.53" }, { "page_index": 222, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_053.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_053.png", "page_index": 222, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:17+07:00" }, "raw_text": "VP-Complete Problems Propositional Logic Review Il Nguyen An Khuong. The following lemma helps us to prove a problem Le Hong Trang, Huynh Tuong Nguyen NP-complete using another NP-complete problem. Tran Van Hoai Lemma: lf Y E NP and X

, which is right-associative. Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.21" }, { "page_index": 224, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_053.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_053.png", "page_index": 224, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:21+07:00" }, "raw_text": "Recall: Partial-while Rule Program Verification Nguyen An Khuong BK TP.HCM Contents Core Programming Language (vABD C (wD Hoare Triples; Partial and Total Correctness Partial-while Proof Calculus for Partial Correctness (vD while B{C } (VA-BD Practical Aspects of Correctness Proofs Correctness of the Factorial Function Proof Calculus fo Total Correctness Homeworks 1f.53" }, { "page_index": 225, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_054.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_054.png", "page_index": 225, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:24+07:00" }, "raw_text": "Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang Propositional Calculus Declarative Sentences BK 2 Propositional Calculus: Natural Deduction TP.HCM 3 Propositional Logic as a Formal Language Contents Introduction Declarative Sentences Semantics of Propositional Logic Natural Deduction Sequents Rules for natural deduction 5 Conjunctive Normal Form Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.54" }, { "page_index": 226, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_054.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_054.png", "page_index": 226, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:27+07:00" }, "raw_text": "Example Propositional Logic Review II : Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen. Tran Van Hoai X1 V x2, X1 V x2 2 V X3, X3 V x4, 1 V x2. BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.10" }, { "page_index": 227, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_054.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_054.png", "page_index": 227, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:31+07:00" }, "raw_text": "Parse Trees Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Vx((P(x)->Q(x))S(x,y)) has parse tree Contents Va Predicate Logic Motivation, Syntax, - Proof Theory A Need for Richer Language Predicate Logic as Formal Language S Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Q y Logic - Soundness and Completeness of x x Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.22" }, { "page_index": 228, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_054.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_054.png", "page_index": 228, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:34+07:00" }, "raw_text": "Factorial Example Program Verification Nguyen An Khuong BK TP.HCM We shall show that the following Core program Fac1 meets this Contents specification: Core Programming Language y = 1; Hoare Triples;Partial and Total Correctness z = 0; while (z != Proof Calculus for x Z: Partial Correctness Thus, to show: Practical Aspects of Correctness Proofs (TD Fac1 (y=x!D Correctness of the FactorialFunction Proof Calculus for Total Correctness Homeworks 1f.54" }, { "page_index": 229, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_055.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_055.png", "page_index": 229, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:38+07:00" }, "raw_text": "Recap: Logical Connectives Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM -: negation of p is denoted by -p V: disjunction of p and r is denoted by p V r, meaning at least Contents one of the two statements is true Introduction A: conjunction of p and r is denoted by p r, meaning both are Declarative Sentences true. Natural Deduction Sequents ?: implication between p and r is denoted by p -> r, meaning Rules for natural deduction Basic and Derived Rules that r is a logical consequence of p. Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.55" }, { "page_index": 230, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_055.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_055.png", "page_index": 230, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:41+07:00" }, "raw_text": "Example Propositional Logic Review II : Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen. Tran Van Hoai X1 V x2, X1 V x2, X2 V X3 X3 V X4, X1 V x2: : Let's try to set 1 = 0. Then the formula simplifies to: BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.10" }, { "page_index": 231, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_055.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_055.png", "page_index": 231, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:44+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Every son of my father is my brother Predicates BK TP.HCM S(x,y): x is a son of y B(x,y): x is a brother of y Contents Predicate Logic Motivation, Syntax. Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.23" }, { "page_index": 232, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_055.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_055.png", "page_index": 232, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:47+07:00" }, "raw_text": "Partial Correctness of Fac1 Program Verification Nguyen An Khuong BK TP.HCM 0y = z!D Contents while ( z != x ) { Core Programming 0y = z!Az Z xD Invariant Language (y(z+1)=(z+1)!D Implied Hoare TriplesPartial and Total Correctness Proof Calculus for (yz =z!D Assignment Partial Correctness Practical Aspects of Correctness Proofs (y = z!D Assignment Correctness of the FactorialFunction } Proof Calculus for 0y= z!A-(z x)D Partial-while Total Correctness 0y = x!D Implied Homeworks 1f.55" }, { "page_index": 233, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_056.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_056.png", "page_index": 233, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:50+07:00" }, "raw_text": "Formal itemize Required Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang Use of Meta-Language BK TP.HCM When we describe rules such as [LEM] V-o Contents we mean that letters such as can be replaced by any formula Introduction Declarative Sentences But what exactly is the set of formulas that can be used for o? Natural Deduction Allowed Sequents Rules for natural deduction (p (-q)) Basic and Derived Rules Intuitionistic Logic Formal Language Not allowed Semantics Meaning of Logical )p q-( Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.56" }, { "page_index": 234, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_056.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_056.png", "page_index": 234, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:53+07:00" }, "raw_text": "Example Propositional Logic Review II : Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen. Tran Van Hoai X1 V x2, X1 V x2, X2 V X3 X3 V X4, X1 V x2: : Let's try to set 1 = 0. Then the formula simplifies to: BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.10" }, { "page_index": 235, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_056.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_056.png", "page_index": 235, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:56+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Every son of my father is my brother Predicates BK TP.HCM S(x,y): x is a son of y B(x,y): x is a brother of y Contents Predicate Logic Motivation, Syntax, Functions Proof Theory Need for Richer Language Predicate Logic as Formal m: constant for \"me' Language Proof Theory of Predicate Logic f(x): father of x Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.23" }, { "page_index": 236, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_056.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_056.png", "page_index": 236, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:35:59+07:00" }, "raw_text": "Partial Correctness of Fac1 Program Verification Nguyen An Khuong BK TP.HCM T C(1=O!)D Implied Contents y = 1; Core Programming 0y = O!D Assignment Language z = 0; Hoare TriplesPartial and Total Correctness 0y = z!D Assignment Proof Calculus for while ( z != x ) Partial Correctness Practical Aspects of Correctness Proofs Correctness of the } FactorialFunction 0y= z!A-(z x)D Partial-while Proof Calculus for Total Correctness 0y = x!D Implied Homeworks 1f.56" }, { "page_index": 237, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_057.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_057.png", "page_index": 237, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:02+07:00" }, "raw_text": "Definition of Well-formed Formulas Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang BK Definition TP.HCM . Every propositional atom p,9,r,. .. and p1,P2,P3, ... is a Contents well-formed formula Introduction If is a well-formed formula, then so is (- Declarative Sentences . If and are well-formed formulas, then so is ( Natural Deduction Sequents - If and are well-formed formulas, then so is ( V ). Rules for natural deduction Basic and Derived Rules . If and are well-formed formulas, then so is ( - Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.57" }, { "page_index": 238, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_057.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_057.png", "page_index": 238, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:05+07:00" }, "raw_text": "Example Propositional Logic Review Il : Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen. Tran Van Hoai 1 V x2, X1 V x2, X2 V X3, X3 V X4, 1 V X2 Let's try to set 1 = 0. Then the formula simplifies to: BK TP.HCM T, 2, X2 V X3, X3 V X4, 2. where T denotes the value \"Truth\". Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.10" }, { "page_index": 239, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_057.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_057.png", "page_index": 239, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:09+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Every son of my father is my brother Predicates BK TP.HCM S(x,y): x is a son of y B(x,y): x is a brother of y Contents Predicate Logic Motivation, Syntax, Functions Proof Theory Need for Richer Language Predicate Logic as Formal m: constant for \"me' Language Proof Theory of Predicate Logic f(x): father of x Quantifier Equivalences Semantics of Predicate Logic The sentence in predicate logic Soundness and Completeness of Predicate Logic Undecidability of Vx(S(x,f(m)) - B(x,m)) Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.23" }, { "page_index": 240, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_057.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_057.png", "page_index": 240, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:12+07:00" }, "raw_text": "Program Verification Nguyen An Khuong Core Programming Language BK TP.HCM 2 Hoare Triples Partial and Total Correctness Contents Proof Calculus for Partial Correctness Core Programming Language Hoare Triples; Partial 4 Practical Aspects of Correctness Proofs and Total Correctness Proof Calculus for Partial Correctness 5 Correctness of the Factorial Function Practical Aspects of Correctness Proofs Correctness of the Factorial Function 6 Proof Calculus for Total Correctness Proof Calculus for Total Correctness Homeworks 7 Homeworks 1f.57" }, { "page_index": 241, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_058.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_058.png", "page_index": 241, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:14+07:00" }, "raw_text": "Definition very restrictive Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Hong Trang How about this formula? BK TP.HCM pA q V r Contents Usually, this is understood to mean Introduction Declarative Sentences ((pA(-q)) V r) Natural Deduction Sequents Rules for natural deduction .but for the formal treatment of this section and the first Basic and Derived Rules Intuitionistic Logic homework, we insist on the strict definition, and exclude such Formal Language formulas. Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.58" }, { "page_index": 242, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_058.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_058.png", "page_index": 242, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:18+07:00" }, "raw_text": "Example Propositional Logic Review Il : Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen. Tran Van Hoai 1 V x2, X1 V x2, X2 V X3, X3 V X4, 1 V X2 Let's try to set 1 = 0. Then the formula simplifies to BK TP.HCM T. 2, X2 V X3, X3 V X4, 2. where T denotes the value \"Truth\". Contents We are now forced to assign x2 = 1 (as there is a Introduction Quick review unit-clause), and the formula simplifies to Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.10" }, { "page_index": 243, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_058.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_058.png", "page_index": 243, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:21+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Every son of my father is my brother Predicates BK TP.HCM S(x,y): x is a son of y B(x,y): x is a brother of y Contents Predicate Logic Motivation, Syntax, Functions Proof Theory Need for Richer Language Predicate Logic as Formal m: constant for \"me\" Language Proof Theory of Predicate Logic f(x): father of x Quantifier Equivalences Semantics of Predicate Logic The sentence in predicate logic Soundness and Completeness of Predicate Logic Undecidability of Vx(S(x,f(m)) - B(x,m)) Predicate Logic Compactness of Predicate Calculus Does this formula hold? Homeworks and Next Week Plan? 1c.23" }, { "page_index": 244, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_058.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_058.png", "page_index": 244, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:24+07:00" }, "raw_text": "Ideas for Total Correctness Program Verification Nguyen An Khuong BK TP.HCM Contents The only source of non-termination is the while command Core Programming - If we can show that the value of an integer expression Language decreases in each iteration, but never becomes negative, we Hoare Triples;Partial and Total Correctness have proven termination. Proof Calculus for Partial Correctness Why? Well-foundedness of natural numbers Practical Aspects of . We shall include this argument in a new version of the while Correctness Proofs rule. Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.58" }, { "page_index": 245, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_059.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_059.png", "page_index": 245, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:27+07:00" }, "raw_text": "Backus Naur Form: A more compact definition Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM Backus Naur Form for propositional formulas Contents Introduction o ::=pol(o )l(V o)(o- o Declarative Sentences Natural Deductior where p stands for any atomic proposition Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.59" }, { "page_index": 246, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_059.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_059.png", "page_index": 246, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:31+07:00" }, "raw_text": "Example Propositional Logic Review Il : Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen. Tran Van Hoai 1 V x2, X1 V x2, X2 V X3, X3 V X4, 1 V X2 Let's try to set 1 = 0. Then the formula simplifies to BK TP.HCM T. 2, X2 V X3, X3 V X4, 2. where T denotes the value \"Truth\". Contents We are now forced to assign x2 = 1 (as there is a Introduction Quick review unit-clause), and the formula simplifies to Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.10" }, { "page_index": 247, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_059.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_059.png", "page_index": 247, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:33+07:00" }, "raw_text": "Equality as Predicate Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Equality is a common predicate, usually used in infix notation. BK TP.HCM =E P Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.24" }, { "page_index": 248, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_059.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_059.png", "page_index": 248, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:36+07:00" }, "raw_text": "Rules for Partial Correctness (continued) Program Verification Nguyen An Khuong BK TP.HCM 0ABD C 1wD Contents Partial-while Core Programming (vD while B {C} (vA-BD Language Hoare Triples;Partial and Total Correctness Proof Calculus for Partial Correctness Practical Aspects of Correctness Proofs vABA0(pA(q V(-r)))) Contents Introduction Declarative Sentences is well-formed? Natural Deduction Answer: We look for the only applicable rule in the definition (the Sequents Rules for natural deduction last rule in this case), and proceed on the parts. Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.60" }, { "page_index": 250, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_060.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_060.png", "page_index": 250, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:44+07:00" }, "raw_text": "Example Propositional Logic Review Il : Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen. Tran Van Hoai 1 V x2, X1 V X2, X2 V X3, X3 V X4, 1 V X2 Let's try to set 1 = 0. Then the formula simplifies to: BK TP.HCM T. 2, X2 V X3, X3 V 4, 2. where T denotes the value \"Truth\". Contents We are now forced to assign x2 = 1 (as there is a Introduction Quick review unit-clause), and the formula simplifies to Boolean Satisfiability (SAT) P and NP T, T, X3 X3 V X4, 0, 2-SAT is in P An example where 0 is the empty clause which denotes contradiction UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.10" }, { "page_index": 251, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_060.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_060.png", "page_index": 251, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:47+07:00" }, "raw_text": "Equality as Predicate Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Equality is a common predicate, usually used in infix notation BK TP.HCM =E P Contents Predicate Logic Example Motivation, Syntax, Proof Theory Need for Richer Language Instead of the formula Predicate Logic as Formal Language Proof Theory of Predicate =(f(x),g(x)) Logic Quantifier Equivalences Semantics of Predicate we usually write the formula Logic Soundness and Completeness of f(x) = g(x) Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.24" }, { "page_index": 252, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_060.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_060.png", "page_index": 252, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:50+07:00" }, "raw_text": "Factorial Example (Again!) Program Verification Nguyen An Khuong BK TP.HCM Contents Core Programming Language y = 1; 0; Hoare Triples; Partial z = and Total Correctness while (z != x) 1; = * z; } Proof Calculus for Partial Correctness What could be a good variant E? Practical Aspects of Correctness Proofs Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.60" }, { "page_index": 253, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_061.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_061.png", "page_index": 253, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:53+07:00" }, "raw_text": "Parse trees Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang A formula BK (((p) Aq) ->(pA(q V(-r)))) TP.HCM ...and its parse tree: Contents Introduction A A Declarative Sentences Natural Deduction q p Sequents Rules for natural deduction Basic and Derived Rules p q Intuitionistic Logic Formal Language r Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.61" }, { "page_index": 254, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_061.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_061.png", "page_index": 254, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:36:58+07:00" }, "raw_text": "Example Propositional Logic Review Il : Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen. Tran Van Hoai x1 V x2, X1 V X2, X2 V X3, X3 V X4, 1 V X2 Let's try to set 1 = 0. Then the formula simplifies to: BK TP.HCM T. 2, X2 V X3, X3 V 4, 2. where T denotes the value \"Truth\". Contents We are now forced to assign 2 = 1 (as there is a Introduction Quick review unit-clause), and the formula simplifies to Boolean Satisfiability (SAT) P and NP T, T, X31 X3 V X4, 0, 2-SAT is in P An example where is the empty clause which denotes contradiction UNSAT So we have to backtrack to the last free step Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.10" }, { "page_index": 255, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_061.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_061.png", "page_index": 255, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:00+07:00" }, "raw_text": "Free and Bound Variables Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Consider the formula BK TP.HCM Vx((P(x)->Q(x))AS(x,y)) Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.25" }, { "page_index": 256, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_061.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_061.png", "page_index": 256, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:03+07:00" }, "raw_text": "Factorial Example (Again!) Program Verification Nguyen An Khuong BK TP.HCM Contents y = 1; Core Programming Language z = 0; while (z != x) Hoare Triples; Partial and Total Correctness What could be a good variant E? Proof Calculus for Partial Correctness Practical Aspects of E must strictly decrease in the loop, but not become negative Correctness Proofs Correctness of the Factorial Function Proof Calculus for Total Correctness Homeworks 1f.61" }, { "page_index": 257, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_062.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_062.png", "page_index": 257, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:07+07:00" }, "raw_text": "Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang Propositional Calculus: Declarative Sentences Propositional Calculus: Natural Deduction BK TP.HCM 3 Propositional Logic as a Formal Language Contents Introduction Semantics of Propositional Logic Declarative Sentences Meaning of Logical Connectives Natural Deduction Sequents Preview: Soundness and Completeness Rules for natural deduction Basic and Derived Rules Intuitionistic Logic 5 Conjunctive Normal Form Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.62" }, { "page_index": 258, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_062.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_062.png", "page_index": 258, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:11+07:00" }, "raw_text": "Example Propositional Logic Review Il Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen Tran Van Hoai X1 V x2, X1 V X2, X2 V X3, X3 V X4, X1 V X2: Let's try to set 1 = 0. Then the formula simplifies to: BK TP.HCM T. 2, X2 V x3, X3 V 4, 2. where T denotes the value \"Truth\". Contents We are now forced to assign x2 = 1 (as there is a Introduction Quick review unit-clause), and the formula simplifies to Boolean Satisfiability (SAT) P and NP T, T, X31 X3 V X4, 0, 2-SAT is in P An example where is the empty clause which denotes contradiction. UNSAT So we have to backtrack to the last free step Graphical View of 2-SAT SAT Solvers . Let's try x1 = 1: WalkSAT: Idea DPLL. Idea 2, T, 2 V X3, X3 V 4, T. A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.10" }, { "page_index": 259, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_062.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_062.png", "page_index": 259, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:14+07:00" }, "raw_text": "Free and Bound Variables Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Consider the formula BK TP.HCM Vx((P(x) -Q(x)) S(x,y)) Contents What is the relationship between variable \"binder\" x and Predicate Logic occurrences of x? Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.25" }, { "page_index": 260, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_062.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_062.png", "page_index": 260, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:17+07:00" }, "raw_text": "Factorial Example (Again!) Program Verification Nguyen An Khuong BK TP.HCM 1; Contents 0; Core Programming while (z 1= x z Z y Z Language What could be a good variant E? HoareTriples;Partial and Total Correctness Proof Calculus for Partial Correctness E must strictly decrease in the loop, but not become negative Practical Aspects of Correctness Proofs Answer: Correctness of the Factorial Function x - z Proof Calculus for Total Correctness Homeworks 1f.62" }, { "page_index": 261, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_063.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_063.png", "page_index": 261, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:21+07:00" }, "raw_text": "Meaning of propositional formula Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Meaning as mathematical object We define the meaning of formulas as a function that maps formulas and valuations to truth values BK TP.HCM Approach Contents We define this mapping based on the structure of the formula Introduction using the meaning of their logical connectives Declarative Sentences Natural Deduction Truth Values Sequents Rules for natural deduction The set of truth values contains two elements T and F, where T Basic and Derived Rules Intuitionistic Logic represents \"true\" and F represents \"false\" Formal Language Semantics Valuations Meaning of Logical Connectives Preview: Soundness and A va/uation or mode/ of a formula δ is an assignment of each Completeness propositional atom in δ to a truth value. Normal Form Homeworks and Next Week Plan? 1a.63" }, { "page_index": 262, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_063.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_063.png", "page_index": 262, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:25+07:00" }, "raw_text": "Example Propositional Logic Review Il Consider the following 2-CNF formula consisting of the Nguyen An Khuong. Le Hong Trang, following clauses: Huynh Tuong Nguyen Tran Van Hoai X1 V x2, X1 V X2, X2 V X3, X3 V X4, 1 V 2: Let's try to set 1 = 0. Then the formula simplifies to: BK TP.HCM T. 2 X2 V x3, X3 V 4, x2. where T denotes the value \"Truth\". Contents We are now forced to assign 2 = 1 (as there is a Introduction Quick review unit-clause), and the formula simplifies to Boolean Satisfiability (SAT) P and NP T, T, X31 X3 V X4, 0, 2-SAT is in P An example where is the empty clause which denotes contradiction. UNSAT So we have to backtrack to the last free step Graphical View of 2-SAT SAT Solvers . Let's try x1 = 1: WalkSAT: Idea DPLL. Idea X2, T, 2 V X3, X3 V X4, T. A Linear Solver A Cubic Solver . We are now forced to set 2 = 1: Homeworks and Next Week Plan? T. T, X3 X3 V X4, T. 1b.10" }, { "page_index": 263, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_063.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_063.png", "page_index": 263, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:29+07:00" }, "raw_text": "Free and Bound Variables Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Consider the formula BK TP.HCM Vx((P(x)-Q(x))S(x,y)) Contents What is the relationship between variable \"binder\" x and Predicate Logic occurrences of x? Motivation, Syntax, Proof Theory Va Need for Richer Language - Predicate Logic as Formal Language A Proof Theory of Predicate Logic Quantifier Equivalences S Semantics of Predicate Logic Soundness and P Q y Completeness of - Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.25" }, { "page_index": 264, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_063.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_063.png", "page_index": 264, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:32+07:00" }, "raw_text": "Total Correctness of Fac1 Program Verification Nguyen An Khuong BK TP.HCM 0y= z!A0< x- zD Contents while ( z != x ) { Core Programming (y=z!Az x A0(-P(x) V Q(y)) Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.26" }, { "page_index": 268, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_064.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_f/slide_064.png", "page_index": 268, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:37:47+07:00" }, "raw_text": "Total Correctness of Fac1 Program Verification Nguyen An Khuong BK TP.HCM qx < OD d(1=O!A0-q q V -p (p->-q)->(q V-p) T T F F F T T Contents T F F T T F F Introduction Declarative Sentences F T T F T T T Natural Deduction F F T T T T T Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.67" }, { "page_index": 277, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_067.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_067.png", "page_index": 277, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:22+07:00" }, "raw_text": "Algorithm() Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows BK Algorithm(@) TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return . Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 278, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_067.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_067.png", "page_index": 278, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:25+07:00" }, "raw_text": "Substitution Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Variables are p/aceholders. Rep/acing them by terms is called BK substitution TP.HCM Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.27" }, { "page_index": 279, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_068.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_068.png", "page_index": 279, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:29+07:00" }, "raw_text": "Validity and Satisfiability Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM Validity A formula is va/id if it computes T for all its valuations Contents Introduction Declarative Sentences Satisfiability Natural Deduction A formula is satisfiab/e if it computes T for at least one of its Sequents Rules for natural deduction valuations Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.68" }, { "page_index": 280, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_068.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_068.png", "page_index": 280, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:31+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm() TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return g. Contents Introduction (2) Choose an unassigned variable i. Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 281, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_068.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_068.png", "page_index": 281, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:34+07:00" }, "raw_text": "Substitution Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Variables are p/aceholders. Rep/acing them by terms is called BK substitution. TP.HCM Definition Contents Given a variable , a term t and a formula , we define x = tld Predicate Logic to be the formula obtained by replacing each free occurrence of Motivation, Syntax, Proof Theory variable x in o with t. Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.27" }, { "page_index": 282, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_069.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_069.png", "page_index": 282, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:38+07:00" }, "raw_text": "Semantic Entailment, Soundness and Completeness of Propositional Logic Review I Propositional Logic Nguyen An Khuong. Tran Van Hoai. Huynh Tuong Nguyen Le Höng Trang Semantic Entailment If, for all valuations in which all 1, 2,: : On evaluate to T, the BK formula evaluates to T as well, we say that 1, 2, : : : , n TP.HCM semantically entail V, written: Contents 1, $2, : . ., n = Introduction Declarative Sentences Natural Deduction Soundness Sequents Rules for natural deduction Let $1, $2,...,$n and be propositional formulas. If Basic and Derived Rules Intuitionistic Logic $1,$2,.. .,$n F valid (has a proof), then $1,$2,...,$n F Formal Language Semantics Meaning of Logical Completeness Connectives Preview:Soundness and Let $1,$2, ...,$n and be propositional formulas. If Compieteness Normal Form 1,2,: . .,n , then 1,2, .. .,$n F valid (has a proof Homeworks and Next Week Plan? 1a.69" }, { "page_index": 283, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_069.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_069.png", "page_index": 283, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:41+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm() TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return o. Contents Introduction (2) Choose an unassigned variable xi. Quick review (@) (Try xi = 1) Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 284, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_069.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_069.png", "page_index": 284, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:46+07:00" }, "raw_text": "Substitution Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Variables are p/aceholders. Rep/acing them by terms is called BK substitution. TP.HCM Definition Contents Given a variable , a term t and a formula , we define [x = tlo Predicate Logic to be the formula obtained by replacing each free occurrence of Motivation, Syntax, Proof Theory variable x in o with t. Need for Richer Language Predicate Logic as Formal Language Example Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate [x = f(x,y)](Vx(P(x) Q(x))) -(P(x) V Q(y) Logic Soundness and Completeness of =Vx(P(x) Q(x)) ->(-P(f(x,y)) V Q(y)) Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.27" }, { "page_index": 285, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_070.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_070.png", "page_index": 285, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:49+07:00" }, "raw_text": "Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang Propositional Calculus: Declarative Sentences BK 2 Propositional Calculus: Natural Deduction TP.HCM 3 Propositional Logic as a Formal Language Contents Introduction Declarative Sentences Semantics of Propositional Logic Natural Deduction Sequents Rules for natural deduction Conjunctive Normal Form Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.70" }, { "page_index": 286, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_070.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_070.png", "page_index": 286, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:52+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm() TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return o. Contents Introduction (2) Choose an unassigned variable xi. Quick review (@) (Try xi = 1) Boolean Satisfiability (SAT) P and NP - Set oj =1,q' Simplify(g,xi 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 287, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_070.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_070.png", "page_index": 287, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:38:56+07:00" }, "raw_text": "A Note on Notation Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Instead of Contents Predicate Logic the textbook uses the notation Motivation, Syntax. Proof Theory Need for Richer Language s[t/x] Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.28" }, { "page_index": 288, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_071.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_071.png", "page_index": 288, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:00+07:00" }, "raw_text": "Conjunctive Normal Form Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Definition Le Höng Trang A literal L is either an atom p or the negation of an atom -p. A formula C is in conjunctive norma/ form (CNF) if it is a BK conjunction of clauses, where each clause is a disjunction of TP.HCM literals: Contents L ::= p-p, Introduction D ::= LLVD, Declarative Sentences C Natural Deduction DDAC. Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Examples Formal Language Semantics (-p V q V r) A(-q V r) A(-r) is in CNF. Meaning of Logical Connectives Preview: Soundness and (-p V q V r) A((pA -q) V r) A(-r) is not in CNF Completeness Normal Form (p V q V r) A -(-q V r) A(Gr) is not in CNF Homeworks and Next Week Plan? 1a.71" }, { "page_index": 289, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_071.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_071.png", "page_index": 289, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:04+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm() TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return o. Contents Introduction (2) Choose an unassigned variable xi: Quick review (@) (Try xi = 1) Boolean Satisfiability (SAT) P and NP - Set oi = 1, g' Simplify(o,xi 2-SAT is in P - o' Unit Clause Propagation(') An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 290, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_071.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_071.png", "page_index": 290, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:07+07:00" }, "raw_text": "A Note on Notation Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Instead of Contents Predicate Logic the textbook uses the notation Motivation, Syntax, Proof Theory Need for Richer Language s[t/x] Predicate Logic as Formal Language Proof Theory of Predicate Logic we find the order of arguments in the latter notation hard to Quantifier Equivalences remember) Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.28" }, { "page_index": 291, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_072.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_072.png", "page_index": 291, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:11+07:00" }, "raw_text": "Usefulness of CNF Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Lemma Huynh Tuong Nguyen. Le Höng Trang A disjunction of literals L1 V L2 V :: : V Em is valid iff there are 1< i,j < m such that Li is -Lj BK How to disprove TP.HCM (-q V p V r) A(-p V r) A q? Contents Introduction Disprove any of: Declarative Sentences Natural Deduction F(-q V pV r) F(-p V r) F q. Sequents Rules for natural deduction Basic and Derived Rules How to prove Intuitionistic Logic Formal Language Semantics (-q V p V q) (p V r-p) (r V -r)? Meaning of Logical Connectives Preview: Soundness and Prove all of: Completeness Normal Form (q V p V q) F(p V r-p) F (r v -r). Homeworks and Next Week Plan? 1a.72" }, { "page_index": 292, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_072.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_072.png", "page_index": 292, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:14+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm(@) TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return o. Contents (2) Choose an unassigned variable x. Introduction Quick review (@) (Try xi = 1) Boolean Satisfiability (SAT) P and NP - Set oi =1, ' < Simplify(o,xi) 2-SAT is in P - δ' Unit Clause Propagation(') An example - If δ' does not contain 0 goto (1) UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 293, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_072.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_072.png", "page_index": 293, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:18+07:00" }, "raw_text": "Example as Parse Tree Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen [x= f(x,y)]((Vx(P(x) AQ(x))) ->(-P(x) V Q(y))) BK TP.HCM =(Vx(P(x) Q(x))) -(-P(f(x,y) V Q(y) Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.29" }, { "page_index": 294, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_073.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_073.png", "page_index": 294, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:21+07:00" }, "raw_text": "Usefulness of CNF (cont.) and Transformation to CNF Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Huynh Tuong Nguyen Le Höng Trang Proposition BK Let be a formula of propositional logic. Then ° is satisfiable iff TP.HCM -o is not valid. Contents Satisfiability test Introduction Declarative Sentences We can test satisfiability of by transforming -o into CNF, and Natural Deduction show that some clause is not valid Sequents Rules for natural deduction Basic and Derived Rules Theorem-Transformation to CNF Intuitionistic Logic Formal Language Every formula in the propositional calculus can be transformed Semantics into an equivalent formula in CNF Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.73" }, { "page_index": 295, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_073.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_073.png", "page_index": 295, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:25+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows BK Algorithm() TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return . Contents (2) Choose an unassigned variable x. Introduction Quick review (a) (Try xi = 1) Boolean Satisfiability (SAT) P and NP - Set oi = 1, g' Simplify(o,xi 2-SAT is in P - $' Unit Clause Propagation(@') An example - If @' does not contain goto (1 UNSAT (b) (Try xi=0) Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 296, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_073.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_073.png", "page_index": 296, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:28+07:00" }, "raw_text": "Example as Parse Tree Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen [x= f(x,y)]((Vx(P(x) Q(x))) ->(-P(x) V Q(y))) BK TP.HCM =(Vx(P(x) Q(x))) -(-P(f(x,y) V Q(y Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language V x V Predicate Logic as Formal Language 1 Proof Theory of Predicate A Q Logic Quantifier Equivalences Semantics of Predicate p P y Logic Soundness and Completeness of x x Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.29" }, { "page_index": 297, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_074.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_074.png", "page_index": 297, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:32+07:00" }, "raw_text": "Algorithm for CNF Transformation Propositional Logic Review I Nguyen An Khuong. Tran Van Hoai, Eliminate implication using: Huynh Tuong Nguyen Le Höng Trang A- B=-AVB. Push all negations inward using De Morgan's laws: BK TP.HCM (AAB)=(-AV-B) Contents -(AV B)=(-A-B) Introduction Declarative Sentences Natural Deduction Eliminate double negations using the equivalence --A = A. Sequents 3 Rules for natural deduction 4 The formula now consists of disjunctions and conjunctions of Basic and Derived Rules Intuitionistic Logic literals. Use the distributive laws to eliminate conjunctions Formal Language within disjunctions: Semantics Meaning of Logical Connectives AV(BAC)=(AVB)A(AVC Preview: Soundness and Completeness Normal Form Homeworks and Next (AB) VC=(AVC) (BVC) Week Plan? 1a.74" }, { "page_index": 298, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_074.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_074.png", "page_index": 298, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:35+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm(@) TP.HCM (0) Initialize empty assignment = * (1) If all variables are assigned return . Contents (2) Choose an unassigned variable xi. Introduction Quick review (a) (Try xi = 1) Boolean Satisfiability (SAT) P and NP - Set oi = 1, $' - Simplify(,xi 2-SAT is in P - $' Unit Clause Propagation(@') An example - If $' does not contain goto (1 UNSAT (b) (Try xi = 0) Graphical View of 2-SAT SAT Solvers - Unassign variables from step (a) WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 299, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_074.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_074.png", "page_index": 299, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:39+07:00" }, "raw_text": "Example as Parse Tree Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Contents V Predicate Logic Motivation, Syntax, A Proof Theory Need for Richer Language Predicate Logic as Formal P Language P y Proof Theory of Predicate Logic Quantifier Equivalences x f Semantics of Predicate Logic x y Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.30" }, { "page_index": 300, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_075.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_075.png", "page_index": 300, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:41+07:00" }, "raw_text": "Example Propositional Logic Review I Nguyen An Khuong Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM (-p->-q)->(p->q) (p V -q) V(p V q) Contents = (G-p A q) V(p V q) Introduction Declarative Sentences (p A q) V (p V q) Natural Deductior (-p V -p V q) A(q V -p V q Sequents Rules for natural deduction T. Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.75" }, { "page_index": 301, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_075.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_075.png", "page_index": 301, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:45+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm(@) TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return . Contents (2) Choose an unassigned variable xi. Introduction Quick review (@) (Try xi = 1) Boolean Satisfiability (SAT) P and NP - Set oi = 1, $' - Simplify(,xi 2-SAT is in P - @' Unit Clause Propagation(') An example - If $' does not contain goto (1 UNSAT (b) (Try xi=0) Graphical View of 2-SAT SAT Solvers - Unassign variables from step (a) WalkSAT: Idea - Set o; = 0, ' - Simplify(,xi DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 302, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_075.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_075.png", "page_index": 302, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:48+07:00" }, "raw_text": "Capturing in [x => t] Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Problem t contains variable y and ε occurs under the scope of Vy in @ BK TP.HCM Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.31" }, { "page_index": 303, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_076.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_076.png", "page_index": 303, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:52+07:00" }, "raw_text": "Homeworks Propositional Logic Review 1 I. Write down the explanations (in Vietnamese, or in English if Nguyen An Khuong. Tran Van Hoai possible) of the following terms, find examples for each term, Huynh Tuong Nguyen Le Höng Trang what are the differences between them: fallacy, contradiction, paradox, counterexample; 2) premise, assumption, axiom, hypothesis, conjecture; BK 3) tautology, valid, contradiction, satisfiable; TP.HCM 4) soundness, completeness; 5) sequent, consequence, implication, (semantic) entailment; Contents 6) argument, variable, arity; Introduction Il. What are the differences between the following notations: Declarative Sentences ', '-', 'F', '-'? And what are the differences between Natural Deduction the following notations: ' ', '', 'HF', '=', '=-'? Find Sequents Rules for natural deduction examples to illustrate these differences. Basic and Derived Rules Intuitionistic Logic Ill. It is recommended that you should do as much as you can Formal Language ALL marked exercises in [2] (notice that sample solutions for Semantics these exercises are available in [3D. For this lecture, the Meaning of Logical Connectives following are recommended exercises [2]: Preview: Soundness and Completeness 1.1: 2d), 2g); Normal Form 1.2: 1d), 1g), 1m, 1q, 1u), 1w),3a),3b, 3c),3f), 3g), 3I),3o); Homeworks and Next 1.4: 12d): Week Plan? 1.5: 3b), 3c), 7c 1a.76" }, { "page_index": 304, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_076.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_076.png", "page_index": 304, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:56+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm(@) TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return . Contents (2) Choose an unassigned variable xi. Introduction Quick review (@) (Try xi = 1) Boolean Satisfiability (SAT) P and NP - Set oi = 1, $' - Simplify(,xi 2-SAT is in P - @' Unit Clause Propagation(') An example - If $' does not contain goto (1 UNSAT (b) (Try xi =0) Graphical View of 2-SAT SAT Solvers - Unassign variables from step (a) WalkSAT: Idea - Set oi = 0, ' - Simplify(,xi). DPLL. Idea - δ' Unit Clause Propagation(') A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 305, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_076.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_076.png", "page_index": 305, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:39:59+07:00" }, "raw_text": "Capturing in [x => t]d Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Problem t contains variable y and ε occurs under the scope of Vy in @ BK TP.HCM Example Contents Predicate Logic Motivation, Syntax, [x= f(y,y)](S(x) Vy(P(x) -> Q(y))) Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalence Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.31" }, { "page_index": 306, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_077.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_a/slide_077.png", "page_index": 306, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:02+07:00" }, "raw_text": "Next Week? Propositional Logic Review I Nguyen An Khuong, Tran Van Hoai, Huynh Tuong Nguyen. Le Höng Trang BK TP.HCM Exercises Session; Contents Introduction 2, Section 1.6]: SAT Solvers; Declarative Sentences Application of SAT Solving Natural Deduction Sequents Rules for natural deduction Basic and Derived Rules Intuitionistic Logic Formal Language Semantics Meaning of Logical Connectives Preview: Soundness and Completeness Normal Form Homeworks and Next Week Plan? 1a.77" }, { "page_index": 307, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_077.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_077.png", "page_index": 307, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:05+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm(@) TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return o. Contents (2) Choose an unassigned variable xi. Introduction Quick review (@) (Try xi = 1) Boolean Satisfiability (SAT) P and NP - Set oi = 1, ' - Simplify(,xi) 2-SAT is in P - @' Unit Clause Propagation(') An example - If $' does not contain goto (1)) UNSAT (b) (Try xi =0) Graphical View of 2-SAT SAT Solvers - Unassign variables from step (a) WalkSAT: Idea - Set 0i = 0, $' - Simplify(δ,xi) DPLL. Idea - δ' Unit Clause Propagation() A Linear Solver - If $' does not contain goto (1)) A Cubic Solver Homeworks and Next Week Plan? 1b.11" }, { "page_index": 308, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_077.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_077.png", "page_index": 308, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:09+07:00" }, "raw_text": "Capturing in [x = t]g Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Problem t contains variable y and ε occurs under the scope of Vy in $ BK TP.HCM Example Contents Predicate Logic Motivation, Syntax, [x= f(y,y)](S(x) Vy(P(x) -> Q(y)) Proof Theory Need for Richer Language A Predicate Logic as Formal Language Proof Theory of Predicate S Vy Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and P Completeness of Predicate Logic 1 Undecidability of Predicate Logic y Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.31" }, { "page_index": 309, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_078.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_078.png", "page_index": 309, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:13+07:00" }, "raw_text": "Algorithm() Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Abstracting the above example, we present an algorithm that Tran Van Hoai attempts to satisfy a 2-CNF formula as follows. BK Algorithm() TP.HCM (0) Initialize empty assignment = *n (1) If all variables are assigned return o. Contents (2) Choose an unassigned variable xi. Introduction Quick review (@) (Try xi = 1) Boolean Satisfiability (SAT) P and NP - Set oi = 1, ' - Simplify(,xi) 2-SAT is in P - δ' Unit Clause Propagation() An example - If $' does not contain goto (1)) UNSAT (b) (Try xi =0) Graphical View of 2-SAT SAT Solvers - Unassign variables from step (a) WalkSAT: Idea - Set 0i =0, $' - Simplify(δ,xi). DPLL. Idea - δ' Unit Clause Propagation(') A Linear Solver - If $' does not contain 0 goto (1 A Cubic Solver Homeworks and Next 3) Halt with \"UNSAT\" Week Plan? 1b.11" }, { "page_index": 310, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_078.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_078.png", "page_index": 310, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:16+07:00" }, "raw_text": "Avoiding Capturing Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK Definition TP.HCM Given a term t, a variable x and a formula o, we say that t is free for x in o if no free x leaf in o occurs in the scope of Vy or =y for Contents Predicate Logic any variable y occurring in t. Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.32" }, { "page_index": 311, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_079.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_079.png", "page_index": 311, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:18+07:00" }, "raw_text": "Simplify(δ,l;) Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Simplify(,li) Tran Van Hoai : V clause C E &: BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 312, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_079.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_079.png", "page_index": 312, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:21+07:00" }, "raw_text": "Avoiding Capturing Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK Definition TP.HCM Given a term t, a variable and a formula o, we say that t is free for in o if no free x leaf in occurs in the scope of Vy or 3y for Contents Predicate Logic any variable y occurring in t. Motivation, Syntax, Proof Theory Need for Richer Language Free-ness as precondition Predicate Logic as Formal Language Proof Theory of Predicate In order to compute x = t]o, we demand that t is free for x in Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.32" }, { "page_index": 313, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_080.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_080.png", "page_index": 313, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:23+07:00" }, "raw_text": "Simplify(δ,;) Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Simplify(,li) Tran Van Hoai : V clause C E : - If li E C, remove C BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 314, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_080.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_080.png", "page_index": 314, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:27+07:00" }, "raw_text": "Avoiding Capturing Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK Definition TP.HCM Given a term t, a variable x and a formula o, we say that t is free for c in o if no free x leaf in o occurs in the scope of Vy or =y for Contents Predicate Logic any variable y occurring in t. Motivation, Syntax, Proof Theory Need for Richer Language Free-ness as precondition Predicate Logic as Formal Language Proof Theory of Predicate In order to compute [x = tl&, we demand that t is free for x in δ Logic Quantifier Equivalences Semantics of Predicate What if not? Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.32" }, { "page_index": 315, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_081.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_081.png", "page_index": 315, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:28+07:00" }, "raw_text": "Simplify(δ,;) Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Simplify(,) Tran Van Hoai : V clause C E &: - If li E C, remove C BK - IfliEC,C-Cli. TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 316, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_081.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_081.png", "page_index": 316, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:31+07:00" }, "raw_text": "Avoiding Capturing Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK Definition TP.HCM Given a term t, a variable ε and a formula δ, we say that t is free for in o if no free x leaf in o occurs in the scope of Vy or y for Contents Predicate Logic any variable y occurring in t. Motivation, Syntax, Proof Theory Need for Richer Language Free-ness as precondition Predicate Logic as Formal Language Proof Theory of Predicate In order to compute x = tl&, we demand that t is free for x in δ Logic Quantifier Equivalences Semantics of Predicate What if not? Logic Soundness and Rename the bound variable! Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.32" }, { "page_index": 317, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_082.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_082.png", "page_index": 317, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:34+07:00" }, "raw_text": "Simplify(δ,;) Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Simplify(,i) Tran Van Hoai : V clause C E : - If li E C, remove C BK TP.HCM - IfliEC,CCli. - Otherwise, copy C as is. Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 318, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_082.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_082.png", "page_index": 318, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:36+07:00" }, "raw_text": "Example of Renaming Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM [x= f(y,y)](S(x) Vy(P(x) -> Q(y))) Contents Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.33" }, { "page_index": 319, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_083.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_083.png", "page_index": 319, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:39+07:00" }, "raw_text": "Simplify(δ,li) Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Simplify(,li) Tran Van Hoai : V clause C E : - If li E C, remove C BK TP.HCM -Ifli EC,CCli - Otherwise, copy C as is Contents Output the modified formula Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 320, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_083.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_083.png", "page_index": 320, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:42+07:00" }, "raw_text": "Example of Renaming Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM [x= f(y,y)](S(x) Vy(P(x) -> Q(y))) Contents Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.33" }, { "page_index": 321, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_084.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_084.png", "page_index": 321, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:44+07:00" }, "raw_text": "Simplify(δ,li) Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Simplify(,li) Tran Van Hoai : V clause C E : - If li E C, remove C BK TP.HCM -Ifli EC,CCli - Otherwise, copy C as is Contents Output the modified formula Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 322, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_084.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_084.png", "page_index": 322, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:47+07:00" }, "raw_text": "Example of Renaming Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM [x= f(y,y)](S(x) Vy(P(x)->Q(y))) Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language [x= f(y,y)](S(x) AVz(P(x)->Q(z)) Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.33" }, { "page_index": 323, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_085.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_085.png", "page_index": 323, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:49+07:00" }, "raw_text": "Simplify(δ,;) Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Simplify(,li) Tran Van Hoai : V clause C E : - If li E C, remove C BK TP.HCM -Ifli EC,CCli - Otherwise, copy C as is Contents Output the modified formula Introduction Unit Clause Propagation() Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 324, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_085.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_085.png", "page_index": 324, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:52+07:00" }, "raw_text": "Example of Renaming Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM [x= f(y,y)](S(x) Vy(P(x)->Q(y))) Contents 4 Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language [x= f(y,y)](S(x) AVz(P(x)-Q(z)) Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences 4 Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.33" }, { "page_index": 325, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_086.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_086.png", "page_index": 325, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:54+07:00" }, "raw_text": "Simplify(δ,; Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Simplify(,li) Tran Van Hoai : V clause C E : - If li E C, remove C BK TP.HCM -Ifli EC,CCli - Otherwise, copy C as is Contents Output the modified formula Introduction Unit Clause Propagation() Quick review Boolean Satisfiability (SAT) : While unit clause l: P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 326, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_086.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_086.png", "page_index": 326, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:40:57+07:00" }, "raw_text": "Example of Renaming Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM [x= f(y,y)](S(x) Vy(P(x)->Q(y))) Contents 4 Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language [x= f(y,y)](S(x) AVz(P(x) ->Q(z)) Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and S(f(y,y))Vz(P(f(y,y))->Q(z)) Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.33" }, { "page_index": 327, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_087.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_087.png", "page_index": 327, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:00+07:00" }, "raw_text": "Simplify(δ,; Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Simplify(,li) Tran Van Hoai : V clause C E : - If li E C, remove C BK TP.HCM If li E C,CCli - Otherwise, copy C as is Contents Output the modified formula Introduction Unit Clause Propagation() Quick review Boolean Satisfiability (SAT) o While 3 unit clause l;: P and NP 2-SAT is in P - Update o: if li = xi set oi = 1, else (li = xi) set oi = O. An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 328, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_087.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_087.png", "page_index": 328, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:03+07:00" }, "raw_text": "Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language BK TP.HCM Proof Theory of Predicate Logic Quantifier Equivalences Contents Predicate Logic: Semantics of Predicate Logic Motivation, Syntax, 2 Proof Theory Need for Richer Language Predicate Logic as Formal Language 3 Soundness and Completeness of Predicate Logic Proof Theory of Predicate Logic Quantifier Equivalences 4 Undecidability of Predicate Logic Semantics of Predicate Logic Soundness and Completeness of 5 Compactness of Predicate Calculus Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.34" }, { "page_index": 329, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_088.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_088.png", "page_index": 329, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:07+07:00" }, "raw_text": "Simplify(δ,; Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Simplify(,li) Tran Van Hoai : V clause C E : - If li E C, remove C BK TP.HCM If li E C,CCli - Otherwise, copy C as is. Contents Output the modified formula Introduction Unit Clause Propagation() Quick review Boolean Satisfiability (SAT) o While 3 unit clause l;: P and NP 2-SAT is in P Update o: if li = xi set oi = 1, else (li = xi) set oi = 0. An example - Simplify($,li). UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 330, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_088.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_088.png", "page_index": 330, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:10+07:00" }, "raw_text": "Natural Deduction for Predicate Logic Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Relationship between propositional and predicate logic If we consider propositions as nullary predicates, propositional logic BK TP.HCM is a sub-language of predicate logic Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.35" }, { "page_index": 331, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_089.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_089.png", "page_index": 331, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:12+07:00" }, "raw_text": "Simplify(δ,; Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Simplify(,li) Tran Van Hoai : V clause C E : - If li E C, remove C BK TP.HCM If li E C,CCli - Otherwise, copy C as is. Contents Output the modified formula Introduction Unit Clause Propagation() Quick review Boolean Satisfiability (SAT) o While 3 unit clause l;: P and NP 2-SAT is in P Update o: if li = xi set oi = 1, else (li = xi) set oi = 0. An example - Simplify($,li). UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.12" }, { "page_index": 332, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_089.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_089.png", "page_index": 332, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:16+07:00" }, "raw_text": "Natural Deduction for Predicate Logic Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Relationship between propositional and predicate logic If we consider propositions as nullary predicates, propositional logic BK TP.HCM is a sub-language of predicate logic Inheriting natural deduction Contents Predicate Logic We can translate the rules for natural deduction in propositiona Motivation, Syntax, Proof Theory logic directly to predicate logic Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.35" }, { "page_index": 333, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_090.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_090.png", "page_index": 333, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:20+07:00" }, "raw_text": "Simplify(δ,; Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Simplify(,l) Tran Van Hoai : V clause C E δ: If li E C, remove C BK - 1f li E C,Ct2]6 Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.36" }, { "page_index": 337, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_092.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_092.png", "page_index": 337, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:31+07:00" }, "raw_text": "Correctness of the Algorithm Propositional Logic Review II Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. If the algorithm outputs an assignment , then satisfies Tran Van Hoai We will need the following definition: A partial assignment o E{0,1,*}n violate a clause C = li V lj if: oi and oj are BK TP.HCM assigned (i.e.,i,j *) and oi doesn't satisfy li and j doesn't satisfy lj. Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.13" }, { "page_index": 338, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_092.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_092.png", "page_index": 338, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:33+07:00" }, "raw_text": "Properties of Equality Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen We show: BK f(x)= g(x) F h(g(x))= h(f(x)) TP.HCM using Contents Predicate Logic Motivation, Syntax, t1 = t2 [x =t1]d Proof Theory Need for Richer Language Predicate Logic as Formal Language t =t x =t2lg Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.37" }, { "page_index": 339, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_093.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_093.png", "page_index": 339, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:36+07:00" }, "raw_text": "Correctness of the Algorithm Propositional Logic Review II Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. If the algorithm outputs an assignment o, then o satisfies ° Tran Van Hoai We will need the following definition: A partial assignment o E{0,1,*}n violate a clause C =li V lj if: oi and oj are BK TP.HCM assigned (i.e.,i,j *) and oi doesn't satisfy li and j doesn't satisfy t;. The lemma follows from the following invariance. Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.13" }, { "page_index": 340, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_093.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_093.png", "page_index": 340, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:40+07:00" }, "raw_text": "Properties of Equality Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen We show: BK f(x)= g(x) F h(g(x))= h(f(x)) TP.HCM using Contents Predicate Logic Motivation, Syntax, t1 = t2 [x=t1]B Proof Theory Need for Richer Language Predicate Logic as Formal Language t = t Proof Theory of Predicate = t2d Logic Quantifier Equivalences Semantics of Predicate Logic 1 f(x) = g(x) premise Soundness and 2 h(f(x))=h(f(x)) = 2 Completeness of Predicate Logic 3 h(g(x)) = h(f(x)) = e 1,2 Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.37" }, { "page_index": 341, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_094.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_094.png", "page_index": 341, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:43+07:00" }, "raw_text": "Correctness of the Algorithm Propositional Logic Review Il Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs an assignment , then satisfies ° Tran Van Hoai We will need the following definition: A partial assignment o E{0,1,*}n violate a clause C = li V lj if: oi and oj are BK TP.HCM assigned (i.e.,i,j *) and oi doesn't satisfy li and ; doesn't satisfy £;. The lemma follows from the following invariance. Contents Lemma Introduction At the beginning of each iteration, the current partial assignment Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.13" }, { "page_index": 342, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_094.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_094.png", "page_index": 342, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:46+07:00" }, "raw_text": "Rules for Universal Quantification Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Contents Predicate Logic: xc Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.38" }, { "page_index": 343, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_095.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_095.png", "page_index": 343, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:49+07:00" }, "raw_text": "Correctness of the Algorithm Propositional Logic Review Il Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs an assignment , then satisfies ° Tran Van Hoai We will need the following definition: A partial assignment o E{0,1,*}n violate a clause C = li V lj if: oi and oj are BK TP.HCM assigned (i.e.,i,j *) and oi doesn't satisfy li and ; doesn't satisfy lj. The lemma follows from the following invariance. Contents Lemma Introduction At the beginning of each iteration, the current partial assignment Quick review Boolean Satisfiability (SAT) (i) does not violate any of the clauses of C. P and NP 2-SAT is in P An example Chü'ng minh. UNSAT Graphical View of 2-SAT Invariance 2 By induction on i. The basis is trivial as in the first SAT Solvers iteration = *\" and so none of the clauses are violated. WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.13" }, { "page_index": 344, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_095.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_095.png", "page_index": 344, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:52+07:00" }, "raw_text": "Example Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Vxo Contents Predicate Logic [x =t]a Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal We prove: F(m(Duong)),Vx(F(x) - -M(x)) F -M(m(Duong)) Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.39" }, { "page_index": 345, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_096.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_096.png", "page_index": 345, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:41:56+07:00" }, "raw_text": "Correctness of the Algorithm Propositional Logic Review Il Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs an assignment , then satisfies ° Tran Van Hoai We will need the following definition: A partial assignment o E{0,1,*}n violate a clause C = li V lj if: oi and oj are BK TP.HCM assigned (i.e.,0i,j *) and oi doesn't satisfy li and j doesn't satisfy £;. The lemma follows from the following invariance. Contents Lemma Introduction At the beginning of each iteration, the current partial assignment Quick review Boolean Satisfiability (SAT) (i) does not violate any of the clauses of C. P and NP 2-SAT is in P An example Chü'ng minh. UNSAT Graphical View of 2-SAT Invariance 2 By induction on i. The basis is trivial as in the first SAT Solvers iteration = * and so none of the clauses are violated. Step: WalkSAT: Idea DPLL. Idea we'll prove that none of the clauses C are violated by o(i+1). If A Linear Solver both variables of C were assigned before the last iteration, then. A Cubic Solver by the induction hypothesis, (i) doesn't violate C, and therefore, Homeworks and Next Week Plan? algorithm finds a contradiction. 1b.13" }, { "page_index": 346, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_096.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_096.png", "page_index": 346, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:00+07:00" }, "raw_text": "Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Vco Contents V. Predicate Logic [x =t]a Motivation, Syntax, Proof Theory Need for Richer Language We prove: F(m(Duong)),Vx(F(x) - -M(x)) F -M(m(Duong)) Predicate Logic as Formal Language Proof Theory of Predicate Logic 1 F(m(Duong)) premise Quantifier Equivalences 2 Vx(F(x) --M(x) premise Semantics of Predicate Logic 3 F(m(Duong)) -> -M(m(Duong)) Vx e 2 Soundness and 4 -M(m(Duong)) > e 3,1 Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.39" }, { "page_index": 347, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_097.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_097.png", "page_index": 347, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:02+07:00" }, "raw_text": "Correctness of the Algorithm (cont.) Propositional Logic Review II Lemma Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen. If the algorithm outputs UNSAT, then o is unsatisfiable Tran Van Hoai BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.14" }, { "page_index": 348, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_097.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_097.png", "page_index": 348, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:05+07:00" }, "raw_text": "Rules for Universal Quantification Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen If we manage to establish a formula $ about a fresh variable xo, BK TP.HCM we can assume Vco Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theoryof Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.40" }, { "page_index": 349, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_098.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_098.png", "page_index": 349, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:08+07:00" }, "raw_text": "Correctness of the Algorithm (cont.) Propositional Logic Review II Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. If the algorithm outputs UNSAT, then is unsatisfiable Tran Van Hoai Chu'ng minh. BK TP.HCM Let °' be the formula at the beginning of the iteration in which A halts, and let xi be the variable chosen at step (2) of Contents this last iteration. Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.14" }, { "page_index": 350, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_098.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_098.png", "page_index": 350, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:11+07:00" }, "raw_text": "Rules for Universal Quantification Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen If we manage to establish a formula about a fresh variable xo, BK TP.HCM we can assume Vao Contents Predicate Logic xo Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language x = xo Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.40" }, { "page_index": 351, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_099.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_099.png", "page_index": 351, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:14+07:00" }, "raw_text": "Correctness of the Algorithm (cont.) Propositional Logic Review II Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs UNSAT, then o is unsatisfiable Tran Van Hoai Chú'ng minh. BK TP.HCM Let ' be the formula at the beginning of the iteration in which A halts, and let xi be the variable chosen at step (2) of Contents this last iteration. Introduction . Note that ' is a 2-CNF formula and ' C (i.e., all the Quick review Boolean Satisfiability (SAT) clauses of $' appear as clauses in ) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.14" }, { "page_index": 352, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_099.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_099.png", "page_index": 352, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:17+07:00" }, "raw_text": "Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK xo TP.HCM [x = xo]8 Contents Predicate Logic Motivation,Syntax Vx(P(x) -> Q(x),VxP(x) FVxQ(x) via Proof Theory Need for Richer Language Vxo Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.41" }, { "page_index": 353, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_100.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_100.png", "page_index": 353, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:20+07:00" }, "raw_text": "Correctness of the Algorithm (cont.) Propositional Logic Review II Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs UNSAT, then is unsatisfiable Tran Van Hoai Chú'ng minh. BK TP.HCM Let δ' be the formula at the beginning of the iteration in which A halts, and let xi be the variable chosen at step (2) of Contents this last iteration. Introduction . Note that ' is a 2-CNF formula and ' C (i.e., all the Quick review Boolean Satisfiability (SAT) clauses of °' appear as clauses in ). P and NP Hence, it suffices to show that ' is unsatisfiable. 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.14" }, { "page_index": 354, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_100.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_100.png", "page_index": 354, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:24+07:00" }, "raw_text": "Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK xo TP.HCM [x = xo]d Contents Predicate Logic Motivation, Syntax, Vx(P(x) -Q(x)),VxP(x)FVxQ(x) via Proof Theory Need for Richer Language Vxo Predicate Logic as Formal Language Proof Theory of Predicate 1 Vx(P(x) ->Q(x)) premise Logic Quantifier Equivalences 2 VxP(x) premise Semantics of Predicate Logic 3 xo P(xo) ->Q(xo) Vx e 1 Soundness and 4 P(xo) Vx e 2 Completeness of Predicate Logic 5 Q(xo) > e 3,4 Undecidability of Predicate Logic 6 VxQ(x) Vx i 3-5 Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.41" }, { "page_index": 355, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_101.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_101.png", "page_index": 355, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:28+07:00" }, "raw_text": "Correctness of the Algorithm (cont.) Propositional Logic Review II Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs UNSAT, then is unsatisfiable Tran Van Hoai Chú'ng minh. BK TP.HCM Let δ' be the formula at the beginning of the iteration in which A halts, and let xi be the variable chosen at step (2) of Contents this last iteration. Introduction . Note that ' is a 2-CNF formula and δ' C (i.e., all the Quick review Boolean Satisfiability (SAT) clauses of @' appear as clauses in ). P and NP Hence, it suffices to show that ' is unsatisfiable. 2-SAT is in P An example Let $o=Simplify(',xi = 0) and $1=Simplify(',x; =1). It UNSAT Graphical View of 2-SAT suffices to show that both δo and 1 are unsatisfiable. SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.14" }, { "page_index": 356, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_101.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_101.png", "page_index": 356, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:31+07:00" }, "raw_text": "Rules for Existential Quantification Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen x=t]G BK TP.HCM 3x i 3xo Contents Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.42" }, { "page_index": 357, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_102.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_102.png", "page_index": 357, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:34+07:00" }, "raw_text": "Correctness of the Algorithm (cont.) Propositional Logic Review II Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs UNSAT, then is unsatisfiable Tran Van Hoai Chü'ng minh. BK TP.HCM Let °' be the formula at the beginning of the iteration in which A halts, and let xi be the variable chosen at step (2) of Contents this last iteration. Introduction . Note that ' is a 2-CNF formula and ' C (i.e., all the Quick review Boolean Satisfiability (SAT) clauses of ' appear as clauses in ) P and NP Hence, it suffices to show that o' is unsatisfiable. 2-SAT is in P An example Let $o=Simplify(',xi = 0) and o1=Simplify(',xi = 1). It UNSAT Graphical View of 2-SAT suffices to show that both $o and 1 are unsatisfiable. SAT Solvers : Recall that the formula UCP($o) and the formula UCP(@1) WalkSAT: Idea DPLL. Idea contain a contradiction. A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.14" }, { "page_index": 358, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_102.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_102.png", "page_index": 358, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:38+07:00" }, "raw_text": "Rules for Existential Quantification Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen x=t]G BK TP.HCM Hx i 3x Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal xo x= xo]g Language Proof Theory of Predicate Logic Quantifier Equivalences x Semantics of Predicate Logic Soundness and Completeness of 3e Predicate Logic x Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.42" }, { "page_index": 359, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_103.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_103.png", "page_index": 359, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:41+07:00" }, "raw_text": "Correctness of the Algorithm (cont.) Propositional Logic Review Il Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs UNSAT, then is unsatisfiable Tran Van Hoai Chü'ng minh. BK TP.HCM Let °' be the formula at the beginning of the iteration in which A halts, and let xi be the variable chosen at step (2) of Contents this last iteration. Introduction . Note that ' is a 2-CNF formula and ' C (i.e., all the Quick review Boolean Satisfiability (SAT) clauses of ' appear as clauses in ). P and NP Hence, it suffices to show that ' is unsatisfiable. 2-SAT is in P An example Let o=Simplify(',xi =0) and $1=Simplify(',xi =1). It UNSAT Graphical View of 2-SAT suffices to show that both o and 1 are unsatisfiable. SAT Solvers : Recall that the formula UCP($o) and the formula UCP($1) WalkSAT: Idea DPLL. Idea contain a contradiction. A Linear Solver A Cubic Solver The proof now follows by noting that if UCP() contains a Homeworks and Next Week Plan? contradiction, then V is UNSAT 1b.14" }, { "page_index": 360, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_103.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_103.png", "page_index": 360, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:44+07:00" }, "raw_text": "Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Vx(P(x) ->Q(x)),3xP(x) F 3xQ(x) Contents 1 Vx(P(x) ->Q(x)) Predicate Logic premise Motivation, Syntax, 2 3xP(x) Proof Theory premise Need for Richer Language 3 x0 P(xo) assumption Predicate Logic as Formal Language 4 P(xo) ->Q(xo) Vx e 1 Proof Theory of Predicate Logic 5 Q(xo) > e 4,3 Quantifier Equivalences 6 3xQ(x) 3x i 5 Semantics of Predicate Logic 7 3xQ(x) 3x e 2,3-6 Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.43" }, { "page_index": 361, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_104.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_104.png", "page_index": 361, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:48+07:00" }, "raw_text": "Correctness of the Algorithm (cont.) Propositional Logic Review Il Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs UNSAT, then is unsatisfiable Tran Van Hoai Chü'ng minh. BK TP.HCM Let °' be the formula at the beginning of the iteration in which A halts, and let xi be the variable chosen at step (2) of Contents this last iteration. Introduction . Note that ' is a 2-CNF formula and ' C (i.e., all the Quick review Boolean Satisfiability (SAT) clauses of ' appear as clauses in ). P and NP Hence, it suffices to show that ' is unsatisfiable. 2-SAT is in P An example Let o=Simplify(',xi =0) and $1=Simplify(',xi =1). It UNSAT Graphical View of 2-SAT suffices to show that both o and 1 are unsatisfiable. SAT Solvers : Recall that the formula UCP($o) and the formula UCP($1) WalkSAT: Idea DPLL. Idea contain a contradiction. A Linear Solver A Cubic Solver The proof now follows by noting that if UCP() contains a Homeworks and Next Week Plan? contradiction, then V is UNSAT 1b.14" }, { "page_index": 362, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_104.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_104.png", "page_index": 362, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:51+07:00" }, "raw_text": "Examples of Quantifier Equivalences Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Vx6 HF 3x- 3x 4F Vx- Contents Predicate Logic 3x3y 4F 3y3xo Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Assume is not free in : Proof Theory of Predicate Logic Quantifier Equivaiences Semantics of Predicate Logic Vxo A W 4F Vx(oA) Soundness and Completeness of 3x(w-o HF w->3xo Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.44" }, { "page_index": 363, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_105.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_105.png", "page_index": 363, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:54+07:00" }, "raw_text": "Correctness of the Algorithm (cont.) Propositional Logic Review Il Lemma Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen If the algorithm outputs UNSAT, then is unsatisfiable Tran Van Hoai Chü'ng minh. BK TP.HCM Let δ' be the formula at the beginning of the iteration in which A halts, and let xi be the variable chosen at step (2) of Contents this last iteration. Introduction . Note that ' is a 2-CNF formula and ' C (i.e., all the Quick review Boolean Satisfiability (SAT) clauses of ' appear as clauses in ). P and NP Hence, it suffices to show that o' is unsatisfiable. 2-SAT is in P An example Let o=Simplify(',xi =0) and $1=Simplify(',xi =1). It UNSAT Graphical View of 2-SAT suffices to show that both $o and 1 are unsatisfiable. SAT Solvers . Recall that the formula UCP($o) and the formula UCP($1) WalkSAT: Idea DPLL. Idea contain a contradiction. A Linear Solver A Cubic Solver The proof now follows by noting that if UCP() contains a Homeworks and Next Week Plan? contradiction, then y is UNSAT. 1b.14" }, { "page_index": 364, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_105.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_105.png", "page_index": 364, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:42:58+07:00" }, "raw_text": "Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen 1 Predicate Logic Motivation, Syntax,Proof Theory BK TP.HCM Semantics of Predicate Logic Contents 3 Soundness and Completeness of Predicate Logic Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Undecidability of Predicate Logic Predicate Logic as Formal Language Proof Theory of Predicate Logic 5 Compactness of Predicate Calculus Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.45" }, { "page_index": 365, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_106.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_106.png", "page_index": 365, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:00+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review II Nguyen An Khuong. . For a 2-CNF formula $, define the implication graph G = G Le Hong Trang Huynh Tuong Nguyen. as follows: Tran Van Hoai BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.15" }, { "page_index": 366, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_106.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_106.png", "page_index": 366, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:04+07:00" }, "raw_text": "Models Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Definition Let F contain function symbols and P contain predicate symbols Contents A model M for (F.P) consists of: Predicate Logic 1 A non-empty set A, the universe; Motivation, Syntax, Proof Theory @ for each nullary function symbol f E F a concrete element Need for Richer Language Predicate Logic as Formal fM E A; Language Proof Theory of Predicate Logic 3 for each f e F with arity n > 0, a concrete function Quantifier Equivalences fM:A\"-A; Semantics of Predicate Logic 4 for each P e P with arity n > 0,a set PM C An. Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.46" }, { "page_index": 367, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_107.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_107.png", "page_index": 367, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:07+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review II Nguyen An Khuong. : For a 2-CNF formula , define the implication graph G = G Le Hong Trang Huynh Tuong Nguyen. as follows: Tran Van Hoai nodes 1,1,X2,2,...;n,n BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.15" }, { "page_index": 368, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_107.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_107.png", "page_index": 368, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:09+07:00" }, "raw_text": "Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Let F={e,} and P ={<} Let model M for (F,P) be defined as follows: Contents Predicate Logic @ Let A be the set of binary strings over the alphabet {0,1} Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate the strings s1 and s2: and Logic Quantifier Equivalences Q let M be defined such that s1 M s2 iff s1 is a prefix of s2. Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.47" }, { "page_index": 369, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_108.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_108.png", "page_index": 369, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:12+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review II Nguyen An Khuong, : For a 2-CNF formula , define the implication graph G = G Le Hong Trang, Huynh Tuong Nguyen as follows: Tran Van Hoai . nodes 1,1,X2,2,- ,Xn,Xn for a clause li V l; define the edges: BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.15" }, { "page_index": 370, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_108.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_108.png", "page_index": 370, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:15+07:00" }, "raw_text": "Example (continued) Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Let A be the set of binary strings over the alphabet {0,1}: BK TP.HCM the strings s1 and s2; and Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Some Elements of A Predicate Logic as Formal Language Proof Theory of Predicate 10001 Logic Quantifier Equivalences F Semantics of Predicate Logic 1010.M 1100 Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.48" }, { "page_index": 371, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_109.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_109.png", "page_index": 371, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:19+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review II Nguyen An Khuong, : For a 2-CNF formula , define the implication graph G = G Le Hong Trang, Huynh Tuong Nguyen. as follows: Tran Van Hoai nodes X1,1,2,2, ,Xn,Xn for a clause li V l; define the edges: li->lj BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.15" }, { "page_index": 372, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_109.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_109.png", "page_index": 372, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:22+07:00" }, "raw_text": "Example (continued) Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Let A be the set of binary strings over the alphabet {0,1}: BK TP.HCM the strings s1 and s2; and Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Some Elements of A Predicate Logic as Formal Language Proof Theory of Predicate 10001 Logic Quantifier Equivalences Semantics of Predicate Logic 1010 .M 1100 = 10101100 Soundness and Completeness of Predicate Logic € Undecidability of 000.M Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.48" }, { "page_index": 373, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_110.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_110.png", "page_index": 373, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:25+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review II Nguyen An Khuong, : For a 2-CNF formula , define the implication graph G = G Le Hong Trang, Huynh Tuong Nguyen. as follows: Tran Van Hoai nodes X1,1,2,2,.. ,Xn,Xn for a clause li V l; define the edges: li->lj BK TP.HCM lj -li Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.15" }, { "page_index": 374, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_110.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_110.png", "page_index": 374, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:29+07:00" }, "raw_text": "Example (continued) Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Let A be the set of binary strings over the alphabet {0,1}: BK TP.HCM the strings s1 and s2; and Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Some Elements of A Predicate Logic as Formal Language Proof Theory of Predicate 10001 Logic Quantifier Equivalences Semantics of Predicate Logic 1010 .M 1100 = 10101100 Soundness and Completeness of Predicate Logic € 000.M c=000 Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.48" }, { "page_index": 375, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_111.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_111.png", "page_index": 375, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:32+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review II Nguyen An Khuong. : For a 2-CNF formula δ, define the implication graph G = G Le Hong Trang, Huynh Tuong Nguyen as follows: Tran Van Hoai nodes 1,1,2,2,- ,Xn,Xn for a clause li V l; define the edges: BK lr-> lj TP.HCM lj -li Main property: Let o be a satisfying assignment Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.15" }, { "page_index": 376, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_111.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_111.png", "page_index": 376, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:36+07:00" }, "raw_text": "Equality Revisited Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Interpretation of equality Contents Usually, we require that the equality predicate = is interpreted as Predicate Logic same-ness. Motivation, Syntax, Proof Theory Need for Richer Language Extensionality restriction Predicate Logic as Formal Language Proof Theory of Predicate This means that allowable models are restricted to those in which Logic a =M b holds if and only if a and b are the same elements of the Quantifier Equivalences Semantics of Predicate model's universe. Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.49" }, { "page_index": 377, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_112.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_112.png", "page_index": 377, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:39+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review Il Nguyen An Khuong. . For a 2-CNF formula δ, define the implication graph G = G Le Hong Trang, Huynh Tuong Nguyen. as follows: Tran Van Hoai . nodes x1,X1,2,2,. Xn;Xn for a clause li V l; define the edges: lr-> lj BK TP.HCM lj -r li Main property: Let o be a satisfying assignment. Contents If o satisfies a node , then o satisfies all nodes u achievable Introduction from v. Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.15" }, { "page_index": 378, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_112.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_112.png", "page_index": 378, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:42+07:00" }, "raw_text": "Example (continued) Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK Let A be the set of binary strings over the alphabet {0,1} TP.HCM let eM = €, the empty string; @ let .M be defined such that s1 .M s2 is the concatenation of Contents Predicate Logic the strings s1 and s2; and Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.50" }, { "page_index": 379, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_113.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_113.png", "page_index": 379, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:45+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review Il Nguyen An Khuong. : For a 2-CNF formula , define the implication graph G = G Le Hong Trang, Huynh Tuong Nguyen. as follows: Tran Van Hoai nodes x1,X1,X2,X2, Xn,Xn for a clause li V l; define the edges: lr-rlj BK TP.HCM lj -r li Main property: Let o be a satisfying assignment. Contents If o satisfies a node , then o satisfies all nodes u achievable Introduction from v. Quick review Boolean Satisfiability (SAT) The property can be proven by induction on the length of the P and NP 2-SAT is in P path. An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.15" }, { "page_index": 380, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_113.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_113.png", "page_index": 380, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:49+07:00" }, "raw_text": "Example (continued) Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK Let A be the set of binary strings over the alphabet {0,1} TP.HCM let eM = e, the empty string; @ let .M be defined such that s1 .M s2 is the concatenation of Contents Predicate Logic the strings s1 and s2: and Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Equality in M Logic Quantifier Equivalences 000=M 000 Semantics of Predicate Logic : 001 M 100 Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.50" }, { "page_index": 381, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_114.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_114.png", "page_index": 381, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:52+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review Il Nguyen An Khuong. : For a 2-CNF formula , define the implication graph G = G Le Hong Trang, Huynh Tuong Nguyen. as follows: Tran Van Hoai nodes x1,X1,X2,X2, Xn,Xn for a clause li V l; define the edges: lr-rlj BK TP.HCM lj -r li Main property: Let o be a satisfying assignment. Contents If o satisfies a node , then o satisfies all nodes u achievable Introduction from v. Quick review Boolean Satisfiability (SAT) The property can be proven by induction on the length of the P and NP 2-SAT is in P path. An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.15" }, { "page_index": 382, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_114.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_114.png", "page_index": 382, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:54+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM LetF={z,s} and P={<} Contents Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.51" }, { "page_index": 383, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_115.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_115.png", "page_index": 383, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:43:58+07:00" }, "raw_text": "Graphical View of 2-SAT Propositional Logic Review Il Nguyen An Khuong. : For a 2-CNF formula , define the implication graph G = G Le Hong Trang, Huynh Tuong Nguyen as follows: Tran Van Hoai nodes 1,1,2,T2, ,Xn,Xr for a clause li V l; define the edges: lr-rlj BK TP.HCM lj -r li Main property: Let o be a satisfying assignment. Contents If g satisfies a node , then o satisfies all nodes u achievable Introduction from v. Quick review Boolean Satisfiability (SAT) The property can be proven by induction on the length of the P and NP 2-SAT is in P path. An example UNSAT Theorem Graphical View of 2SAT SAT Solvers is satisfiable iff the graph G does not contain a \"contradiction WalkSAT: Idea DPLL. Idea path\" of the form: A Linear Solver A Cubic Solver li-...-li-...-li Homeworks and Next Week Plan? 1b.15" }, { "page_index": 384, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_115.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_115.png", "page_index": 384, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:44:00+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Let F={z,s} and P={<} Let model M for (F,P) be defined as follows: Contents Predicate Logic 1 Let A be the set of natural numbers; Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.51" }, { "page_index": 385, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_116.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_116.png", "page_index": 385, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:44:02+07:00" }, "raw_text": "Proof for the previous theorem Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, 1 (E contradiction path = is UNSAT): Tran Van Hoai BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.16" }, { "page_index": 386, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_116.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_116.png", "page_index": 386, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:44:05+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Let F={z,s} and P ={<} Let model M for (F,P) be defined as follows: Contents Predicate Logic 1 Let A be the set of natural numbers; Motivation, Syntax. Proof Theory let zM = 0; Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.51" }, { "page_index": 387, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_117.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_117.png", "page_index": 387, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:44:09+07:00" }, "raw_text": "Proof for the previous theorem Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen. 1 (= contradiction path = is UNSAT): Tran Van Hoai - Take a potential assignment o. BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.16" }, { "page_index": 388, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_117.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_117.png", "page_index": 388, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:44:12+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Let F={z,s} and P ={<} Let model M for (F,P) be defined as follows: Contents Predicate Logic Let A be the set of natural numbers; Motivation, Syntax, Proof Theory let zM = 0; Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.51" }, { "page_index": 389, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_118.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_118.png", "page_index": 389, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:44:15+07:00" }, "raw_text": "Proof for the previous theorem Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, 1 (E contradiction path = is UNSAT): Huynh Tuong Nguyen. Tran Van Hoai - Take a potential assignment o. - If satisfies i, then by Property it must satisfy li Contradiction. BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.16" }, { "page_index": 390, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_118.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_118.png", "page_index": 390, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:44:19+07:00" }, "raw_text": "Another Example Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Let F={z,s} and P={<} Let model M for (F,P) be defined as follows: Contents Predicate Logic Let A be the set of natural numbers; Motivation, Syntax, Proof Theory let zM = 0; Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate 4 let 3 contradiction path): Introduction If 6 is UNSAT = Algorithm Halts Quick review > for some c, we have: Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.16" }, { "page_index": 394, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_120.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_120.png", "page_index": 394, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:44:34+07:00" }, "raw_text": "How To Handle Free Variables? Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Idea We can give meaning to formulas with free variables by providing Contents an environment (lookup table) that assigns variables to elements Predicate Logic of our universe: Motivation, Syntax, Proof Theory I : var - A. Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Environment extension Quantifier Equivalences We define environment extension such that l[ ? a] is the Semantics of Predicate Logic environment that maps x to a and any other variable y to l(y) Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.52" }, { "page_index": 395, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_121.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_121.png", "page_index": 395, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:44:38+07:00" }, "raw_text": "Proof for the previous theorem Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. 1 (E contradiction path = is UNSAT) Tran Van Hoai Take a potential assignment o. If satisfies li, then by Property it must satisfy l: Contradiction. BK TP.HCM If satisfies li, then by Property it must satisfy l: Contradiction. Contents 2 ( is UNSAT = 3 contradiction path): Introduction If 6 is UNSAT = Algorithm Halts Quick review -> for some ci we have: Boolean Satisfiability (SAT) P and NP (a) lj... 3 contradiction path): 2 Introduction If o is UNSAT = Algorithm Halts Quick review > for some c; we have: Boolean Satisfiability (SAT) P and NP (@) lj<... 3 contradiction path): 2 Introduction If o is UNSAT = Algorithm Halts Quick review > for some c; we have: Boolean Satisfiability (SAT) P and NP (@) lj<... for some c; we have: Boolean Satisfiability (SAT) P and NP (@) lj<... [, is an edge, then l; - l; is also an Graphical View of 2-SAT edge. SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.16" }, { "page_index": 402, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_124.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_124.png", "page_index": 402, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:01+07:00" }, "raw_text": "Satisfaction Relation (continued) Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM in case o has the form -, if M does not hold Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.54" }, { "page_index": 403, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_125.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_125.png", "page_index": 403, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:04+07:00" }, "raw_text": "Proof for the previous theorem Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, (E contradiction path = is UNSAT) Huynh Tuong Nguyen Tran Van Hoai Take a potential assignment o. If satisfies li, then by Property it must satisfy l: Contradiction. BK TP.HCM If o satisfies l, then by Property it must satisfy li Contradiction. Contents ( is UNSAT => 3 contradiction path): 2 Introduction If o is UNSAT = Algorithm Halts Quick review > for some c; we have: Boolean Satisfiability (SAT) P and NP (@) lj... lj is an edge, then l, -> , is also an Graphical View of 2-SAT SAT Solvers edge. WalkSAT: Idea By reversing edges and negating DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.16" }, { "page_index": 404, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_125.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_125.png", "page_index": 404, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:08+07:00" }, "raw_text": "Satisfaction Relation (continued) Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM in case o has the form -, if M = does not hold Contents in case o has the form W1 V V2, if M Fi W1 holds or Predicate Logic M Fi 2 holds; Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.54" }, { "page_index": 405, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_126.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_126.png", "page_index": 405, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:12+07:00" }, "raw_text": "Proof for the previous theorem Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, (E contradiction path = is UNSAT) Huynh Tuong Nguyen Tran Van Hoai Take a potential assignment o. If satisfies li, then by Property it must satisfy l: Contradiction. BK TP.HCM If o satisfies l, then by Property it must satisfy t: Contradiction. Contents @ ( is UNSAT = 3 contradiction path): Introduction If o is UNSAT = Algorithm Halts Quick review > for some c, we have: Boolean Satisfiability (SAT) P and NP (@) lj... lj is an edge, then l, -> , is also an Graphical View of 2SAT SAT Solvers edge. WalkSAT: Idea By reversing edges and negating: DPLL. Idea A Linear Solver (a) =xi-...-lj-...-xi A Cubic Solver Homeworks and Next Week Plan? 1b.16" }, { "page_index": 406, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_126.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_126.png", "page_index": 406, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:15+07:00" }, "raw_text": "Satisfaction Relation (continued) Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM in case has the form -y, if M = does not hold Contents in case δ has the form 1 V V2, if M Fl 1 holds or Predicate Logic M Fi V2 holds; Motivation, Syntax, Proof Theory in case has the form W1 W2, if M 1 holds and Need for Richer Language Predicate Logic as Formal M Fi W2 holds; and Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.54" }, { "page_index": 407, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_127.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_127.png", "page_index": 407, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:19+07:00" }, "raw_text": "Proof for the previous theorem Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, (E contradiction path = is UNSAT) Huynh Tuong Nguyen Tran Van Hoai Take a potential assignment o. If satisfies li, then by Property it must satisfy l Contradiction. BK TP.HCM If o satisfies l, then by Property it must satisfy t: Contradiction. Contents @ ( is UNSAT = 3 contradiction path): Introduction If δ is UNSAT = Algorithm Halts Quick review > for some ε; we have: Boolean Satisfiability (SAT) P and NP (@) lj... lj is an edge, then l, -> , is also an Graphical View of 2SAT SAT Solvers edge. WalkSAT: Idea By reversing edges and negating: DPLL. Idea A Linear Solver a) =xi-...-lj-...-i A Cubic Solver Homeworks and Next Week Plan? 1b.16" }, { "page_index": 408, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_127.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_127.png", "page_index": 408, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:22+07:00" }, "raw_text": "Satisfaction Relation (continued) Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM in case o has the form -, if M W does not hold; Contents in case o has the form 1 V W2, if M Fz 1 holds or Predicate Logic M Fi W2 holds; Motivation, Syntax, Proof Theory in case & has the form V1 V2,if M i W1 holds and Need for Richer Language Predicate Logic as Formal M Fl 2 holds; and Language Proof Theory of Predicate Logic in case o has the form 1 -> 2, if Mz 1 holds whenever Quantifier Equivalences M Fi 2 holds. Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.54" }, { "page_index": 409, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_128.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_128.png", "page_index": 409, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:27+07:00" }, "raw_text": "Proof for the previous theorem Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, (E contradiction path = is UNSAT) Huynh Tuong Nguyen Tran Van Hoai Take a potential assignment o. If satisfies li, then by Property it must satisfy l Contradiction. BK TP.HCM If o satisfies li, then by Property it must satisfy : Contradiction. Contents @ ( is UNSAT = 3 contradiction path): Introduction If δ is UNSAT = Algorithm Halts Quick review > for some ε; we have: Boolean Satisfiability (SAT) P and NP (@) lj...xi-...-l 2-SAT is in P (b) lk<...i-...-lk An example UNSAT In our graph, if li -> lj is an edge, then l, -> is also an Graphicai View of 2-SAT SAT Solvers edge. WalkSAT: Idea By reversing edges and negating: DPLL. Idea A Linear Solver (a) =xi-... -l-...-Xi A Cubic Solver Homeworks and Next Week Plan? Therefore, there exists a contradiction path 1b.16" }, { "page_index": 410, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_128.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_128.png", "page_index": 410, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:30+07:00" }, "raw_text": "Satisfaction of Closed Formulas Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Contents If a formula o has no free variables, we call o a sentence Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.55" }, { "page_index": 411, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_129.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_129.png", "page_index": 411, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:33+07:00" }, "raw_text": "Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen Tran Van Hoai 1 lntroduction BK 2-SAT is in P TP.HCM Contents SAT Solvers Introduction WalkSAT: Idea Quick review DPLL: Idea Boolean Satisfiability (SAT) P and NP A Linear Solver 2-SAT is in P A Cubic Solver An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.17" }, { "page_index": 412, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_129.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_129.png", "page_index": 412, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:36+07:00" }, "raw_text": "Satisfaction of Closed Formulas Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Contents If a formula o has no free variables, we call o a sentence Predicate Logic Motivation, Syntax, M $ holds or does not hold regardless of the choice of l. Thus Proof Theory Need for Richer Language we write M F o or M Z o. Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.55" }, { "page_index": 413, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_130.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_130.png", "page_index": 413, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:39+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen - Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT. Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.18" }, { "page_index": 414, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_130.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_130.png", "page_index": 414, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:41+07:00" }, "raw_text": "Semantic Entailment and Satisfiability Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Let I be a possibly infinite set of formulas in predicate logic and a formula BK TP.HCM Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.56" }, { "page_index": 415, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_131.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_131.png", "page_index": 415, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:44+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. . Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found. Details: In each step, choose an unsatisfied clause (clause BK selection), and \"flip\" one of its variables (variable selection) TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT. Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.18" }, { "page_index": 416, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_131.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_131.png", "page_index": 416, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:47+07:00" }, "raw_text": "Semantic Entailment and Satisfiability Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Let T be a possibly infinite set of formulas in predicate logic and a formula BK TP.HCM Entailment r iff for all models M and environments l, whenever Contents MFz o holds for alI o E I, then M=z W Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.56" }, { "page_index": 417, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_132.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_132.png", "page_index": 417, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:50+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. . Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found Details: In each step, choose an unsatisfied clause (clause BK selection), and \"flip\" one of its variables (variable selection) TP.HCM WalkSAT: Details Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT. Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.18" }, { "page_index": 418, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_132.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_132.png", "page_index": 418, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:54+07:00" }, "raw_text": "Semantic Entailment and Satisfiability Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Let I be a possibly infinite set of formulas in predicate logic and a formula. BK TP.HCM Entailment I = iff for all models M and environments l, whenever Contents M Fz o holds for all o E T,then M i W Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Satisfiability of Formulas Predicate Logic as Formal Language is satisfiable iff there is some model M and some environment l Proof Theory of Predicate Logic such that M j W holds Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.56" }, { "page_index": 419, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_133.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_133.png", "page_index": 419, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:45:57+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen . Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found Details: In each step, choose an unsatisfied clause (clause BK selection), and \"flip\" one of its variables (variable selection) TP.HCM WalkSAT:Details Contents Introduction Termination criterion: No unsatisfied clauses are left Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT.ldea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.18" }, { "page_index": 420, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_133.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_133.png", "page_index": 420, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:01+07:00" }, "raw_text": "Semantic Entailment and Satisfiability Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Let I be a possibly infinite set of formulas in predicate logic and / a formula BK TP.HCM Entailment I = iff for all models M and environments l, whenever Contents M Fi o holds for all b E T,then M Fi W Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Satisfiability of Formulas Predicate Logic as Formal Language is satisfiable iff there is some model M and some environment [ Proof Theory of Predicate Logic such that M j W holds Quantifier Equivalences Semantics of Predicate Logic Satisfiability of Formula Sets Soundness and Completeness of Predicate Logic I is satisfiable iff there is some model M and some environment l Undecidability of such that M F δ,for all δ E T. Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.56" }, { "page_index": 421, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_134.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_134.png", "page_index": 421, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:04+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. - Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found. Details: In each step, choose an unsatisfied clause (clause BK selection), and \"flip\" one of its variables (variable selection) TP.HCM WalkSAT: Details Contents Introduction Termination criterion: No unsatisfied clauses are left Quick review Boolean Satisfiability (SAT) Clause selection: Choose a random unsatisfied clause P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT. Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.18" }, { "page_index": 422, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_134.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_134.png", "page_index": 422, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:08+07:00" }, "raw_text": "Semantic Entailment and Satisfiability Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Let I be a possibly infinite set of formulas in predicate logic and Contents a formula Predicate Logic Motivation, Syntax, Validity Proof Theory Need for Richer Language is valid iff for all models M and environments l, we have Predicate Logic as Formal Language M Fi v. Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.57" }, { "page_index": 423, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_135.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_135.png", "page_index": 423, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:11+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. . Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found. Details: In each step, choose an unsatisfied clause (clause BK selection), and \"flip\" one of its variables (variable selection) TP.HCM WalkSAT:Details Contents Introduction : Termination criterion: No unsatisfied clauses are left Quick review Boolean Satisfiability (SAT Clause selection: Choose a random unsatisfied clause P and NP 2-SAT is in P . Variable selection: An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT. Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.18" }, { "page_index": 424, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_135.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_135.png", "page_index": 424, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:14+07:00" }, "raw_text": "The Problem with Predicate Logic Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK Entailment ranges over models TP.HCM Semantic entailment between sentences: 1, 2, : . : , n requires that in all models that satisfy $1, $2, : . : , $n, the sentence Contents W is satisfied. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.58" }, { "page_index": 425, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_136.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_136.png", "page_index": 425, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:18+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen : Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found Details: In each step, choose an unsatisfied clause (clause BK selection), and \"flip\" one of its variables (variable selection) TP.HCM WalkSAT:Details Contents Introduction Termination criterion: No unsatisfied clauses are left Quick review Boolean Satisfiability (SAT) Clause selection: Choose a random unsatisfied clause P and NP 2-SAT is in P . Variable selection: An example UNSAT : If there are variables that when flipped make no currently Graphical View of 2-SAT satisfied clause unsatisfied, flip one which makes the most SAT Solvers unsatisfied clauses satisfied WaikSAT.Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.18" }, { "page_index": 426, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_136.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_136.png", "page_index": 426, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:21+07:00" }, "raw_text": "The Problem with Predicate Logic Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Entailment ranges over models BK TP.HCM Semantic entailment between sentences: 1, 2, : . : , n requires that in all models that satisfy $1, 2, : . :, $n, the sentence Contents W is satisfied. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language How to effectively argue about all possible models? Predicate Logic as Formal Language Usually the number of models is infinite; it is very hard to argue Proof Theory of Predicate Logic on the semantic level in predicate logic. Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.58" }, { "page_index": 427, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_137.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_137.png", "page_index": 427, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:25+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen : Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found Details: In each step, choose an unsatisfied clause (clause BK selection), and \"flip\" one of its variables (variable selection) TP.HCM WalkSAT:Details Contents Introduction Termination criterion: No unsatisfied clauses are left Quick review Boolean Satisfiability (SAT) Clause selection: Choose a random unsatisfied clause P and NP 2-SAT is in P . Variable selection: An example UNSAT : If there are variables that when flipped make no currently Graphical View of 2-SAT satisfied clause unsatisfied, flip one which makes the most SAT Solvers unsatisfied clauses satisfied. WalkSAT. Idea Otherwise, make a choice with a certain probability between: DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.18" }, { "page_index": 428, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_137.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_137.png", "page_index": 428, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:29+07:00" }, "raw_text": "The Problem with Predicate Logic Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK Entailment ranges over models TP.HCM Semantic entailment between sentences: 1, 2, : : : , n requires that in al/ models that satisfy 1, 2,: . : , n, the sentence Contents is satisfied. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language How to effectively argue about all possible models? Predicate Logic as Formal Language Usually the number of models is infinite; it is very hard to argue Proof Theory of Predicate Logic on the semantic level in predicate logic. Quantifier Equivalences Semantics of Predicate Logic Idea from propositional logic Soundness and Completeness of Can we use natural deduction for showing entailment? Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.58" }, { "page_index": 429, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_138.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_138.png", "page_index": 429, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:33+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen . Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found Details: In each step, choose an unsatisfied clause (clause BK selection), and \"flip\" one of its variables (variable selection) TP.HCM WalkSAT: Details Contents Introduction Termination criterion: No unsatisfied clauses are left Quick review Boolean Satisfiability (SAT) Clause selection: Choose a random unsatisfied clause P and NP 2-SAT is in P . Variable selection: An example UNSAT : If there are variables that when flipped make no currently Graphical View of 2-SAT satisfied clause unsatisfied, flip one which makes the most SAT Solvers unsatisfied clauses satisfied. WalkSAT. Idea Otherwise, make a choice with a certain probability between: DPLL. Idea A Linear Solver . picking a random variable, and A Cubic Solver Homeworks and Next Week Plan? 1b.18" }, { "page_index": 430, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_138.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_138.png", "page_index": 430, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:36+07:00" }, "raw_text": "Central Result of Natural Deduction Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Contents 1, . . ., n = Predicate Logic Motivation, Syntax, iff Proof Theory Need for Richer Language Predicate Logic as Formal 1,...,n F Language Proof Theory of Predicate Logic Quantifier Equivalences proven by Kurt Gödel, in 1929 in his doctoral dissertation Semantics of Predicate Logic Soundness and Compieteness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.59" }, { "page_index": 431, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_139.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_139.png", "page_index": 431, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:41+07:00" }, "raw_text": "WalkSAT: An Incomplete Solver Propositional Logic Review Il Nguyen An Khuong, Le Hong Trang, Huynh Tuong Nguyen : Idea: Start with a random truth assignment, and then Tran Van Hoai iteratively improve the assignment until model is found . Details: In each step, choose an unsatisfied clause (clause BK selection), and \"flip\" one of its variables (variable selection) TP.HCM WalkSAT: Details Contents Introduction Termination criterion: No unsatisfied clauses are left Quick review Boolean Satisfiability (SAT) Clause selection: Choose a random unsatisfied clause P and NP 2-SAT is in P . Variable selection: An example UNSAT . If there are variables that when flipped make no currently Graphical View of 2-SAT satisfied clause unsatisfied, flip one which makes the most SAT Solvers unsatisfied clauses satisfied WalkSAT. Idea Otherwise, make a choice with a certain probability between: DPLL. Idea A Linear Solver : picking a random variable, and A Cubic Solver picking a variable that when flipped minimizes the number of Homeworks and Next unsatisfied clauses. Week Plan? 1b.18" }, { "page_index": 432, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_139.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_139.png", "page_index": 432, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:43+07:00" }, "raw_text": "Recall: Decidability Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Decision problems BK A decision problem is a question in some formal system with a TP.HCM yes-or-no answer. Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.60" }, { "page_index": 433, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_140.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_140.png", "page_index": 433, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:45+07:00" }, "raw_text": "DPLL: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai BK TP.HCM : Simplify formula based on Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.19" }, { "page_index": 434, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_140.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_140.png", "page_index": 434, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:49+07:00" }, "raw_text": "Recall: Decidability Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Decision problems BK A decision prob/em is a question in some formal system with a TP.HCM yes-or-no answer. Contents Decidability Predicate Logic Motivation, Syntax, Proof Theory Decision problems for which there is an algorithm that returns Need for Richer Language yes\" whenever the answer to the problem is \"yes\", and that Predicate Logic as Formal Language Proof Theory of Predicate returns \"no\" whenever the answer to the problem is \"no\", are called Logic decidable. Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.60" }, { "page_index": 435, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_141.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_141.png", "page_index": 435, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:52+07:00" }, "raw_text": "DPLL: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen. Tran Van Hoai BK TP.HCM : Simplify formula based on pure literal elimination and Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.19" }, { "page_index": 436, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_141.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_141.png", "page_index": 436, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:55+07:00" }, "raw_text": "Recall: Decidability Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Decision problems BK TP.HCM A decision prob/em is a question in some formal system with a yes-or-no answer. Contents Decidability Predicate Logic Motivation, Syntax, Proof Theory Decision problems for which there is an algorithm that returns Need for Richer Language yes\" whenever the answer to the problem is \"yes\", and that Predicate Logic as Formal Language Proof Theory of Predicate returns \"no\" whenever the answer to the problem is \"no\", are called Logic decidable. Quantifier Equivalences Semantics of Predicate Logic Decidability of satisfiability Soundness and Completeness of Predicate Logic The question, whether a given propositional formula is satisifiable Undecidability of is decidable Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.60" }, { "page_index": 437, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_142.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_142.png", "page_index": 437, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:46:58+07:00" }, "raw_text": "DPLL: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen Tran Van Hoai BK TP.HCM Simplify formula based on pure literal elimination and unit Contents Introduction propagation Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.19" }, { "page_index": 438, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_142.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_142.png", "page_index": 438, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:01+07:00" }, "raw_text": "Undecidability of Predicate Logic Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula o in that language, decides whether o. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.61" }, { "page_index": 439, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_143.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_143.png", "page_index": 439, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:03+07:00" }, "raw_text": "DPLL: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai BK TP.HCM Simplify formula based on pure literal elimination and unit Contents Introduction propagation Quick review If not done, pick an atom p and split: p or -p Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.19" }, { "page_index": 440, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_143.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_143.png", "page_index": 440, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:06+07:00" }, "raw_text": "Undecidability of Predicate Logic Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula o in that language, decides whether . Predicate Logic Motivation, Syntax, Proof Theory Proof Need for Richer Language Predicate Logic as Formal Language Establish that the Post Correspondence Problem (PCP) is Proof Theory of Predicate Logic undecidable (here only as sketch). Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.61" }, { "page_index": 441, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_144.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_144.png", "page_index": 441, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:09+07:00" }, "raw_text": "A Linear Solver: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen. Tran Van Hoai BK TP.HCM Transform formula to tree of conjunctions and negations Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.20" }, { "page_index": 442, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_144.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_144.png", "page_index": 442, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:12+07:00" }, "raw_text": "Undecidability of Predicate Logic Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula in that language, decides whether . Predicate Logic Motivation, Syntax, Proof Theory Proof Need for Richer Language Predicate Logic as Formal Language Establish that the Post Correspondence Problem (PCP) is Proof Theory of Predicate Logic undecidable (here only as sketch). Quantifier Equivalences Translate an arbitrary PCP, say C, to a formula . Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.61" }, { "page_index": 443, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_145.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_145.png", "page_index": 443, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:15+07:00" }, "raw_text": "A Linear Solver: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK TP.HCM Transform formula to tree of conjunctions and negations Transform tree into graph Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.20" }, { "page_index": 444, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_145.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_145.png", "page_index": 444, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:19+07:00" }, "raw_text": "Undecidability of Predicate Logic Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula in that language, decides whether . Predicate Logic Motivation, Syntax, Proof Theory Proof Need for Richer Language Predicate Logic as Formal Language Establish that the Post Correspondence Problem (PCP) is Proof Theory of Predicate Logic undecidable (here only as sketch). Quantifier Equivalences Translate an arbitrary PCP, say C, to a formula . Semantics of Predicate Logic Establish that holds if and only if C has a solution Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.61" }, { "page_index": 445, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_146.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_146.png", "page_index": 445, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:21+07:00" }, "raw_text": "A Linear Solver: ldea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK TP.HCM Transform formula to tree of conjunctions and negations Transform tree into graph Contents Introduction Mark the top of the tree as T Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.20" }, { "page_index": 446, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_146.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_146.png", "page_index": 446, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:24+07:00" }, "raw_text": "Undecidability of Predicate Logic Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula o in that language, decides whether . Predicate Logic Motivation, Syntax, Proof Theory Proof Need for Richer Language Predicate Logic as Formal Language : Establish that the Post Correspondence Problem (PCP) is Proof Theory of Predicate Logic undecidable (here only as sketch). Quantifier Equivalences Translate an arbitrary PCP, say C, to a formula Semantics of Predicate Logic Establish that = holds if and only if C has a solution. Soundness and Completeness of : Conclude that validity of pred. logic formulas is undecidable. Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.61" }, { "page_index": 447, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_147.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_147.png", "page_index": 447, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:28+07:00" }, "raw_text": "A Linear Solver: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK TP.HCM Transform formula to tree of conjunctions and negations Transform tree into graph Contents Introduction Mark the top of the tree as T Quick review . Propagate constraints using obvious rules Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.20" }, { "page_index": 448, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_147.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_147.png", "page_index": 448, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:30+07:00" }, "raw_text": "Post Correspondence Problem Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Informally Can we line up copies of the cards such that the top row spells out Contents the same sequence as the bottom row? Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.62" }, { "page_index": 449, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_148.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_148.png", "page_index": 449, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:33+07:00" }, "raw_text": "A Linear Solver: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK TP.HCM Transform formula to tree of conjunctions and negations Transform tree into graph. Contents Introduction Mark the top of the tree as T. Quick review . Propagate constraints using obvious rules. Boolean Satisfiability (SAT) P and NP If all leaves are marked, check that corresponding assignment 2-SAT is in P An example makes the formula true. UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.20" }, { "page_index": 450, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_148.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_148.png", "page_index": 450, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:37+07:00" }, "raw_text": "Post Correspondence Problem Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Informally Can we line up copies of the cards such that the top row spells out Contents the same sequence as the bottom row? Predicate Logic Motivation, Syntax, Proof Theory Formally Need for Richer Language Predicate Logic as Formal Language Given a finite sequence of pairs (s1,t1,(s2,t2,:..,(sk,tk) such Proof Theory of Predicate Logic that all s, and t, are binary strings of positive length, is there a Quantifier Equivalences sequence of indices i1,i2,. ..,in with n 1 such that the Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.62" }, { "page_index": 451, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_149.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_149.png", "page_index": 451, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:39+07:00" }, "raw_text": "Transformation Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai p BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.21" }, { "page_index": 452, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_149.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_149.png", "page_index": 452, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:43+07:00" }, "raw_text": "Undecidability of Post Correspondence Problem Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Turing machines BK Basic abstract symbol-manipulating devices that can simulate in TP.HCM prinicple any computer algorithm. The input is a string of symbols on a tape, and the machine \"accepts\" the input string, if it reaches Contents one of a number of accepting states. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.63" }, { "page_index": 453, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_150.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_150.png", "page_index": 453, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:47+07:00" }, "raw_text": "Transformation Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai T(p) p BK T($1 $2) T(1)T(2) TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.21" }, { "page_index": 454, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_150.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_150.png", "page_index": 454, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:50+07:00" }, "raw_text": "Undecidability of Post Correspondence Problem Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Turing machines BK Basic abstract symbol-manipulating devices that can simulate in TP.HCM prinicple any computer algorithm. The input is a string of symbols on a tape, and the machine \"accepts\" the input string, if it reaches Contents one of a number of accepting states. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Termination of Programs is Undecidable Predicate Logic as Formal Language It is undecidable, whether program with input terminates Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.63" }, { "page_index": 455, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_151.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_151.png", "page_index": 455, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:52+07:00" }, "raw_text": "Transformation Propositional Logic Review II Nguyen An Khuong, Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai T(p) p BK T($1 $2) = T($1)T($2) TP.HCM T(-) = -$(T) Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.21" }, { "page_index": 456, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_151.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_151.png", "page_index": 456, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:56+07:00" }, "raw_text": "Undecidability of Post Correspondence Problem Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Turing machines BK Basic abstract symbol-manipulating devices that can simulate in TP.HCM prinicple any computer algorithm. The input is a string of symbols on a tape, and the machine \"accepts\" the input string, if it reaches Contents one of a number of accepting states. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Termination of Programs is Undecidable Predicate Logic as Formal Language It is undecidable, whether program with input terminates Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Proof idea Logic For a Turing machine with a given input, construct a PCP such Soundness and Completeness of that a solution of the PCP exists if and only if the Turing machine Predicate Logic Undecidability of accepts the solution. Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.63" }, { "page_index": 457, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_152.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_152.png", "page_index": 457, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:47:59+07:00" }, "raw_text": "Transformation Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai T(p) p BK T(1 $2) = T($1)T($2) TP.HCM T(-$) = -$(T) T($1 -$2) = -(T($1) -T(@2) Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.21" }, { "page_index": 458, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_152.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_152.png", "page_index": 458, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:02+07:00" }, "raw_text": "Translate Post Correspondence Problem to Formula Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Bits as Functions Represent bits 0 and 1 by functions fo and f1 Contents Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.64" }, { "page_index": 459, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_153.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_153.png", "page_index": 459, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:05+07:00" }, "raw_text": "Transformation Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai T(p) p BK T($1 $2) = T(1)T(2) TP.HCM T(-) = -$(T) T($1 -$2) = -(T($1) -T($2)) Contents Introduction T($1 V $2) = -(-T(1) -T(2)) Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.21" }, { "page_index": 460, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_153.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_153.png", "page_index": 460, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:08+07:00" }, "raw_text": "Translate Post Correspondence Problem to Formula Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Bits as Functions Represent bits 0 and 1 by functions fo and f1 Contents Predicate Logic Motivation, Syntax, Strings as Terms Proof Theory Need for Richer Language Represent the empty string by a constant e. Predicate Logic as Formal Language The string b1b2 ...bi corresponds to the term Proof Theory of Predicate Logic Quantifier Equivalences fb(fb-1...(fb2(fb(e)))...) Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.64" }, { "page_index": 461, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_154.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_154.png", "page_index": 461, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:11+07:00" }, "raw_text": "Transformation Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai T(p) = p BK T($1 $2) = T($1)T($2) TP.HCM T(-) = -$(T) T($1 -$2) -(T(1) -T($2)) Contents = Introduction T($1 V $2) = -(-T(1) -T(2)) Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example Example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.21" }, { "page_index": 462, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_154.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_154.png", "page_index": 462, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:13+07:00" }, "raw_text": "Towards a Formula for a PCP Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM S1 S2 Sk Let C be the problem Contents t1 t2 tk Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.65" }, { "page_index": 463, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_155.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_155.png", "page_index": 463, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:17+07:00" }, "raw_text": "Transformation Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai T(p) = p BK T($1 $2) = T($1)T($2) TP.HCM T(-) = -$(T) T($1 -$2) -(T($1)-T($2)) Contents = Introduction T($1 V $2) = -(-T(1) -T(2)) Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P Example An example UNSAT Graphical View of 2-SAT SAT Solvers o=pA-(q V-p) WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.21" }, { "page_index": 464, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_155.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_155.png", "page_index": 464, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:20+07:00" }, "raw_text": "Towards a Formula for a PCP Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM s1 S2 Sk Let C be the problem Contents t1 t2 tk Predicate Logic Motivation, Syntax, Proof Theory Idea Need for Richer Language P(s,t) holds iff there is a sequence of indices (i1,i2,:..,im) such Predicate Logic as Formal Language that s is Si Siz ...sim and t is titiz ...tim. Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.65" }, { "page_index": 465, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_156.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_156.png", "page_index": 465, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:23+07:00" }, "raw_text": "Transformation Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai T(p) = p BK T($1 $2) = T($1)T($2) TP.HCM T(-) -(T) = T($1 -$2) -(T($1)-T($2)) Contents = Introduction T($1 V $2) = -(-T($1) -T($2)) Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example Example UNSAT Graphical View of 2SAT SAT Solvers o=pA-(q V-p) WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver T(o) =pA-(q -p Homeworks and Next Week Plan? 1b.21" }, { "page_index": 466, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_156.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_156.png", "page_index": 466, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:26+07:00" }, "raw_text": "The Formula o Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM = $1 $2 - $3, where k Contents P(fs;(e),ft;(e)) Predicate Logic Motivation, Syntax, Proof Theory i=1 Need for Richer Language k Predicate Logic as Formal Language VvVw(P(v,w)- P(fs;(v),ft;(w))) Proof Theory of Predicate Logic i=1 Quantifier Equivalences 3zP(z,z) Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.66" }, { "page_index": 467, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_157.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_157.png", "page_index": 467, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:29+07:00" }, "raw_text": "Binary Decision Tree: Example Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen. Tran Van Hoai BK TP.HCM Contents See Example 1.48 and Figure 1.12 on page 70 Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.22" }, { "page_index": 468, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_157.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_157.png", "page_index": 468, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:32+07:00" }, "raw_text": "Undecidability of Predicate Logic Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen BK So Far TP.HCM Post correspondence problem is undecidable Constructed c for Post correspondence problem C Contents Predicate Logic Motivation, Syntax, To Show Proof Theory Need for Richer Language Oc holds if and only if C has a solution. Predicate Logic as Formal Language Proof Theory of Predicate Logic Proof Quantifier Equivalences Proof via construction of δc. Formally construct an interpretation Semantics of Predicate Logic of strings and show that whenever there is a solution, the formula Soundness and Completeness of Oc holds and vice versa. Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.67" }, { "page_index": 469, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_158.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_158.png", "page_index": 469, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:34+07:00" }, "raw_text": "Problem Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai BK TP.HCM Contents What happens to formulas of the kind -(@1 2)? Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Soiver A Cubic Solver Homeworks and Next Week Plan? 1b.23" }, { "page_index": 470, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_158.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_158.png", "page_index": 470, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:38+07:00" }, "raw_text": "Summary of Undecidability Proof Advanced Predicate Logic Nguyen An Khuong, Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula o in that language, decides whether o. Predicate Logic Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.68" }, { "page_index": 471, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_159.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_159.png", "page_index": 471, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:40+07:00" }, "raw_text": "A Cubic Solver: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai BK Improve the linear solver as follows TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.24" }, { "page_index": 472, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_159.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_159.png", "page_index": 472, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:42+07:00" }, "raw_text": "Summary of Undecidability Proof Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula o in that language, decides whether . Predicate Logic Motivation, Syntax, Proof Theory Proof Need for Richer Language Predicate Logic as Formal Language Establish that the Post Correspondence Problem (PCP) is Proof Theory of Predicate Logic undecidable Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.68" }, { "page_index": 473, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_160.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_160.png", "page_index": 473, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:44+07:00" }, "raw_text": "A Cubic Solver: ldea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai BK Improve the linear solver as follows TP.HCM : Run linear solver Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.24" }, { "page_index": 474, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_160.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_160.png", "page_index": 474, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:48+07:00" }, "raw_text": "Summary of Undecidability Proof Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula in that language, decides whether . Predicate Logic Motivation, Syntax, Proof Theory Proof Need for Richer Language Predicate Logic as Formal Language Establish that the Post Correspondence Problem (PCP) is Proof Theory of Predicate Logic undecidable Quantifier Equivalences Translate an arbitrary PCP, say C, to a formula δ Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.68" }, { "page_index": 475, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_161.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_161.png", "page_index": 475, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:51+07:00" }, "raw_text": "A Cubic Solver: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK mprove the linear solver as follows: TP.HCM . Run linear solver : For every node n that is still unmarked: Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.24" }, { "page_index": 476, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_161.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_161.png", "page_index": 476, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:54+07:00" }, "raw_text": "Summary of Undecidability Proof Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula in that language, decides whether . Predicate Logic Motivation, Syntax, Proof Theory Proof Need for Richer Language Predicate Logic as Formal Language : Establish that the Post Correspondence Problem (PCP) is Proof Theory of Predicate Logic undecidable Quantifier Equivalences Translate an arbitrary PCP, say C, to a formula Semantics of Predicate Logic Establish that holds if and only if C has a solution Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.68" }, { "page_index": 477, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_162.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_162.png", "page_index": 477, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:48:58+07:00" }, "raw_text": "A Cubic Solver: Idea Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK Improve the linear solver as follows: TP.HCM . Run linear solver : For every node n that is still unmarked: Contents Introduction . Mark m with T and run linear solver, possibly resulting in Quick review temporary marks. Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.24" }, { "page_index": 478, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_162.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_162.png", "page_index": 478, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:01+07:00" }, "raw_text": "Summary of Undecidability Proof Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Theorem BK TP.HCM The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and Contents any formula o in that language, decides whether . Predicate Logic Motivation, Syntax, Proof Theory Proof Need for Richer Language Predicate Logic as Formal Language : Establish that the Post Correspondence Problem (PCP) is Proof Theory of Predicate Logic undecidable Quantifier Equivalences Translate an arbitrary PCP, say C, to a formula Semantics of Predicate Logic Establish that holds if and only if C has a solution. Soundness and Completeness of . Conclude that validity of pred. logic formulas is undecidable. Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.68" }, { "page_index": 479, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_163.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_163.png", "page_index": 479, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:03+07:00" }, "raw_text": "A Cubic Solver: Idea Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK Improve the linear solver as follows: TP.HCM : Run linear solver : For every node m that is still unmarked: Contents Introduction . Mark m with T and run linear solver, possibly resulting in Quick review temporary marks. Boolean Satisfiability (SAT) Mark n with F and run linear solver, possibly resulting in P and NP 2-SAT is in P temporary marks. An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.24" }, { "page_index": 480, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_163.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_163.png", "page_index": 480, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:07+07:00" }, "raw_text": "Compactness Theorem Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Contents Predicate Logic Let I be a set of sentences of predicate logic. If all finite subsets Motivation, Syntax, Proof Theory of I are satisfiable, then I is satisfiable Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.69" }, { "page_index": 481, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_164.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_164.png", "page_index": 481, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:09+07:00" }, "raw_text": "A Cubic Solver: Idea Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai BK mprove the linear solver as follows: TP.HCM Run linear solver For every node m that is still unmarked: Contents Introduction Mark n with T and run linear solver, possibly resulting in Quick review temporary marks. Boolean Satisfiability (SAT) Mark n with F and run linear solver, possibly resulting in P and NP 2-SAT is in P temporary marks. An example Combine temporary marks, resulting in possibly new UNSAT Graphical View of 2SAT permanent marks SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.24" }, { "page_index": 482, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_164.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_164.png", "page_index": 482, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:11+07:00" }, "raw_text": "Proof of Compactness Theorem Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Assume T is not satisfiable Contents Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.70" }, { "page_index": 483, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_165.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_165.png", "page_index": 483, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:14+07:00" }, "raw_text": "An application of SAT solving: Solve Sudoku Boolean Formula Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen, Tran Van Hoai BK TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.25" }, { "page_index": 484, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_165.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_165.png", "page_index": 484, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:16+07:00" }, "raw_text": "Proof of Compactness Theorem Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Assume I' is not satisfiable Contents We thus have I L Predicate Logic: Motivation, Syntax, Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.70" }, { "page_index": 485, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_166.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_166.png", "page_index": 485, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:18+07:00" }, "raw_text": "An application of SAT solving: Solve Sudoku Boolean Formula Propositional Logic Review II Nguyen An Khuong. Le Hong Trang Huynh Tuong Nguyen. Tran Van Hoai BK At the end of Chapter 0, we saw that TP.HCM Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL. Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.25" }, { "page_index": 486, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_166.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_166.png", "page_index": 486, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:21+07:00" }, "raw_text": "Proof of Compactness Theorem Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Assume I' is not satisfiable Contents We thus have I L Predicate Logic Motivation, Syntax, Via completeness, we have I F L. Proof Theory Need for Richer Language Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.70" }, { "page_index": 487, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_167.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_167.png", "page_index": 487, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:23+07:00" }, "raw_text": "An application of SAT solving: Solve Sudoku Boolean Formula Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai BK At the end of Chapter 0, we saw that TP.HCM o= I A RA C A B Contents Introduction Quick review Boolean Satisfiability (SAT) P and NP 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.25" }, { "page_index": 488, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_167.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_167.png", "page_index": 488, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:25+07:00" }, "raw_text": "Proof of Compactness Theorem Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Assume I is not satisfiable Contents We thus have I L Predicate Logic Motivation, Syntax, Via completeness, we have I L. Proof Theory The proof is finite, thus only uses a finite subset C I of Need for Richer Language Predicate Logic as Formal Language premises. Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.70" }, { "page_index": 489, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_168.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_168.png", "page_index": 489, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:28+07:00" }, "raw_text": "An application of SAT solving: Solve Sudoku Boolean Formula Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK At the end of Chapter 0, we saw that TP.HCM o=I A RA C A B Contents Introduction Quick review o Note that o is in CNF Boolean Satisfiability (SAT P and NP 2-SAT is in P An example UNSAT Graphical View of 2-SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.25" }, { "page_index": 490, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_168.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_168.png", "page_index": 490, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:32+07:00" }, "raw_text": "Proof of Compactness Theorem Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Assume T is not satisfiable Contents We thus have I L Predicate Logic Motivation, Syntax, Via completeness, we have I L. Proof Theory The proof is finite, thus only uses a finite subset C I of Need for Richer Language Predicate Logic as Formal Language premises. Proof Theory of Predicate Thus, F L, and L via soundness Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.70" }, { "page_index": 491, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_169.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_169.png", "page_index": 491, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:34+07:00" }, "raw_text": "An application of SAT solving: Solve Sudoku Boolean Formula Propositional Logic Review Il Nguyen An Khuong, Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai BK At the end of Chapter 0, we saw that TP.HCM o=I R C B Contents Introduction Quick review Note that o is in CNF Boolean Satisfiability (SAT) P and NP can be altered so that it contains exactly 3 literals per 2-SAT is in P An example clause (can be fed to 3-SAT solver) UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.25" }, { "page_index": 492, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_169.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_169.png", "page_index": 492, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:37+07:00" }, "raw_text": "Reachability not Expressible in Predicate Logic Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Contents There is no predicate logic formula $G,u,v with u and as its only Predicate Logic Motivation, Syntax, free variables and R as its only predicate symbol, such that $G,u,v Proof Theory Need for Richer Language holds iff there is a path from u to in G. Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.71" }, { "page_index": 493, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_170.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_170.png", "page_index": 493, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:40+07:00" }, "raw_text": "An application of SAT solving: Solve Sudoku Boolean Formula Propositional Logic Review Il Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen. Tran Van Hoai BK At the end of Chapter 0, we saw that TP.HCM 6= I A RC B Contents Introduction Quick review Note that o is in CNF Boolean Satisfiability (SAT P and NP can be altered so that it contains exactly 3 literals per 2-SAT is in P An example clause (can be fed to 3-SAT solver) UNSAT Graphical View of 2SAT Problem: Solve this 3-SAT problem with a suitable solver? SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.25" }, { "page_index": 494, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_170.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_170.png", "page_index": 494, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:43+07:00" }, "raw_text": "Lowenheim-Skolem Theorem Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Contents Let / be a sentence of predicate logic such that for any natural Predicate Logic Motivation, Syntax, number n > 1 there is a model of / with at least n elements. Proof Theory Need for Richer Language Then / has a model with infinitely many elements Predicate Logic as Formal Language Proof Theory of Predicate Logic Quantifier Equivalences Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.72" }, { "page_index": 495, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_171.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_171.png", "page_index": 495, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:46+07:00" }, "raw_text": "Homeworks and Next Week Plan? Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai Homeworks BK TP.HCM . Read carefully all proofs in this note. : Try to solve the Sudoku in the Introduction note Contents Show that kSAT E NPC for all k > 3 Introduction Quick review Do ALL marked questions of Exercises 1.6 in [2] Boolean Satisfiability (SAT) P and NP Read carefully Subsections 1.6.1 and 1.6.2 in [2] 2-SAT is in P An example UNSAT Graphical View of 2SAT SAT Solvers WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.26" }, { "page_index": 496, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_171.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_171.png", "page_index": 496, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:49+07:00" }, "raw_text": "Homeworks and Next Week Plan? Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Homeworks It is recommended that you should do as much as you can ALL BK marked exercises in [2, Sect. 2.8) (notice that sample solutions for TP.HCM these exercises are available in [3]). For this lecture, the following are recommended exercises [2]: Contents . 2.1: 1a); 2a) Predicate Logic Motivation, Syntax, 2.2: 6 Proof Theory Need for Richer Language 2.3: 1a); 1b); 6a); 6b); 6c); 7b): 9b); 9c); 13d) Predicate Logic as Formal Language Proof Theory of Predicate 2.4: 2); 3); 11a); 11c); 12e); 12f); 12h); 12k) Logic Quantifier Equivalences . 2.5: 1c); 1e). Semantics of Predicate Logic Soundness and Completeness of Predicate Logic Undecidability of Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.73" }, { "page_index": 497, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_172.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_b/slide_172.png", "page_index": 497, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:52+07:00" }, "raw_text": "Homeworks and Next Week Plan? Propositional Logic Review II Nguyen An Khuong. Le Hong Trang, Huynh Tuong Nguyen Tran Van Hoai Homeworks BK TP.HCM . Read carefully all proofs in this note. : Try to solve the Sudoku in the Introduction note Contents Show that kSAT E NPC for all k > 3 Introduction Quick review Do ALL marked questions of Exercises 1.6 in [2] Boolean Satisfiability (SAT) P and NP Read carefully Subsections 1.6.1 and 1.6.2 in [2] 2-SAT is in P An example UNSAT Graphical View of 2-SAT Next Week? SAT Solvers Predicate Logic WalkSAT: Idea DPLL: Idea A Linear Solver A Cubic Solver Homeworks and Next Week Plan? 1b.26" }, { "page_index": 498, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_172.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_172.png", "page_index": 498, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:49:56+07:00" }, "raw_text": "Homeworks and Next Week Plan? Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Homeworks It is recommended that you should do as much as you can ALL BK marked exercises in [2, Sect. 2.8) (notice that sample solutions for TP.HCM these exercises are available in [3D. For this lecture, the following are recommended exercises [2]: Contents . 2.1: 1a); 2a) Predicate Logic Motivation, Syntax, 2.2: 6 Proof Theory Need for Richer Language 2.3: 1a); 1b); 6a); 6b); 6c); 7b): 9b); 9c); 13d Predicate Logic as Formal Language Proof Theory of Predicate 2.4: 2); 3); 11a): 11c); 12e); 12f); 12h); 12k) Logic Quantifier Equivalences . 2.5: 1c); 1e). Semantics of Predicate Logic Soundness and Completeness of Next Weeks? Predicate Logic Undecidability of Exercises Session; Predicate Logic Compactness of Predicate Calculus Homeworks and Next Week Plan? 1c.73" }, { "page_index": 499, "chapter_num": 1, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_173.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_1/Chapter_1_c/slide_173.png", "page_index": 499, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:00+07:00" }, "raw_text": "Homeworks and Next Week Plan? Advanced Predicate Logic Nguyen An Khuong. Huynh Tuong Nguyen Homeworks It is recommended that you should do as much as you can ALL BK marked exercises in [2, Sect. 2.8) (notice that sample solutions for TP.HCM these exercises are available in [3D. For this lecture, the following are recommended exercises [2]: Contents . 2.1: 1a); 2a) Predicate Logic Motivation, Syntax, 2.2: 6 Proof Theory Need for Richer Language 2.3: 1a); 1b); 6a); 6b); 6c); 7b): 9b); 9c); 13d Predicate Logic as Formal Language Proof Theory of Predicate 2.4: 2); 3); 11a): 11c); 12e); 12f); 12h); 12k) Logic Quantifier Equivalences . 2.5: 1c); 1e). Semantics of Predicate Logic Soundness and Completeness of Next Weeks? Predicate Logic Undecidability of Exercises Session; Predicate Logic Compactness of Applications of FoL Predicate Calculus Homeworks and Next Week Plan? 1c.73" }, { "page_index": 500, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_001.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_001.png", "page_index": 500, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:03+07:00" }, "raw_text": "Integer Linear Programming Chapter 3 BK TP.HCM Integer Linear Programming Contents Mathematical Modeling Motivated Examples Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Le Hong Trang, Nguyen An Khuong, Huynh Tuong Nguyen Problems and Faculty of Computer Science and Engineering Homeworks HCMC University of Technology 3.1" }, { "page_index": 501, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_002.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_002.png", "page_index": 501, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:05+07:00" }, "raw_text": "Contents Integer Linear Programming Motivated Examples Linear Classification Problem BK TP.HCM Facility Location Problem Linear and Integer Linear Programs Contents Motivated Examples Simplex Method for Solving LP Classification Facility Location Branch & Bound Method for Solving ILP Linear and Integer Numerical Example 1 - IP Linear Programs Numerical Example 2 - Binary IP Simplex Method Branch & Bound Strategy and Steps Method Exercise: 0-1 Knapsack Problem Numerical Example 1 Numerical Example 2 Strategy and Steps 5) Remarks on Branch & Bound Method Exercise: 0-1 KP How to Branch? Remarks How to Branch? Which Node to Select? Which Node to Select? Rule of Fathoming Rule of Fathoming Software for ILP Software for Integer Programming Problems and Homeworks 6) Problems and Homeworks 3.2" }, { "page_index": 502, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_003.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_003.png", "page_index": 502, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:09+07:00" }, "raw_text": "Outline Integer Linear Programming Motivated Examples Linear Classification Problem BK Facility Location Problem TP.HCM 2 Linear and Integer Linear Programs Contents Motivated Examples 3 Simplex Method for Solving LP Classification Facility Location Branch & Bound Method for Solving ILP Linear and Integer NumericalExample1-IP Linear Programs Simplex Method Numerical Example 2-BinaryP Branch & Bound Strategy and Steps Method Numerical Example 1 Exercise:0-1KnapsackProblem Numerical Example 2 Strategy and Steps @ Remarks on Branch & Bound Method Exercise: 0-1 KP Remarks How to Branch? How to Branch? Which Node to Select? Which Node to Select? Rule of Fathoming Rule of Fathoming Software for ILP Software for Integer Programming Problems and Homeworks Problems and Homeworks 3.3" }, { "page_index": 503, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_004.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_004.png", "page_index": 503, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:12+07:00" }, "raw_text": "Linear Classification Integer Linear Programming Statement Given two sets of points in R\", denoted by{x1,:..,} and BK y1,: . . ,ym}, we need to classify the points. TP.HCM Contents Motivated Examples Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 01 KP Applications in Computer Science Remarks How to Branch? Machine learning Which Node to Select? Rule of Fathoming Pattern recognition, Software for ILP Problems and Data mining Homeworks 3.4" }, { "page_index": 504, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_005.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_005.png", "page_index": 504, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:14+07:00" }, "raw_text": "Linear Classification Integer Linear Programming BK TP.HCM Contents Motivated Examples Classification Facility Location Linear and Integer Linear Programs Simplex Method Use a line (hyperplane in R\") aT'x + b = 0, hence the Branch & Bound Method followings must be satisfied: Numerical Example 1 Numerical Example 2 aTxi+b>O,i=1,...,N and aTyj+b1,i=1,...,N and aTyj+b<-1,j=1,...,M. Problems and Homeworks 3.5" }, { "page_index": 505, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_006.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_006.png", "page_index": 505, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:17+07:00" }, "raw_text": "Linear Classification Integer Linear Programming BK TP.HCM Contents Motivated Examples Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method The classification is better if having ui,Vi Z 0 such that: Numerical Example 1 Numerical Example 2 aTxi+b>1-ui and aTyj+b<-1+vj, Strategy and Steps Exercise: 0-1 KP Remarks where i=1, ..,N and j = 1, ..., M How to Branch? Which Node to Select? Rule of Fathoming j=1 Vj Software for ILP Problems and Homeworks 3.6" }, { "page_index": 506, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_007.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_007.png", "page_index": 506, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:20+07:00" }, "raw_text": "Linear Classification Integer Linear Programming Classification for inseparable sets (SVM form) BK TP.HCM N M min Vj Contents i=1 j=1 Motivated Examples Classification s.t. Facility Location Linear and Integer a'yi+b-1+vi,j=1,...,M, Linear Programs Simplex Method ui,vjO,i=1,...,N and j=1, ..,M Branch & Bound Method or, in vector form Numerical Example 1 Numerical Example 2 1Tu+1Tv Strategy and Steps min Exercise: 0-1 KP Remarks s.t. How to Branch? Which Node to Select? Rule of Fathoming Software for ILP u,v 0. Problems and Homeworks 3.7" }, { "page_index": 507, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_008.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_008.png", "page_index": 507, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:25+07:00" }, "raw_text": "Linear Classification: numerically solving Integer Linear Programming BK n - 2; TP.HCM randn('state',2); N 50: M = 50; Y [1.5+0.9*randn(1,0.6*N), 1.5+0.7*randn(1,0.4*N); Contents 2*(randn(1,0.6*N)+1), 2*(randn(1,0.4*N)-1)]; Motivated Examples x [-1.5+0.9*randn1,0.6*M), -1.5+0.7*randn(1,0.4*M); Classification 2*(randn(1,0.6*M)-1), 2*(randn(1,0.4*M)+1)]; Facility Location T [-1 1: 1 1]: Linear and Integer Linear Programs Y = T*Y; X = T*X; Simplex Method % Solution via CVX Branch & Bound Method cvx_begin Numerical Example 1 variables a(n b(1 u(N v(M) Numerical Example 2 minimize (ones(1,N)*u + ones(1,M)*v) Strategy and Steps Exercise: 0-1 KP x'*a + b >= 1 - u; Remarks Y'*a + b <=-1- v); How to Branch? 0; Which Node to Select? v >= 0; Rule of Fathoming cvx_end Software for ILP Problems and Homeworks 3.8" }, { "page_index": 508, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_009.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_009.png", "page_index": 508, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:33+07:00" }, "raw_text": "Linear Classification: numerically solving Integer Linear Programming BK Approximate linear classification via linear programming TP.HCM 8 Contents 6 Motivated Examples Classification Facility Location O O Linear and Integer C Linear Programs O Simplex Method Branch & Bound O Method O O Numerical Example 1 O O 000 Numerical Example 2 -2 O O Strategy and Steps O O Exercise: 0-1 KP % O Remarks -4 .0 O How to Branch? O Which Node to Select? O O O Rule of Fathoming -6 Software for ILP & 1 Problems and -8 -6 -4 -2 0 2 4 6 8 10 Homeworks 3.9" }, { "page_index": 509, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_010.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_010.png", "page_index": 509, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:35+07:00" }, "raw_text": "Linear Classification Integer Linear Programming BK Classification for inseparable sets (SVM form) TP.HCM 1Tu+1Tv Contents min Motivated Examples s.t. Classification Facility Location y'a+1b-1+v, Linear and Integer Linear Programs u,v 0. Simplex Method Branch & Bound Question: if u = 0 and v = 0? Method Numerical Example 1 Numerical Example 2 min 0 Strategy and Steps Exercise: 0-1 KP s.t. Remarks T How to Branch? y'a+1b<0. Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.10" }, { "page_index": 510, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_011.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_011.png", "page_index": 510, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:39+07:00" }, "raw_text": "Uncapacitated Facility Location Integer Linear Programming Statement BK TP.HCM Given a set of supermarket locations, denoted by I, let J be the set of candidate warehouse locations. The problem is to determine Contents which locations among J should be used to construct a Motivated Examples warehouses such that the total cost is minimum. Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Network resource allocation Problems and Homeworks 3.11" }, { "page_index": 511, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_012.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_012.png", "page_index": 511, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:42+07:00" }, "raw_text": "Uncapacitated Facility Location: variables and objective Integer Linear Programming BK A location j E J is decided by TP.HCM if is chosen for a facility Contents 0 otherwise. Motivated Examples Classification Facility Location The cost associated with constructing each warehouse: f Linear and Integer The fraction of supply received by supermarket i from Linear Programs Simplex Method warehouse j: yij 0. Branch & Bound The cost of transportation between candidate warehouse site Method Numerical Example 1 j and supermarket location i: cij . Numerical Example 2 Strategy and Steps The total cost: Exercise: 01 KP Remarks Zfjxj+ZZ CijYij How to Branch? Which Node to Select? jEJ iEI Rule of Fathoming jEJ Software for ILP Problems and Homeworks 3.12" }, { "page_index": 512, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_013.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_013.png", "page_index": 512, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:45+07:00" }, "raw_text": "Uncapacitated Facility Location: constraints Integer Linear Programming BK TP.HCM Contents The sum of fraction from each warehouse for each Motivated Examples supermarket: Classification Facility Location C Yij = 1, Vi E I. Linear and Integer Linear Programs jEJ Simplex Method The warehouse works if it actually constructed: Branch & Bound Method Numerical Example 1 Yij K xj,Vi E I,j E J. Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.13" }, { "page_index": 513, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_014.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_014.png", "page_index": 513, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:48+07:00" }, "raw_text": "Uncapacitated Facility Location: model Integer Linear Programming BK TP.HCM Contents Zfjxj+ZZ min CijYij Motivated Examples a jEJ jEJ iEI Classification Facility Location s.t. Linear and Integer Linear Programs (1) Simplex Method jEJ Branch & Bound Method Yij K xj,Vi E I,j E J (2) Numerical Example 1 Numerical Example 2 Yij 0,Vi e I,j E J (3) Strategy and Steps Exercise: 0-1 KP Xj E{0,1}. (4) Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.14" }, { "page_index": 514, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_015.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_015.png", "page_index": 514, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:51+07:00" }, "raw_text": "Uncapacitated Facility Location: numerically solving Integer Linear Programming BK TP.HCM % ILP model solved by CVX cvx_begin Contents % variables Motivated Examples variable x(n binary Classification variable y(m,n binary) Facility Location % model Linear and Integer minimize (buildingCost*x + sum(sum(servingCost.*y))) Linear Programs subject to Simplex Method for i = 1:m Branch & Bound sum(y(i,:)) 1; Method Numerical Example 1 for j = 1:n Numerical Example 2 y(i,j) <= x(j); Strategy and Steps end Exercise: 0-1 KP end Remarks cvx_end How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.15" }, { "page_index": 515, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_016.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_016.png", "page_index": 515, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:50:55+07:00" }, "raw_text": "Uncapacitated Facility Location: numerically solving Integer Linear Programming BK TP.HCM Faciltylocation problern bymte gerineer prograning Faciltyloceb orproblem byintog erieer pro Contents Warehouses Warkd.s Warenouees vakes Motivated Examples 2.5 2.5 Classification Facility Location 2 0 Linear and Integer D Linear Programs 1.5 1.5 Simplex Method 1 . Branch & Bound Method 0.5 0.5 Numerical Example 1 Numerical Example 2 0.5 2 5 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.16" }, { "page_index": 516, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_017.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_017.png", "page_index": 516, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:00+07:00" }, "raw_text": "Uncapacitated Facility Location: numerically solving Integer Linear Programming Facilitylocation problern byintegerlinear prograrnning BK Warehouses TP.HCM 4.5 0 Markets 4 Contents MotivatedExamples 3.5 Classification Facility Location 3 Linear and Integer Linear Programs 2.5 Simplex Method Branch & Bound 2 Method Numerical Example 1 Numerical Example 2 1.5 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? 0.5 Which Node to Select? Rule of Fathoming Software for ILP 2 4 5 Problems and Homeworks 3.17" }, { "page_index": 517, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_018.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_018.png", "page_index": 517, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:04+07:00" }, "raw_text": "Outline Integer Linear Programming Motivated Examples Linear Classification Problem BK Facility Location Problem TP.HCM Linear and Integer Linear Programs Contents Motivated Examples 3 Simplex Method for Solving LP Classification Facility Location Branch & Bound Method for Solving ILP Linear and Integer NumericalExample1-IP Linear Programs Simplex Method Numerical Example 2-Binary IP Branch & Bound Strategy and Steps Method Numerical Example 1 Exercise:0-1KnapsackProblem Numerical Example 2 Strategy and Steps @ Remarks on Branch & Bound Method Exercise: 0-1 KP Remarks How to Branch? How to Branch? Which Node to Select? Which Node to Select? Rule of Fathoming Rule of Fathoming Software for ILP Software for Integer Programming Problems and Homeworks Problems and Homeworks 3.18" }, { "page_index": 518, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_019.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_019.png", "page_index": 518, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:08+07:00" }, "raw_text": "Linear Program (LP) Integer Linear Programming BK Recall the standard form of LP. TP.HCM min a Contents s.t. Motivated Examples Classification Ax = b. (5) Facility Location Linear and Integer x >0, (6) Linear Programs Simplex Method where c E R\", A is an m x n matrix with full row rank, and Branch & Bound Method b E Rm Numerical Example 1 Numerical Example 2 A polyhedron is a set of the form {x E R\"Bx > d} for some Strategy and Steps Exercise: 0-1 KP matrix B. Remarks . Let P E R\" be a given polyhedron. A vector x E P is an How to Branch? Which Node to Select? extreme point of P if there does not exist y,z E P, and Rule of Fathoming e (0,1) such that x =Ay +(1-)z. Software for ILP Problems and Homeworks 3.19" }, { "page_index": 519, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_020.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_020.png", "page_index": 519, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:10+07:00" }, "raw_text": "Linear Program in General form Integer Linear Programming BK TP.HCM General form of LP: Contents min a Motivated Examples Classification s.t. Facility Location Ax = b, (7) Linear and Integer Linear Programs Cx < d, (8) Simplex Method (9) Branch & Bound x > 0. Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks Question How to Branch? Is there any way to transform a general form to standard one? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.20" }, { "page_index": 520, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_021.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_021.png", "page_index": 520, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:13+07:00" }, "raw_text": "Integer Linear Program (ILP) Integer Linear Programming An integer linear program is a linear program where variables BK are constrained to be integers. TP.HCM min a Contents s.t. Motivated Examples Classification Ax = b, (10) Facility Location (11) Linear and Integer x > 0 and x e Zn Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Question Numerical Example 2 Strategy and Steps Why do we consider the problem with only equality constraints but Exercise: 0-1 KP Remarks not inequality ones? How to Branch? Which Node to Select? Rule of Fathoming Remark Software for ILP Often a mix is desired of integer and non-integer variables, called Problems and Homeworks Mixed Integer Linear Programs (MILP) 3.21" }, { "page_index": 521, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_022.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_022.png", "page_index": 521, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:17+07:00" }, "raw_text": "Basic Solutions and Extreme Points Integer Linear Programming BK Let S ={x E R\"Ax =b,x > 0},the feasible set of LP. Since TP.HCM A is full row rank, if the feasible set is not empty, then we must m < n. Without loss of generality, we assume that Contents m 0, then How to Branch? B-1b Which Node to Select? is said to be a basic feasible solution. Rule of Fathoming Software for ILP Problems and Homeworks 3.22" }, { "page_index": 522, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_023.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_023.png", "page_index": 522, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:21+07:00" }, "raw_text": "Basic Solutions and Extreme Points Integer Linear Programming BK : E S is an extreme point of S if and on/y if c is a basic TP.HCM feasible solution. Two extreme points are adjacent if they differ in only one Contents basic variable Motivated Examples Classification Theorem (Basic theorem of LP) Facility Location Linear and Integer Linear Programs Consider the linear program min{cT'xAx = b,x > 0}. If S has at Simplex Method least one extreme point and there exists an optimal solution, then Branch & Bound there exists an optimal solution that is an extreme point Method Numerical Example 1 Numerical Example 2 The feasible set of standard form linear program has at least Strategy and Steps one extreme point. Exercise: 0-1 KP Remarks Therefore, we claim that the optimal value of a linear How to Branch? Which Node to Select? program is either -, or is attained an extreme point (basic Rule of Fathoming feasible solution) of the feasible set. Software for ILP Problems and Homeworks 3.23" }, { "page_index": 523, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_024.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_024.png", "page_index": 523, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:24+07:00" }, "raw_text": "A Naive Algorithm for Solving Linear Program Integer Linear Programming BK TP.HCM Let min{cTxAx = b,x > 0} be a bounded linear program. Enumerate all bases B E{1,2,...,n} 0(nm), too Contents Motivated Examples many. Classification Facility Location Compute associated basic solution x Linear and Integer 0 Linear Programs Simplex Method Return the one which has largest objective function value Branch & Bound among the feasible basic solutions. Method Numerical Example 1 . Running time is O(nm .m3) Numerical Example 2 Strategy and Steps Exercise: 01 KP Question Remarks How to Branch? Are there more efficient algorithms? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.24" }, { "page_index": 524, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_025.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_025.png", "page_index": 524, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:28+07:00" }, "raw_text": "Outline Integer Linear Programming Motivated Examples Linear Classification Problem BK Facility Location Problem TP.HCM 2 Linear and Integer Linear Programs Contents Motivated Examples Simplex Method for Solving LP Classification Facility Location Branch & Bound Method for Solving ILP Linear and Integer NumericalExample1-IP Linear Programs Simplex Method Numerical Example 2-BinaryIP Branch & Bound Strategy and Steps Method Numerical Example 1 Exercise:0-1KnapsackProblem Numerical Example 2 Strategy and Steps @ Remarks on Branch & Bound Method Exercise: 0-1 KP Remarks How to Branch? How to Branch? Which Node to Select? Which Node to Select? Rule of Fathoming Rule of Fathoming Software for ILP Software for Integer Programming Problems and Homeworks Problems and Homeworks 3.25" }, { "page_index": 525, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_026.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_026.png", "page_index": 525, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:32+07:00" }, "raw_text": "Simplex Method Integer Linear Programming . Simplex method was invented by George Dantzig BK (1914-2005) (father of linear programming) TP.HCM Suppose we have a basic feasible solution Contents A=(B,N). Motivated Examples Classification LB Facility Location Let x E S be any feasible solution of the LP. Let ε Linear and Integer Linear Programs CB and c : Then,BxB+ Nxn =b and Simplex Method CN Branch & Bound xB=B-1b-B-1Nxn.We have Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 01 KP =cTB-1b-ctB-1Nxn+cTxN Remarks How to Branch? cTx+(cT -cBB-1N)xN. Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks then cTε > cTε and the current extreme point is optimal. 3.26" }, { "page_index": 526, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_027.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_027.png", "page_index": 526, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:35+07:00" }, "raw_text": "Simplex Method Integer Linear Programming BK TP.HCM Otherwise, there must exist an r: < 0, we can let current nonbasic variable ; become a basic one i > 0 (entering Contents variable). Motivated Examples Classification Suitably choosing basic variable to become a nonbasic one Facility Location (/eaving variable), we can get a new basic feasible solution Linear and Integer Linear Programs whose objective value is less than that of the current basic Simplex Method feasible solution . Branch & Bound Method Geometrically, the simplex method moves from one extreme Numerical Example 1 point to one of its adjacent extreme point. Numerical Example 2 Strategy and Steps . Since there are only a finite number of extreme points, the Exercise: 0-1 KP Remarks method terminates finitely at an optima/ so/ution or detects How to Branch? that the problem is infeasib/e or it is unbounded. Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.27" }, { "page_index": 527, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_028.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_028.png", "page_index": 527, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:38+07:00" }, "raw_text": "Geometry of Simplex Method Integer Linear Programming BK TP.HCM Optimal solution Contents Motivated Examples Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Starting Which Node to Select? 1 Rule of Fathoming vertex Software for ILP Problems and Homeworks 3.28" }, { "page_index": 528, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_029.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_029.png", "page_index": 528, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:42+07:00" }, "raw_text": "Simplex Method Integer Linear Programming BK Step 0: Compute an initial basis B and the basic feasible TP.HCM B-1b B-1b solution x Contents Motivated Examples solution. Otherwise goto Step 2. Classification Facility Location : Step 2: Choose j satisfying cT - cBB-1aj < 0, if Linear and Integer Linear Programs aj = B-1aj 0, stop, the LP is infeasible. Otherwise, goto Simplex Method Step 3. Branch & Bound Method Step 3: compute the step size Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP A min aiy ar3 Remarks How to Branch? Which Node to Select? Rule of Fathoming Let x := x + dj, where dj goto Step 1. Software for ILP e Problems and Homeworks 3.29" }, { "page_index": 529, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_030.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_030.png", "page_index": 529, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:45+07:00" }, "raw_text": "Simplex Tableau Integer Linear Programming BK TP.HCM rhs Contents X B XN Motivated Examples B N b Classification CN 0 Facility Location Linear and Integer Linear Programs It implies that Simplex Method Branch & Bound B CN rhs Method Numerical Example 1 1 B-1N B-1b Numerical Example 2 0 -ckB-1b Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.30" }, { "page_index": 530, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_031.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_031.png", "page_index": 530, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:50+07:00" }, "raw_text": "Simplex Method: An example Integer Linear Programming : Consider the following LP BK TP.HCM min - 7x1 - 2x2 Contents s.t. Motivated Examples Classification 1 + 2x2 + 3 = 4 Facility Location 5x1 + x2 + x4 = 20, Linear and Integer Linear Programs 2x1 + 2x2 - X5 = 7, Simplex Method Branch & Bound x > 0. Method Numerical Example 1 Numerical Example 2 The initial tableau should be Strategy and Steps x1 X2 X3 x4 5 rhs Exercise: 0-1 KP Remarks -1 2 1 0 0 4 How to Branch? 5 1 0 1 0 20 Which Node to Select? Rule of Fathoming 2 2 0 0 -1 7 Software for ILP -7 -2 0 0 0 0 Problems and Homeworks 3.31" }, { "page_index": 531, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_032.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_032.png", "page_index": 531, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:51:58+07:00" }, "raw_text": "Simplex Method: example (cont.) Integer Linear Programming . Choose the initial basis to be B = (a1,a3,a4), we have basic B = (1,3,4)7. The simplex tableau is then BK x1 X3 rhs TP.HCM X2 4 5 0 3 1 0 12 23 0 0 1 2 Contents 2 0 Motivated Examples 1 1 0 2 Classification 0 5 0 0 24 Facility Location 2 Linear and Integer The basic feasible solution is Linear Programs xB=(x1,X3,x4)T=(3,7,23)T Simplex Method -? < 0, 5 is chosen entering variable. Branch & Bound : Since T5 = Method 2 Numerical Example 1 = = 1, then 4 is leaving variable. The new basic Numerical Example 2 21 Strategy and Steps variable should be B = (x1,x3, 5)T. The new tableau is Exercise: 01 KP obtained as below. Remarks How to Branch? X1 X2 X3 x4 5 rhs Which Node to Select? Rule of Fathoming 0 11 1 0 8 Software for ILP 0 0 1 1 Problems and Homeworks 1 0 0 4 0 5 0 0 28 3 3.32" }, { "page_index": 532, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_033.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_033.png", "page_index": 532, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:03+07:00" }, "raw_text": "Simplex Method: example (cont.) Integer Linear Programming BK TP.HCM Similarly, choose ? as the entering variable, then Contents Therefore, the tableau is Motivated Examples Classification x1 2 X 3 X4 X5 rhs Facility Location 0 1 5 1 0 40 Linear and Integer 100 Linear Programs 0 0 1 36 Simplex Method 1 0 0 11 Branch & Bound 11 16 Method 0 0 0 30 11 Numerical Example 1 Numerical Example 2 Since ry 3 O, the current basic feasible solution Strategy and Steps Exercise: 01 KP 36 0.0. 75 is optimal with the optimal value is Remarks -30 How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.33" }, { "page_index": 533, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_034.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_034.png", "page_index": 533, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:07+07:00" }, "raw_text": "Outline Integer Linear Programming Motivated Examples Linear Classification Problem BK Facility Location Problem TP.HCM 2 Linear and Integer Linear Programs Contents Motivated Examples Simplex Method for Solving LP Classification Facility Location 4 Branch & Bound Method for Solving ILP Linear and Integer Numerical Example 1 - IP Linear Programs Simplex Method Numerical Example 2 - Binary IP Branch & Bound Strategy and Steps Method Numerical Example 1 Exercise: 0-1 Knapsack Problem Numerical Example 2 Strategy and Steps 5 Remarks on Branch & Bound Method Exercise: 0-1 KP Remarks How to Branch? How to Branch? Which Node to Select? Which Node to Select? Rule of Fathoming Rule of Fathoming Software for ILP Software for Integer Programming Problems and Homeworks Problems and Homeworks 3.34" }, { "page_index": 534, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_035.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_035.png", "page_index": 534, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:09+07:00" }, "raw_text": "Branch & Bound Method - lntroduction Integer Linear Programming BK TP.HCM This is the divide and conquer method. We divide a large problem into a few smaller ones. This is the \"branch\" part Contents Motivated Examples Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.35" }, { "page_index": 535, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_036.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_036.png", "page_index": 535, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:11+07:00" }, "raw_text": "Branch & Bound Method - lntroduction Integer Linear Programming BK TP.HCM This is the divide and conquer method. We divide a large problem into a few smaller ones. This is the \"branch\" part. Contents The conquering part is done by estimate how good a solution Motivated Examples Classification we can get for each smaller problems. Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.35" }, { "page_index": 536, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_037.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_037.png", "page_index": 536, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:14+07:00" }, "raw_text": "Branch & Bound Method - Introduction Integer Linear Programming BK TP.HCM This is the divide and conquer method. We divide a large problem into a few smaller ones. This is the \"branch\" part Contents The conquering part is done by estimate how good a solution Motivated Examples Classification we can get for each smaller problems. Facility Location . To do so, we may have to divide the problem further, until we Linear and Integer Linear Programs get a problem that we can handle, that is the 'bound\" part. Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.35" }, { "page_index": 537, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_038.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_038.png", "page_index": 537, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:17+07:00" }, "raw_text": "Branch & Bound Method - Introduction Integer Linear Programming BK TP.HCM This is the divide and conquer method. We divide a large problem into a few smaller ones. This is the \"branch\" part. Contents The conquering part is done by estimate how good a solution Motivated Examples Classification we can get for each smaller problems. Facility Location - To do so, we may have to divide the problem further, until we Linear and Integer Linear Programs get a problem that we can handle, that is the \"bound\" part. Simplex Method . We will use the linear programming relaxation to estimate the Branch & Bound Method optimal solution of an integer programming Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.35" }, { "page_index": 538, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_039.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_039.png", "page_index": 538, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:20+07:00" }, "raw_text": "Branch & Bound Method - Introduction Integer Linear Programming BK TP.HCM This is the divide and conquer method. We divide a large problem into a few smaller ones. This is the \"branch\" part Contents The conquering part is done by estimate how good a solution Motivated Examples Classification we can get for each smaller problems. Facility Location - To do so, we may have to divide the problem further, until we Linear and Integer Linear Programs get a problem that we can handle, that is the 'bound\" part. Simplex Method : We will use the linear programming relaxation to estimate the Branch & Bound Method optimal solution of an integer programming. Numerical Example 1 . For an integer programming model P, the linear programming Numerical Example 2 Strategy and Steps model we get by dropping the requirement that all variables Exercise: 0-1 KP Remarks must be integers is called the linear programming relaxation How to Branch? of P. Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.35" }, { "page_index": 539, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_040.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_040.png", "page_index": 539, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:23+07:00" }, "raw_text": "Numerical Example 1: Integer Programming Integer Linear Programming BK TP.HCM Contents max(Z = -x1 + 4x2 (12) Motivated Examples Classification Subject to Facility Location (13) Linear and Integer -10x1 + 20x2 22 Linear Programs (14) Simplex Method 5x1 + 10x2 < 49 Branch & Bound Method x1 5 (15) Numerical Example 1 Numerical Example 2 xi>0,xi E Z Vi E{1,2} (16) Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.36" }, { "page_index": 540, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_041.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_041.png", "page_index": 540, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:26+07:00" }, "raw_text": "Numerical Example 1 (cont') : Linear programming relaxation Integer Linear Programming BK TP.HCM With linear programming relaxation, we drop xi E Z Contents Motivated Examples max(Z = -x1 +4x2) (17) Classification Facility Location Subject to Linear and Integer Linear Programs -10x1 + 20x2 < 22 (18) Simplex Method 5x1 +10x2 < 49 (19) Branch & Bound Method Numerical Example 1 x1 < 5 (20) Numerical Example 2 Strategy and Steps xj>0 ViE{1,2} (21) Exercise: 01 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.37" }, { "page_index": 541, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_042.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_042.png", "page_index": 541, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:29+07:00" }, "raw_text": "Numerical Example 1 (cont') Integer Linear Programming BK 4 (3) TP.HCM (3.8.3) Contents 3 Motivated Examples (1) Classification Facility Location Linear and Integer Linear Programs Simplex Method (2) (4) Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps 0 Exercise: 0-1 KP -1 1 2 3 4 6 Remarks How to Branch? Which Node to Select? -1+ Rule of Fathoming Software for ILP Problems and Optimal solution of relaxation is (3.8, 3) with Z = 8.2 Homeworks 3.38" }, { "page_index": 542, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_043.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_043.png", "page_index": 542, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:32+07:00" }, "raw_text": "Numerical Example 1 (cont'): Linear programming relaxation Integer Linear Programming BK TP.HCM X=4 Contents 3 Motivated Examples Classification Facility Location 2 Linear and Integer Linear Programs Simplex Method Branch & Bound Method 0 Numerical Example1 Numerical Example 2 -1 0 1 2 3 4 6 Strategy and Steps Exercise: 01 KP 1 Remarks How to Branch? Which Node to Select? Since optimal solution of relaxation is (3.8, 3), we consider two Rule of Fathoming Software for ILP cases: x1 > 4 and c1 < 3. Problems and Homeworks 3.39" }, { "page_index": 543, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_044.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_044.png", "page_index": 543, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:35+07:00" }, "raw_text": "Example 1 (cont') - LP relaxation: 1 > 4 Integer Linear Programming BK TP.HCM Case X1 > 4: max(Z = -x1 +4x2 (22) Contents Motivated Examples Subject to Classification Facility Location -10x1 + 20x2 22 (23) Linear and Integer Linear Programs 5:1 + 10x2 49 (24) Simplex Method Branch & Bound x1 5 (25) Method Numerical Example 1 x1 > 4 (26) Numerical Example 2 Strategy and Steps Exercise: 0-1 KP x2 0 (27) Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.40" }, { "page_index": 544, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_045.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_045.png", "page_index": 544, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:39+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation:c1 > 4 Integer Linear Programming BK TP.HCM Contents 3 Motivated Examples 2.9 Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method 0 Numerical Example1 Numerical Example 2 -1 0 1 2 3 6 Strategy and Steps Exercise: 0-1 KP -1 Remarks How to Branch? Which Node to Select? has optimal solution at (4, 2.9) with Z = 7.6. Rule of Fathoming Software for ILP Then we consider two cases: x2 3 and :2 2. Problems and Homeworks 3.41" }, { "page_index": 545, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_046.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_046.png", "page_index": 545, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:41+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: 1 > 4,2 > 3 Integer Linear Programming BK TP.HCM Case 1 > 4, 2 > 3: Contents max(Z = -x1 + 4x2 (28) Motivated Examples Classification Subject to Facility Location (29) Linear and Integer -10x1 + 20x2 22 Linear Programs (30) Simplex Method 5x1 + 10x2 < 49 Branch & Bound Method 4 x1 < 5 (31) Numerical Example 1 Numerical Example 2 x2 3 (32) Strategy and Steps Exercise: 0-1 KP has no feasible solution (5x1 +10x2 50) so the IP has no Remarks How to Branch? feasible solution either. Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.42" }, { "page_index": 546, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_047.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_047.png", "page_index": 546, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:44+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: x1 > 4,2 2 Integer Linear Programming BK TP.HCM Case 1 > 4, 2 < 2: Contents Motivated Examples max(Z = -x1 + 4x2 (33) Classification Facility Location Subject to Linear and Integer Linear Programs -10x1 + 20x2 < 22 (34) Simplex Method 5x1 +10x2 < 49 (35) Branch & Bound Method Numerical Example 1 4 x1 5 (36) Numerical Example 2 Strategy and Steps 0 < x2 < 2 (37) Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.43" }, { "page_index": 547, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_048.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_048.png", "page_index": 547, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:48+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: - 1 > 4, x2 < 2 Integer Linear Programming BK TP.HCM Contents 3 Motivated Examples Classification (4.2) Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method 0 Numerical Example1 -1 0 1 2 3 6 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP 1 Remarks How to Branch? has an optimal solution at (4, 2) with Z = 4. Which Node to Select? Rule of Fathoming This is the optimal solution of the IP as well. Currently, the best Software for ILP value of Z for the original IP is Z = 4. Problems and Homeworks 3.44" }, { "page_index": 548, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_049.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_049.png", "page_index": 548, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:51+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: 1 < 3 Integer Linear Programming BK TP.HCM Come back case x1 < 3: Contents Motivated Examples max(Z = -x1 + 4x2 (38) Classification Facility Location Subject to Linear and Integer Linear Programs -10x1 + 20x2 < 22 (39) Simplex Method 5x1 +10x2 < 49 (40) Branch & Bound Method Numerical Example 1 0 x1 3 (41) Numerical Example 2 Strategy and Steps (42) Exercise: 0-1 KP x2 0 Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.45" }, { "page_index": 549, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_050.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_050.png", "page_index": 549, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:54+07:00" }, "raw_text": "Numerical Example 1 (cont'): LP relaxation Integer Linear Programming BK TP.HCM Contents 3 (3.2.6) Motivated Examples Classification Z4 Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method 0 Numerical Example1 Numerical Example 2 -1 0 1 2 3 6 Strategy and Steps Exercise: 0-1 KP -1 Remarks How to Branch? Which Node to Select? has an optimal solution at (3, 2.6) with Z = 7.4. Rule of Fathoming Software for ILP We branch out further to two cases: x2 > 3 and x2 < 2 Problems and Homeworks 3.46" }, { "page_index": 550, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_051.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_051.png", "page_index": 550, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:52:58+07:00" }, "raw_text": "Example 1 (cont') - LP relaxation: 1 3,2 > 3 Integer Linear Programming BK TP.HCM Case x1 < 3,x2 > 3: Contents max(Z = -x1 + 4x2 (43) Motivated Examples Classification Subject to Facility Location (44) Linear and Integer -10x1 + 20x2 < 22 Linear Programs (45) Simplex Method 5x1 + 10x2 < 49 Branch & Bound Method 0< x1< 3 (46) Numerical Example 1 Numerical Example 2 x2 3 (47) Strategy and Steps Exercise: 0-1 KP has no feasible solution (-10x1 +20x2 30 Remarks How to Branch? The lP has no solution either. Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.47" }, { "page_index": 551, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_052.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_052.png", "page_index": 551, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:00+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: - 1 3, x2 2 Integer Linear Programming BK TP.HCM Case 1 < 3, 2 < 2: Contents (48) Motivated Examples max(Z = -x1 + 4x2 Classification Facility Location Subject to Linear and Integer Linear Programs -10x1 + 20x2 < 22 (49) Simplex Method 5x1 +10x2 < 49 (50) Branch & Bound Method Numerical Example 1 0 x1< 3 (51) Numerical Example 2 Strategy and Steps 0 < x2 < 2 (52) Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.48" }, { "page_index": 552, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_053.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_053.png", "page_index": 552, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:03+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: 1 3,2 < 2 Integer Linear Programming BK TP.HCM Contents 3 Motivated Examples Classification 1.8,2 Z=4 Facility Location 2 Linear and Integer Linear Programs Simplex Method Branch & Bound Method 0 Numerical Example1 Numerical Example 2 -1 0 1 2 4 5 6 Strategy and Steps Exercise: 0-1 KP -1 + Remarks How to Branch? Which Node to Select? has an optimal at (1.8, 2) with Z = 6.2. Rule of Fathoming Software for ILP We branch further with two cases: x1 > 2 or x1 < 1 Problems and Homeworks 3.49" }, { "page_index": 553, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_054.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_054.png", "page_index": 553, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:07+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: 1 > 2,2 < 2 Integer Linear Programming BK TP.HCM Case 1 > 2, 2 < 2: Contents Motivated Examples max(Z = -x1 + 4x2 (53) Classification Facility Location Subject to Linear and Integer Linear Programs -10x1 + 20x2 < 22 (54) Simplex Method 5x1 +10x2 < 49 (55) Branch & Bound Method Numerical Example 1 2 < x1 < 3 (56) Numerical Example 2 Strategy and Steps 0 < x2 < 2 (57) Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.50" }, { "page_index": 554, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_055.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_055.png", "page_index": 554, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:10+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: 1 > 2,2 < 2 Integer Linear Programming BK TP.HCM 3 Contents Motivated Examples Z=6 Classification Facility Location 2.2 Linear and Integer Linear Programs Simplex Method Branch & Bound 0 Method Numerical Example1 -1 0 6 Numerical Example 2 Strategy and Steps -1 Exercise: 0-1 KP Remarks How to Branch? has an optimal at (2, 2), with Z = 6. Which Node to Select? Rule of Fathoming Since this is better than the incumbent Z = 4 at (4, 2), this new Software for ILP integer solution is our current best solution Problems and Homeworks 3.51" }, { "page_index": 555, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_056.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_056.png", "page_index": 555, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:13+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: x1 1, 2 2 Integer Linear Programming BK TP.HCM Case 1 1, 2 < 2: Contents Motivated Examples max(Z = -x1 + 4x2 (58) Classification Facility Location Subject to Linear and Integer Linear Programs -10x1 + 20x2 < 22 (59) Simplex Method 5x1 +10x2 < 49 (60) Branch & Bound Method Numerical Example 1 0 < x1 1 (61) Numerical Example 2 Strategy and Steps 0 < x2 < 2 (62) Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.52" }, { "page_index": 556, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_057.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_057.png", "page_index": 556, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:16+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: 1 3,2 < 2 Integer Linear Programming BK TP.HCM Contents 3 Motivated Examples Classification Z=6 Facility Location Z = 5.4 Linear and Integer (1. 1.6) Linear Programs Simplex Method Branch & Bound Method 0 Numerical Example1 -1 0 6 Numerical Example 2 Strategy and Steps Exercise: 01 KP -1 Remarks How to Branch? has an optimal at (1, 1.6) with Z = 5.4. Which Node to Select? Rule of Fathoming Then any integer solution in this region can not give us a solution Software for ILP with the value of Z greater than 5.4. This branch is fathomed Problems and Homeworks 3.53" }, { "page_index": 557, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_058.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_058.png", "page_index": 557, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:19+07:00" }, "raw_text": "Numerical Example 1 (cont') - LP relaxation: conclusion Integer Linear Programming BK TP.HCM Z=8.2 (3.8,3) Contents Motivated Examples Classification Facility Location Z=7.6 (4,2.9) (3,2.6) Z=7.4 Linear and Integer Linear Programs Simplex Method Branch & Bound NF (4,3) Z=4 (4,2) (3,3) NF (1.8,2) Z=6.2 Method Numerical Example1 Numerical Example 2 Strategy and Steps Exercise: 01 KP (2,2) (1,1.6) Remarks Z=6 Z=5.4 How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.54" }, { "page_index": 558, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_059.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_059.png", "page_index": 558, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:22+07:00" }, "raw_text": "Numerical Example 2 - Binary IP Problem Integer Linear Programming BK TP.HCM max(Z = 9x1 + 5x2 +6x3 + 4x4 (63) Contents Motivated Examples Subject to Classification Facility Location 6x1 + 3x2+5x3+ 2x4 < 10 (64) Linear and Integer Linear Programs x3 + x4 1 (65) Simplex Method Branch & Bound -X1 + x3 < 0 (66) Method Numerical Example 1 -X2 + x4 < 0 (67) Numerical Example 2 Strategy and Steps Exercise: 0-1 KP XiE{0,1} (68) Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.55" }, { "page_index": 559, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_060.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_060.png", "page_index": 559, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:25+07:00" }, "raw_text": "Numerical Example 2: LP Relaxation Integer Linear Programming BK TP.HCM max(Z = 9x1 + 5x2 + 6x3 + 4x4 (69) Subject to Contents 6x1 +3x2 +5x3 +2x4 < 10 (70) Motivated Examples Classification (71) Facility Location x3 + x4 < 1 Linear and Integer Linear Programs -X1 + x3 < 0 (72) Simplex Method -x2 + x4 < 0 (73) Branch & Bound Method Numerical Example 1 xi <1,for 1< i< 4 (74) Numerical Example 2 Strategy and Steps 0< xi (75) Exercise: 01 KP Remarks has optimal solution at (@, 1, 0, 1) (why?-HW on Simplex How to Branch? Which Node to Select? Method!) with Z = 16.5 Rule of Fathoming Has two branch, x1 = 0 or x1 = 1 Software for ILP Problems and Homeworks 3.56" }, { "page_index": 560, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_061.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_061.png", "page_index": 560, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:29+07:00" }, "raw_text": "Numerical Example 2 - LP Relaxation: 1 = 0 Integer Linear Programming BK TP.HCM With x1 = 0, problem becomes max(Z = 5x2 + 4c4 (76) Contents Motivated Examples Classification Subject to Facility Location Linear and Integer Linear Programs 3x2 + 2x4 < 10 Simplex Method x3 + x4 < 1 Branch & Bound Method -x2 + x4 0 Numerical Example 1 Numerical Example 2 xi E{0,1} Strategy and Steps Exercise: 01 KP Remarks has the optimal solution at (0, 1, 0, 1) with Z = 9. (Current best How to Branch? solution) Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.57" }, { "page_index": 561, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_062.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_062.png", "page_index": 561, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:31+07:00" }, "raw_text": "Numerical Example 2 - LP Relaxation: 1 = 1 Integer Linear Programming BK With x1 = 1,the LP relaxation TP.HCM max(Z = 9+ 5x2 + 6x3 + 4x4 (77) Contents Subject to Motivated Examples Classification Facility Location 3:2 + 5x3 + 2x4 < 4 Linear and Integer Linear Programs x3+x4 < 1 Simplex Method x3 < 1 Branch & Bound Method Numerical Example 1 -X2 + x4 < 0 Numerical Example 2 Strategy and Steps xi < 1 for 2< i< 4 Exercise: 01 KP 0 The current best solution (1, 1, 0, 0) with Z = 14 is the Software for ILP optimal solution. Problems and Homeworks 3.61" }, { "page_index": 566, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_067.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_067.png", "page_index": 566, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:48+07:00" }, "raw_text": "Numerical Example 2 - LP Relaxation: 1 = 1, x2 = 1 Integer Linear Programming BK max(Z = 14 + 6x3 + 4x4 (80) TP.HCM Contents S.t. 5x3 + 2x4 < 1 Motivated Examples x3 +x4 < 1 Classification Facility Location 0< X3 < 1 Linear and Integer Linear Programs 0< x4 1 Simplex Method Branch & Bound Method Numerical Example 1 With 3 = 0, Z = 16 is still feasible solution at (1, 1, 0, 0.5) Numerical Example 2 Strategy and Steps : With x4 = 0, (1, 1, 0, 0), Z = 14 (new optimal solution) Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming > The current best solution (1, 1, 0, 0) with Z = 14 is the Software for ILP optimal solution. Problems and Homeworks 3.61" }, { "page_index": 567, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_068.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_068.png", "page_index": 567, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:51+07:00" }, "raw_text": "Numerical Example 2 - LP Relaxation: 1 = 1, 2 = 1 Integer Linear Programming BK max(Z = 14 + 6x3 + 4x4) (80) TP.HCM Contents S.t. 5x3 + 2x4 < 1 Motivated Examples x3 + x4 < 1 Classification Facility Location 0< X3 < 1 Linear and Integer Linear Programs 0< x4 1 Simplex Method Branch & Bound Method Numerical Example 1 With x3 = 0, Z = 16 is still feasible solution at (1, 1, 0, 0.5) Numerical Example 2 Strategy and Steps With x4 = 0, (1, 1, 0, 0), Z = 14 (new optimal solution) Exercise: 0-1 KP With x4 = 1,(1, 1, 0, 1) is not feasible Remarks How to Branch? Which Node to Select? Rule of Fathoming > The current best solution (1, 1, 0, 0) with Z = 14 is the Software for ILP optimal solution. Problems and Homeworks 3.61" }, { "page_index": 568, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_069.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_069.png", "page_index": 568, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:54+07:00" }, "raw_text": "Numerical Example 2 - LP Relaxation: 1 = 1, 2 = 1 Integer Linear Programming BK max(Z = 14 + 6x3 + 4x4) (80) TP.HCM Contents S.t. 5x3 + 2x4 < 1 Motivated Examples x3 + x4 < 1 Classification Facility Location 0< X3 < 1 Linear and Integer Linear Programs 0< x4 1 Simplex Method Branch & Bound Method Numerical Example 1 With 3 = 0, Z = 16 is still feasible solution at (1, 1, 0, 0.5) Numerical Example 2 Strategy and Steps With x4 = 0, (1,1, 0, 0), Z = 14 (new optimal solution Exercise: 01 KP With x4 = 1, (1, 1, 0,1) is not feasible Remarks How to Branch? . With x3 = 1, No feasible solution Which Node to Select? Rule of Fathoming > The current best solution (1, 1, 0, 0) with Z = 14 is the Software for ILP optimal solution. Problems and Homeworks 3.61" }, { "page_index": 569, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_070.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_070.png", "page_index": 569, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:53:58+07:00" }, "raw_text": "Numerical Example 2 (cont') - LP relaxation: conclusion Integer Linear Programming BK TP.HCM (0,1,0,1) Z=9 (1,0,0,0) Z = 9 Contents (5,1,0,1) (1,0,0.8,0 Motivated Examples Z=16.5 Z=13.8 Classification Facility Location (1,1,1,0) NF Linear and Integer Linear Programs (1, 0.8, 0, 0.8) Z = 16.2 Simplex Method (1,1,0,0) z=14 Branch & Bound Method (1,1,0.0.5) Numerical Example1 Z=16 Numerical Example 2 Strategy and Steps (1,1,0,0.5) Z=16 (1,1,0,1) NF Exercise: 0-1 KP Remarks (1,1,1,0.5) NF How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.62" }, { "page_index": 570, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_071.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_071.png", "page_index": 570, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:00+07:00" }, "raw_text": "Branch & Bound: General Strategy Integer Linear Programming BK TP.HCM Solve the linear relaxation of the problem. lf the solution is Contents integer, then we are done. Otherwise create two new Motivated Examples subproblems by branching on a fractional variable Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.63" }, { "page_index": 571, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_072.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_072.png", "page_index": 571, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:03+07:00" }, "raw_text": "Branch & Bound: General Strategy Integer Linear Programming BK TP.HCM Solve the linear relaxation of the problem. If the solution is Contents integer, then we are done. Otherwise create two new Motivated Examples subproblems by branching on a fractional variable Classification : A node (subproblem) is not active when any of the following Facility Location Linear and Integer occurs: Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.63" }, { "page_index": 572, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_073.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_073.png", "page_index": 572, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:05+07:00" }, "raw_text": "Branch & Bound: General Strategy Integer Linear Programming BK TP.HCM Solve the linear relaxation of the problem. If the solution is Contents integer, then we are done. Otherwise create two new Motivated Examples subproblems by branching on a fractional variable Classification : A node (subproblem) is not active when any of the following Facility Location Linear and Integer occurs: Linear Programs 1 The node is being branched on; Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.63" }, { "page_index": 573, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_074.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_074.png", "page_index": 573, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:09+07:00" }, "raw_text": "Branch & Bound: General Strategy Integer Linear Programming BK TP.HCM Solve the linear relaxation of the problem. If the solution is Contents integer, then we are done. Otherwise create two new Motivated Examples subproblems by branching on a fractional variable Classification : A node (subproblem) is not active when any of the following Facility Location Linear and Integer occurs: Linear Programs 1 The node is being branched on; Simplex Method 2 The solution is integral; Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.63" }, { "page_index": 574, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_075.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_075.png", "page_index": 574, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:12+07:00" }, "raw_text": "Branch & Bound: General Strategy Integer Linear Programming BK TP.HCM Solve the linear relaxation of the problem. If the solution is Contents integer, then we are done. Otherwise create two new Motivated Examples subproblems by branching on a fractional variable Classification : A node (subproblem) is not active when any of the following Facility Location Linear and Integer occurs: Linear Programs 1 The node is being branched on; Simplex Method The solution is integral; Branch & Bound Method 3 The subproblem is infeasible; Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.63" }, { "page_index": 575, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_076.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_076.png", "page_index": 575, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:15+07:00" }, "raw_text": "Branch & Bound: General Strategy Integer Linear Programming BK TP.HCM Solve the linear relaxation of the problem. If the solution is Contents integer, then we are done. Otherwise create two new Motivated Examples subproblems by branching on a fractional variable Classification : A node (subproblem) is not active when any of the following Facility Location Linear and Integer occurs: Linear Programs 1 The node is being branched on; Simplex Method The solution is integral; Branch & Bound Method The subproblem is infeasible; Numerical Example 1 You can fathom the subproblem by a bounding argument. Numerical Example 2 4 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.63" }, { "page_index": 576, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_077.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_077.png", "page_index": 576, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:19+07:00" }, "raw_text": "Branch & Bound: General Strategy Integer Linear Programming BK TP.HCM Solve the linear relaxation of the problem. If the solution is Contents integer, then we are done. Otherwise create two new Motivated Examples subproblems by branching on a fractional variable Classification : A node (subproblem) is not active when any of the following Facility Location Linear and Integer occurs: Linear Programs 1 The node is being branched on; Simplex Method The solution is integral; Branch & Bound 21 Method 3 The subproblem is infeasible Numerical Example 1 4 You can fathom the subproblem by a bounding argument. Numerical Example 2 Strategy and Steps Choose an active node and branch on a fractional variable Exercise: 0-1 KP Remarks Repeat until there are no active subproblems. How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.63" }, { "page_index": 577, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_078.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_078.png", "page_index": 577, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:22+07:00" }, "raw_text": "Branch & Bound: Steps Integer Linear Programming BK Omitted the integrality constraints from the original problem TP.HCM in order to obtain a relaxation. Thus a LP problem is obtained and solve this LP problem Contents Motivated Examples Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.64" }, { "page_index": 578, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_079.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_079.png", "page_index": 578, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:24+07:00" }, "raw_text": "Branch & Bound: Steps Integer Linear Programming BK Omitted the integrality constraints from the original problem TP.HCM in order to obtain a relaxation. Thus a LP problem is obtained and solve this LP problem Contents . Divide a problem into subproblems Motivated Examples Classification Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.64" }, { "page_index": 579, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_080.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_080.png", "page_index": 579, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:27+07:00" }, "raw_text": "Branch & Bound: Steps Integer Linear Programming BK Omitted the integrality constraints from the original problem TP.HCM in order to obtain a relaxation. Thus a LP problem is obtained and solve this LP problem. Contents Divide a problem into subproblems Motivated Examples Classification : Calculate the LP relaxation of a subproblem Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.64" }, { "page_index": 580, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_081.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_081.png", "page_index": 580, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:30+07:00" }, "raw_text": "Branch & Bound: Steps Integer Linear Programming BK Omitted the integrality constraints from the original problem TP.HCM in order to obtain a relaxation. Thus a LP problem is obtained and solve this LP problem Contents . Divide a problem into subproblems Motivated Examples Classification : Calculate the LP relaxation of a subproblem Facility Location - The LP problem has no feasible solution, done; Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.64" }, { "page_index": 581, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_082.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_082.png", "page_index": 581, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:33+07:00" }, "raw_text": "Branch & Bound: Steps Integer Linear Programming BK Omitted the integrality constraints from the original problem TP.HCM in order to obtain a relaxation. Thus a LP problem is obtained and solve this LP problem Contents . Divide a problem into subproblems Motivated Examples Classification Calculate the LP relaxation of a subproblem Facility Location The LP problem has no feasible solution, done; Linear and Integer Linear Programs The LP problem has an integer optimal solution; done. Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.64" }, { "page_index": 582, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_083.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_083.png", "page_index": 582, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:35+07:00" }, "raw_text": "Branch & Bound: Steps Integer Linear Programming BK Omitted the integrality constraints from the original problem TP.HCM in order to obtain a relaxation. Thus a LP problem is obtained and solve this LP problem Contents Divide a problem into subproblems Motivated Examples Classification : Calculate the LP relaxation of a subproblem Facility Location The LP problem has no feasible solution, done; Linear and Integer Linear Programs The LP problem has an integer optimal solution; done. Simplex Method Compare the optimal solution with the best solution we know Branch & Bound (the incumbent) Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.64" }, { "page_index": 583, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_084.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_084.png", "page_index": 583, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:40+07:00" }, "raw_text": "Branch & Bound: Steps Integer Linear Programming BK Omitted the integrality constraints from the original problem TP.HCM in order to obtain a relaxation. Thus a LP problem is obtained and solve this LP problem Contents Divide a problem into subproblems Motivated Examples Classification : Calculate the LP relaxation of a subproblem Facility Location The LP problem has no feasible solution, done; Linear and Integer Linear Programs The LP problem has an integer optimal solution; done. Simplex Method Compare the optimal solution with the best solution we know Branch & Bound (the incumbent Method The LP problem has an optimal solution that is worse than Numerical Example 1 Numerical Example 2 the incumbent, done. Strategy and Steps In all the cases above, we know all we need to know about Exercise: 0-1 KP Remarks that subproblem. We say that subproblem is fathomed. How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.64" }, { "page_index": 584, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_085.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_085.png", "page_index": 584, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:43+07:00" }, "raw_text": "Branch & Bound: Steps Integer Linear Programming BK Omitted the integrality constraints from the original problem TP.HCM in order to obtain a relaxation. Thus a LP problem is obtained and solve this LP problem Contents Divide a problem into subproblems Motivated Examples Classification : Calculate the LP relaxation of a subproblem Facility Location The LP problem has no feasible solution, done; Linear and Integer Linear Programs . The LP problem has an integer optimal solution; done. Simplex Method Compare the optimal solution with the best solution we know Branch & Bound (the incumbent)) Method Numerical Example 1 The LP problem has an optimal solution that is worse than Numerical Example 2 the incumbent, done. Strategy and Steps In all the cases above, we know all we need to know about Exercise: 0-1 KP Remarks that subproblem. We say that subproblem is fathomed. How to Branch? . The LP problem has an optimal solution that are not all Which Node to Select? Rule of Fathoming integer, better than the incumbent. In this case we would Software for ILP have to divide this subproblem further and repeat. Problems and Homeworks 3.64" }, { "page_index": 585, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_086.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_086.png", "page_index": 585, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:47+07:00" }, "raw_text": "Exercise: An 0-1 Knapsack Problem Instance Integer Linear Programming BK TP.HCM Consider the problem Contents max 8x1 +112 + 6x3 + 4x4 Motivated Examples Classification s.t. Facility Location 51 + 7x2 +4x3 + 3x4 < 14 Linear and Integer Linear Programs x e{0,1}4. Simplex Method Branch & Bound Method The linear relaxation solution is = (1,1.0.5,0) with a value Numerical Example 1 of 22. The solution is not integral. Numerical Example 2 Strategy and Steps Choose xa to branch. The next two subproblems will have Exercise0-1KP Remarks x3 = 0 and x3 = 1, respectively How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.65" }, { "page_index": 586, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_087.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_087.png", "page_index": 586, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:50+07:00" }, "raw_text": "Exercise: An 0-1 Knapsack Problem Instance Integer Linear Programming BK TP.HCM The tree search. Z = 22 (1,1,0.5,0) Contents Motivated Examples Classification Facility Location Z = 21.65 (1,1,0,0.67) (1,0.71,1,0) Z = 21.85 Linear and Integer Linear Programs Simplex Method Branch & Bound Z = 18 (1,1,1,1 (0.6,1,1.0) Z = 21.8 Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise0-1KP Z = 21 (0,1,1,1) (1,1,1,0) NF Remarks How to Branch? Theo ptimal solution is x = (0,1,1,1) Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.66" }, { "page_index": 587, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_088.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_088.png", "page_index": 587, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:53+07:00" }, "raw_text": "Outline Integer Linear Programming Motivated Examples Linear Classification Problem BK Facility Location Problem TP.HCM 2 Linear and Integer Linear Programs Contents Motivated Examples 3 Simplex Method for Solving LP Classification Facility Location Branch & Bound Method for Solving ILP Linear and Integer NumericalExample1-IP Linear Programs Simplex Method Numerical Example 2-BinaryIP Branch & Bound Strategy and Steps Method Numerical Example 1 Exercise:0-1 Knapsack Problem Numerical Example 2 Strategy and Steps 5 Remarks on Branch & Bound Method Exercise: 0-1 KP How to Branch? Remarks How to Branch? Which Node to Select? Which Node to Select? Rule of Fathoming Rule of Fathoming Software for ILP Software for Integer Programming Problems and Homeworks Problems and Homeworks 3.67" }, { "page_index": 588, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_089.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_089.png", "page_index": 588, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:54:57+07:00" }, "raw_text": "How to Branch? Integer Linear Programming : We want to divide the current problem into two or more BK subproblems that are easier than the original. A commonly TP.HCM used branching method: Contents Xi= 1 - u; Remarks Y'*a + b <= -(1 - v); How to Branch? u >= 0; Which Node to Select? v >= 0; Rule of Fathoming cvx_end Software for ILP Problems and Homeworks 3.77" }, { "page_index": 598, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_099.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_099.png", "page_index": 598, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:55:34+07:00" }, "raw_text": "Solving the Linear Classification Problem using CVX Integer Linear Programming BK Approximate linear classification via linear programming TP.HCM 8 Contents 6 Motivated Examples Classification 4 O Facility Location O Linear and Integer C C Linear Programs 2 O Simplex Method Branch & Bound O Method H O O Numerical Example 1 O O 000 Numerical Example 2 -2 O O Strategy and Steps O Exercise: 0-1 KP 00 O Remarks .0 How to Branch? O Which Node to Select? O O O Rule of Fathoming -6 Software for ILP & 1 Problems and -8 -6 -4 -2 0 2 4 6 8 10 Homeworks 3.78" }, { "page_index": 599, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_100.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_100.png", "page_index": 599, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:55:38+07:00" }, "raw_text": "Outline Integer Linear Programming Motivated Examples Linear Classification Problem BK Facility Location Problem TP.HCM 2 Linear and Integer Linear Programs Contents Motivated Examples 3 Simplex Method for Solving LP Classification Facility Location 4 Branch & Bound Method for Solving ILP Linear and Integer NumericalExample1-IP Linear Programs Simplex Method Numerical Example 2-BinaryIP Branch & Bound Strategy and Steps Method Numerical Example 1 Exercise:0-1 Knapsack Problem Numerical Example 2 Strategy and Steps @ Remarks on Branch & Bound Method Exercise: 0-1 KP How to Branch? Remarks How to Branch? Which Node to Select? Which Node to Select? Rule of Fathoming Rule of Fathoming Software for ILP Software for Integer Programming Problems and Homeworks 6 Problems and Homeworks 3.79" }, { "page_index": 600, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_101.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_101.png", "page_index": 600, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:55:42+07:00" }, "raw_text": "Transportation Problem Linear Programming Model Integer Linear Programming Statement of an instance BK Suppose a company has 2 factories F1,F, and 9 retail outlets TP.HCM R1, . . ., Rg: : the total supply of the product from each factory F1 is ai; Contents the total demand for the product at each outlet R; is bj; Motivated Examples Classification : The cost of sending one unit of the product from factory F: Facility Location to outlet Rj is equal to Cij' Linear and Integer Linear Programs where i=1,2 and =1,2, .. .,9. Simplex Method The problem is to determine a transportation scheme between the Branch & Bound Method factories and the outlets so as to minimize the total transportation Numerical Example 1 Numerical Example 2 cost, subject to the specified supply and demand constraints.. Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.80" }, { "page_index": 601, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_102.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_102.png", "page_index": 601, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:55:46+07:00" }, "raw_text": "Transportation Problem Linear Programming Model Integer Linear Programming Statement of an instance BK Suppose a company has 2 factories F1,F2 and 9 retail outlets TP.HCM R1, .. .,Rg: . the total supply of the product from each factory F1 is ai: Contents the total demand for the product at each outlet R; is bj; Motivated Examples Classification . The cost of sending one unit of the product from factory F: Facility Location to outlet Rj is equal to Cij' Linear and Integer Linear Programs where i=1,2 and j =1,2,.. .,9. Simplex Method The problem is to determine a transportation scheme between the Branch & Bound Method factories and the outlets so as to minimize the total transportation Numerical Example 1 Numerical Example 2 cost, subject to the specified supply and demand constraints.. Strategy and Steps Exercise: 0-1 KP Remarks : Objective: minimum cost of transporting How to Branch? . Constraint: total supply of the factories and, total demand for Which Node to Select? Rule of Fathoming the product of the outlets. Software for ILP Problems and . Variable: the size of the shipment from F; to Ri, where Homeworks i=1,2andj=1,2,...,9. 3.80" }, { "page_index": 602, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_103.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_103.png", "page_index": 602, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:55:50+07:00" }, "raw_text": "Transportation Problem: Mathematical Formulation Integer Linear Programming BK TP.HCM Variable: xij 0. Contents Ri X11? Motivated Examples Classification x2y? Facility Location C11 F Linear and Integer C12 R2 Linear Programs C19 X22? Simplex Method Branch & Bound Method C21 Numerical Example 1 Numerical Example 2 C22 2ig? Strategy and Steps F2 C29 Exercise: 0-1 KP 29? Rg Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.81" }, { "page_index": 603, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_104.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_104.png", "page_index": 603, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:55:55+07:00" }, "raw_text": "Transportation Problem: Mathematical Formulation Integer Linear Programming BK TP.HCM Variable: xi: Z 0. Contents R T11? : Objective Motivated Examples Classification x2? * transportation cost from F: tc Facility Location C11 C12 X12? Rj: Cijxij Linear and Integer : objective function: Zij Cij ij . Linear Programs R2 C19 X22? Simplex Method Branch & Bound Method C21 Numerical Example 1 Numerical Example 2 C22 2ig? Strategy and Steps F2 C29 Exercise: 0-1 KP 29? Rg Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.81" }, { "page_index": 604, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_105.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_105.png", "page_index": 604, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:55:59+07:00" }, "raw_text": "Transportation Problem: Mathematical Formulation Integer Linear Programming BK TP.HCM Variable: xi 0. Contents R T11? . Objective Motivated Examples Classification x2? transportation cost from F; tc Facility Location C11 F C12 X12? Rj: Cijxij Linear and Integer : objective function: Cij Cijxij . Linear Programs R2 C19 X22? Simplex Method Constraints Branch & Bound Method C21 total supply of F: : Numerical Example 1 Numerical Example 2 C22 Tig? total demand of Rj : Strategy and Steps F2 C29 Exercise: 0-1 KP 29? D=1 xij bj. Rg Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.81" }, { "page_index": 605, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_106.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_106.png", "page_index": 605, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:04+07:00" }, "raw_text": "Transportation Problem: Mathematical Formulation (cont.) Integer Linear Programming BK TP.HCM min Ca jij Contents R a ij Motivated Examples x1? subject to Classification x2? Facility Location C11 F 9 Linear and Integer C12 x12? Linear Programs R2 xijai, i=1,2, (81) C19 Simplex Method j=1 Branch & Bound 2 Method C21 Numerical Example1 xij bj, j=1,...,9, (82 Numerical Example 2 C22 x19? Strategy and Steps i=1 F2 C29 Exercise: 0-1 KP 29? Rg xij 0,i=1,2, j=1,...,9. Remarks (83) How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.82" }, { "page_index": 606, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_107.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_107.png", "page_index": 606, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:07+07:00" }, "raw_text": "Mathematical Formulation Integer Linear Programming BK TP.HCM Contents Motivated Examples Steps Classification Facility Location Linear and Integer Define decision variables Linear Programs Determine the objective function Simplex Method Branch & Bound Establish constraints Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.83" }, { "page_index": 607, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_108.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_108.png", "page_index": 607, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:11+07:00" }, "raw_text": "Knapsack Problems - Integer Linear Programming Model Integer Linear Programming There are many different knapsack problems. The first and classical one is the binary knapsack problem. BK Binary (0-1) knapsack problem TP.HCM A tourist is planning a tour in the mountains. He has a lot of Contents objects which may be useful during the tour. For example ice pick Motivated Examples and can opener can be among the objects. We suppose that Classification Facility Location : Each object has a positive value and a positive weight, the Linear and Integer value is the degree of contribution of the object to the Linear Programs success of the tour; Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.84" }, { "page_index": 608, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_109.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_109.png", "page_index": 608, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:14+07:00" }, "raw_text": "Knapsack Problems - Integer Linear Programming Model Integer Linear Programming There are many different knapsack problems. The first and classical one is the binary knapsack problem. BK Binary (0-1) knapsack problem TP.HCM A tourist is planning a tour in the mountains. He has a lot of Contents objects which may be useful during the tour. For example ice pick Motivated Examples and can opener can be among the objects. We suppose that Classification Facility Location . Each object has a positive value and a positive weight, the Linear and Integer value is the degree of contribution of the object to the Linear Programs success of the tour; Simplex Method Branch & Bound The objects are independent from each other (e.g. can and Method Numerical Example 1 can opener are not independent as any of them without the Numerical Example 2 other one has limited value); Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.84" }, { "page_index": 609, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_110.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_110.png", "page_index": 609, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:19+07:00" }, "raw_text": "Knapsack Problems - Integer Linear Programming Model Integer Linear Programming There are many different knapsack problems. The first and classical one is the binary knapsack problem. BK Binary (0-1) knapsack problem TP.HCM A tourist is planning a tour in the mountains. He has a lot of Contents objects which may be useful during the tour. For example ice pick Motivated Examples and can opener can be among the objects. We suppose that Classification Facility Location . Each object has a positive value and a positive weight, the Linear and Integer value is the degree of contribution of the object to the Linear Programs success of the tour; Simplex Method Branch & Bound The objects are independent from each other (e.g. can and Method Numerical Example 1 can opener are not independent as any of them without the Numerical Example 2 other one has limited value); Strategy and Steps Exercise: 0-1 KP The knapsack of the tourist is strong and large enough to Remarks contain all possible objects; How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.84" }, { "page_index": 610, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_111.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_111.png", "page_index": 610, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:23+07:00" }, "raw_text": "Knapsack Problems - Integer Linear Programming Model Integer Linear Programming There are many different knapsack problems. The first and classical one is the binary knapsack problem. BK Binary (0-1) knapsack problem TP.HCM A tourist is planning a tour in the mountains. He has a lot of Contents objects which may be useful during the tour. For example ice pick Motivated Examples and can opener can be among the objects. We suppose that Classification Facility Location . Each object has a positive value and a positive weight, the Linear and Integer value is the degree of contribution of the object to the Linear Programs success of the tour; Simplex Method Branch & Bound The objects are independent from each other (e.g. can and Method Numerical Example 1 can opener are not independent as any of them without the Numerical Example 2 other one has limited value); Strategy and Steps Exercise: 0-1 KP The knapsack of the tourist is strong and large enough to Remarks contain all possible objects; How to Branch? Which Node to Select? The strength of the tourist makes possible to bring only a Rule of Fathoming Software for ILP limited total weight; Problems and Homeworks 3.84" }, { "page_index": 611, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_112.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_112.png", "page_index": 611, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:29+07:00" }, "raw_text": "Knapsack Problems - Integer Linear Programming Model Integer Linear Programming There are many different knapsack problems. The first and classical one is the binary knapsack problem. BK Binary (0-1) knapsack problem TP.HCM A tourist is planning a tour in the mountains. He has a lot of Contents objects which may be useful during the tour. For example ice pick Motivated Examples and can opener can be among the objects. We suppose that Classification Facility Location . Each object has a positive value and a positive weight, the Linear and Integer value is the degree of contribution of the object to the Linear Programs success of the tour; Simplex Method Branch & Bound The objects are independent from each other (e.g. can and Method Numerical Example 1 can opener are not independent as any of them without the Numerical Example 2 other one has limited value); Strategy and Steps Exercise: 0-1 KP The knapsack of the tourist is strong and large enough to Remarks contain all possible objects; How to Branch? Which Node to Select? The strength of the tourist makes possible to bring only a Rule of Fathoming Software for ILP limited total weight; Problems and Homeworks : But within this weight limit the tourist want to achieve the maximal total value 3.84" }, { "page_index": 612, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_113.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_113.png", "page_index": 612, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:32+07:00" }, "raw_text": "Mathematical Formulation Integer Linear Programming The following notations are used to the mathematical formulation of the problem: BK TP.HCM n the number of objects; j the index of the objects; Contents Wj the weight of object j; Motivated Examples Vj the value of object j; Classification Facility Location b the maximal weight what the tourist can bring Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.85" }, { "page_index": 613, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_114.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_114.png", "page_index": 613, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:36+07:00" }, "raw_text": "Mathematical Formulation Integer Linear Programming The following notations are used to the mathematical formulation of the problem: BK TP.HCM n the number of objects; the index of the objects; Contents Wj the weight of object j; Motivated Examples Vj the value of object j; Classification Facility Location b the maximal weight what the tourist can bring Linear and Integer Linear Programs For each object j a so-called binary or zero-one decision Simplex Method variable, say x, is introduced: Branch & Bound Method Numerical Example 1 if object j is present on the tour Numerical Example 2 Strategy and Steps if object j isn't present on the tour. Exercise: 0-1 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.85" }, { "page_index": 614, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_115.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_115.png", "page_index": 614, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:40+07:00" }, "raw_text": "Mathematical Formulation Integer Linear Programming The following notations are used to the mathematical formulation of the problem: BK TP.HCM n the number of objects; the index of the objects; Contents Wj the weight of object j; Motivated Examples Vj the value of object j; Classification Facility Location b the maximal weight what the tourist can bring Linear and Integer Linear Programs For each object j a so-called binary or zero-one decision Simplex Method variable, say x, is introduced: Branch & Bound Method Numerical Example 1 if object j is present on the tour Numerical Example 2 Strategy and Steps 0 if object j isn't present on the tour. Exercise: 0-1 KP Remarks Notice that How to Branch? Which Node to Select? Rule of Fathoming Wj if object j is present on the tour, Software for ILP W j X j if object j isn't present on the tour Problems and 0 Homeworks is the weight of the object in the knapsack 3.85" }, { "page_index": 615, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_116.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_116.png", "page_index": 615, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:43+07:00" }, "raw_text": "Mathematical Formulation (cont.) Integer Linear Programming Similarly ;j is the value of the object on the tour. The total weight in the knapsack is BK TP.HCM n Vj X j Contents j=1 Motivated Examples Classification which may not exceed the weight limit Facility Location Linear and Integer Linear Programs Simplex Method Branch & Bound Method Numerical Example 1 Numerical Example 2 Strategy and Steps Exercise: 01 KP Remarks How to Branch? Which Node to Select? Rule of Fathoming Software for ILP Problems and Homeworks 3.86" }, { "page_index": 616, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_117.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_117.png", "page_index": 616, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:47+07:00" }, "raw_text": "Mathematical Formulation (cont.) Integer Linear Programming Similarly Uj j is the value of the object on the tour. The total weight in the knapsack is BK TP.HCM 2 Vj Xj Contents j=1 Motivated Examples Classification which may not exceed the weight limit Facility Location Linear and Integer Hence the mathematical form of the problem is as follows Linear Programs Simplex Method Branch & Bound Vj X j (84) Method max Numerical Example 1 c j=1 Numerical Example 2 Strategy and Steps subject to Exercise: 0-1 KP Remarks n C How to Branch? WjXj S b, (85) Which Node to Select? Rule of Fathoming .j=1 Software for ILP xj =0 or 1, j =1,.. (86) .n Problems and Homeworks 3.86" }, { "page_index": 617, "chapter_num": 3, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_118.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_3/slide_118.png", "page_index": 617, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:56:51+07:00" }, "raw_text": "Solving the problem in this case Integer Linear Programming The difficulty of the problem is caused by the integrality requirement. If constraint (86) is substituted by the relaxed BK constraint, i.e. by TP.HCM 01, Deterministic finite automata iteration or star operation Recognized languages L*=ULi=LULUL2U...ULU... Determinisation Minimization We will use also the notation + DFAs combination Some applications L+=UL=LULU...ULiU... The union, product and iteration are called regular operations 4.11" }, { "page_index": 649, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_017.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_017.png", "page_index": 649, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:58:41+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong.LHTrang BK TP.HCM Let E={a,b,c}, L1 ={ab,aa,b},L2={b,ca,bac} Contents Motivation a L1 UL2 =?, Alphabets, words and L1 nL2=?. languages b Regular expression or L1L2 =?, rationnal expressior Non-deterministic L1L2 =?, finite automata L2L1 =?. Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.12" }, { "page_index": 650, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_018.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_018.png", "page_index": 650, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:58:44+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Let E={a,b,c},L1 ={ab,aa,b},L2={b,ca,bac} Contents Motivation a L1 U L2 ={ab,aa,b,ca,bac} Alphabets,words and L1NL2={b}, languages Regular expression or L1L2 ={ab,aa} rationnal expressior Non-deterministic @ L1L2 = {abb, aab, bb, abca, aaca, bca, abbac, aabac, bbac} finite automata @ L2L1 = {bab, baa, bb, caab, caaa, cab, bacab, bacaa, bacb} Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.12" }, { "page_index": 651, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_019.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_019.png", "page_index": 651, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:58:47+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Let E={a,b,c}, L1 ={ab,aa,b},L2={b,ca,bac} @ L1 UL2={ab,aa,b,ca,bac} Contents Motivation b L1NL2={b} Alphabets,words and L1L2 ={ab,aa} languages Regular expression or L1L2 = {abb, aab, bb, abca, aaca, bca, abbac, aabac, bbac} rationnal expressior e L2L1 = {bab, baa, bb, caab, caaa, cab, bacab, bacaa, bacb} Non-deterministic finite automata Deterministic finite automata Let E={a,b,c} and L={ab,aa,b,ca,bac} Recognized languages L2 =? Determinisation Minimization DFAs combination Some applications 4.12" }, { "page_index": 652, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_020.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_020.png", "page_index": 652, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:58:50+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c},L1 ={ab,aa,b},L2={b,ca,bac} BK @ L1 UL2 ={ab,aa,b,ca,bac} TP.HCM b L1NL2={b}, L1L2={ab,aa} Contents d L1L2 ={abb, aab,bb, abca, aaca, bca, abbac, aabac, bbac} Motivation Alphabets, words and @ L2Li ={bab, baa, bb, caab, caaa, cab, bacab, bacaa, bacb} languages Regular expression or rationnal expression Let E={a,b,c} and L={ab,aa,b,ca,bac} Non-deterministic finite automata L2 = u., with u, E L including the following strings: Deterministic finite automata abab, abaa, abb, abca, abbac, Recognized languages aaab, aaaa, aab, aaca, aabac, Determinisation Minimization bab, baa, bb, bca, bbac, DFAs combination caab, caaa, cab, caca, cabac Some applications bacab, bacaa, bacb, bacca, bacbac. 4.12" }, { "page_index": 653, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_021.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_021.png", "page_index": 653, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:58:53+07:00" }, "raw_text": "Exercise Automata TVHoai,HTNguyen NAKhuong, LHTrang Let E={a,b,c} BK Give at least 5 strings for each of the following languages TP.HCM all strings with exactly one 'a'. Contents all strings of even length Motivation all strings which the number of appearances of 'b' is divisible by 3. Alphabets,words and languages all strings ending with 'a'. Regular expression or rationnal expression all non-empty strings not ending with 'a'. Non-deterministic finite automata all strings with at least one 'a'. Deterministic finite automata all strings with at most one 'a'. Recognized languages all strings without any 'a'. Determinisation all strings including at least one 'a' and whose the first appearance Minimization 9. of 'a' is not followed by 'c'. DFAs combination Some applications 4.13" }, { "page_index": 654, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_022.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_022.png", "page_index": 654, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:58:55+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c} and L={ab,aa,b,ca,bac} BK Which of the following strings are in L*? TP.HCM S1 Contents Motivation s3 bbb, Alphabets,words and languages $4 aab, Regular expression or rationnal expression S5 cc, Non-deterministic s6 aaaabaaaa = a4ba4 finite automata Deterministic finite s7 cabbbbaaaaaaaaab = cab4a b, automata Recognized languages s baaaaabaaaab = ba5bab, Determinisation $9 baaaaabaac = ba5ba2c, Minimization DFAs combination 1 baca. Some applications 4.14" }, { "page_index": 655, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_023.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_023.png", "page_index": 655, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:58:58+07:00" }, "raw_text": "Regular expressions Automata TVHoai, HTNguyen, NAKhuong, LHTrang Regular expressions (biéu thüc chinh quy) Permit to specify a language with strings consist of letters and &, BK parentheses O, operating symbols +, :, *. This string can be TP.HCM empty, denoted . Contents Regular operations on the languages Motivation Alphabets, words and languages union U or + Regular expression or rationnalexpression product of concatenation Non-deterministic finite automata transitive closure * Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.15" }, { "page_index": 656, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_024.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_024.png", "page_index": 656, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:01+07:00" }, "raw_text": "Regular expressions Automata TVHoai, HTNguyen, NAKhuong, LHTrang Regular expressions (biéu thúc chinh quy) Permit to specify a language with strings consist of letters and &, BK parentheses O, operating symbols +, :, *. This string can be TP.HCM empty, denoted . Contents Regular operations on the languages Motivation Alphabets, words and languages union U or + Regular expression or rationnalexpression product of concatenation Non-deterministic transitive closure * finite automata Deterministic finite automata Recognized languages Example on the aphabet set ={a,b} Determinisation Minimization (a + b)* represent all the strings DFAs combination a*(ba*)* represent the same language Some applications (a + b)*aab represent all strings ending with aab 4.15" }, { "page_index": 657, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_025.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_025.png", "page_index": 657, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:04+07:00" }, "raw_text": "Regular expressions Automata TVHoai, HTNguyen, NAKhuong, LHTrang is a regular expression representing the empty language ε is a regular expression representing language {ε}. BK TP.HCM If a E E, then a is a regular expression representing language {a} If , y are regular expressions representing languages X and Y Contents respectively, then (x + y), (xy), x* are regular expression Motivation representing languages X UY, XY and X* respectively Alphabets, words and languages Regular expression or rationnalexpression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.16" }, { "page_index": 658, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_026.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_026.png", "page_index": 658, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:10+07:00" }, "raw_text": "Regular expressions Automata TVHoai, HTNguyen, NAKhuong, LHTrang is a regular expression representing the empty language a is a regular expression representing language {}. BK TP.HCM If a E E, then a is a regular expression representing language {a} If , y are regular expressions representing languages X and Y Contents respectively, then (x + y), (xy), x* are regular expression Motivation representing languages XUY, XY and X* respectively Alphabets, words and languages x+y = y+x Regular expression or (x+y) + z rationnalexpression x+(y+z) Non-deterministic (xy)z x(yz) finite automata Deterministic finite (x+y)z xz+yz automata Recognized languages x(y+ z) = xy+xz Determinisation (x+y)* = (x*+y)* (x+y*)* = (x*+y*)* Minimization (x + y)* (x*y*)* DFAs combination = Some applications (x*)* = :x* x*x - xx* xx*+E = x* 4.16" }, { "page_index": 659, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_027.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_027.png", "page_index": 659, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:12+07:00" }, "raw_text": "Regular expressions Automata TVHoai, HTNguyen, NAKhuong,LHTrang BK TP.HCM Contents Motivation Kleene's theorem Alphabets, words and Language L C * is regular if and only if there exists a regular languages Regular expression or expression over representing language L. rationnalexpression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.17" }, { "page_index": 660, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_028.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_028.png", "page_index": 660, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:15+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen. NAKhuong, LHTrang BK Let E={a,b,c} TP.HCM Give at least 3 words for each language represented by the Following regular expressions Contents Motivation E1 = a*+ b*. Alphabets, words and languages E2 = a*b+ b*a, Regular expression or E3 = b(ca +ac)(aa)* + a*(a + b) rationnalexpression Non-deterministic 4 E4 =(a*b+b*a)*: finite automata Deterministic finite automata Example Recognized languages Determinisation a*b={b, ab, a2b, ab,..., aaa... ab}, Minimization DFAs combination Some applications 4.18" }, { "page_index": 661, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_029.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_029.png", "page_index": 661, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:19+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c} BK Determine regular expression presenting for each of the following TP.HCM languages. all strings with exactly one 'a'. Contents Motivation all strings of even length Alphabets, words and all strings which the number of appearances of 'b' is divisible by 3 languages Regular expression or all strings ending with 'a'. rationnalexpression Non-deterministic all non-empty strings not ending with 'a'. finite automata all strings with at least one 'a'. Deterministic finite automata all strings with at most one 'a'. Recognized languages Determinisation all strings without any 'a'. Minimization 9 all strings including at least one 'a' and whose the first appearance DFAs combination of 'a' is not followed by a 'c'. Some applications 4.19" }, { "page_index": 662, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_030.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_030.png", "page_index": 662, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:22+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c} and L={ab,aa,b,ca,bac} BK Which languages represented by the following regular expressions TP.HCM are in I*? E1 =a* +ba, Contents Motivation E2 = b* + a*aba*, Alphabets, words and E3 = aab + cab*ac, languages Regular expression or E4 = b(ca + ac)(aa)* + a*(a + b) rationnalexpression E5 = (a4ba3)2*c, Non-deterministic finite automata E6 = b+ac (b+ = bb*). Deterministic finite automata E7 = (b+ c)ab+ ba(c+ a6)* Recognized languages Determinisation 8 Es =(b+ c)*ab + a(c + a)*. Minimization DFAs combination Some applications 4.20" }, { "page_index": 663, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_031.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_031.png", "page_index": 663, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:25+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c} and L={ab,aa,b,ca,bac} BK Which languages represented by the following regular expressions TP.HCM are in I*? 1 E1 =a* +ba, Contents Motivation E2 = b* + a*aba*, Alphabets, words and E3 = aab + cab*ac, languages Regular expression or E4 = b(ca + ac)(aa)* + a*(a + b rationnalexpression E5 = (a4ba3)2*c, Non-deterministic finite automata E6 = b+ac (b+ = bb*) Deterministic finite automata E7 = (b+ c)ab+ ba(c + a6)* Recognized languages Determinisation 8 E8 =(b+ c)*ab + a(c + a)*. Minimization DFAs combination Define a (simple) regular expression representing the language L*: Some applications 4.20" }, { "page_index": 664, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_032.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_032.png", "page_index": 664, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:28+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Simplify each of the following regular expressions Contents E1 = b + ab* + aa*b + aa*ab*, Motivation E2 = a*(b+ ab*), Alphabets, words and languages E3 = e + ab+ abab(ab)* Regular expression or E4 = (ba)* + a(ba)* +(ba)*b + a(ba)*b rationnal expression Non-deterministic E5 = aa(b* + a) + a(ab* + aa), finite automata E6 = (a*(ba)*)*(b+ ε). Deterministic finite automata Er = a(a+ 6)* + aa(a+ b)* + aaa(a +b)* Recognized languages Determinisation Minimization DFAs combination Some applications 4.21" }, { "page_index": 665, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_033.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_033.png", "page_index": 665, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:31+07:00" }, "raw_text": "Finite automata Automata TVHoai, HTNguyen. NAKhuong, LHTrang Finite automata (Automat hüu han) BK The aim is representation of a process system TP.HCM It consists of states (including an initial state and one or several (or one) final/accepting states) and transitions Contents (events). Motivation Alphabets, words and The number of states must be finite languages Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.22" }, { "page_index": 666, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_034.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_034.png", "page_index": 666, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:33+07:00" }, "raw_text": "Finite automata Automata TVHoai, HTNguyen, NAKhuong, LHTrang Finite automata (Automat hüu han) BK The aim is representation of a process system TP.HCM It consists of states (including an initial state and one or several (or one) final/accepting states) and transitions Contents (events). Motivation Alphabets, words and The number of states must be finite languages Regular expression or rationnal expression b Non-deterministic finite automata Deterministic finite automata qo q1 Recognized languages Determinisation Minimization DFAscombination Regular expression Some applications b*(a+ b) 4.22" }, { "page_index": 667, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_035.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_035.png", "page_index": 667, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:37+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b} Which of the strings a3b, BK 1 TP.HCM aba2b, ab2ab3a, Contents abas, Motivation Alphabets, words and ab4ab, languages 5 Regular expression or ba5ba b, rationnal expression Non-deterministic ba5b2. finite automata bab2 a Deterministic finite automata are accepted by the following finite automata? Recognized languages Determinisation a b Minimization b DFAs combination a Some applications qo q2 a 4.23" }, { "page_index": 668, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_036.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_036.png", "page_index": 668, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:40+07:00" }, "raw_text": "Exercise Automata TVHoai,HTNguyen NAKhuong, LHTrang Let E={a,b,c} BK TP.HCM Propose FA presenting each of the following languages all strings with exactly one 'a'. Contents all strings of even length Motivation all strings which the number of appearances of 'b' is divisible by 3. Alphabets, words and languages all strings ending with 'a'. Regular expression or rationnal expression all non-empty strings not ending with 'a'. Non-deterministic finite automata all strings with at least one 'a'. Deterministic finite automata all strings with at most one 'a'. Recognized languages all strings without any 'a'. Determinisation all strings including at least one 'a' and whose the first appearance Minimization 9. of 'a' is not followed by a 'c'. DFAs combination Some applications 4.24" }, { "page_index": 669, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_037.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_037.png", "page_index": 669, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:44+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Give regular expression for the following finite automata. BK TP.HCM b 1 Contents a a qo 91 Motivation qo 91 Alphabets,words and languages b Regular expression or a rationnalexpression Non-deterministic finite automata b a qo q1 Deterministic finite 9o q1 automata Recognized languages Determinisation Minimization a DFAs combination a Some applications q2 q2 4.25" }, { "page_index": 670, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_038.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_038.png", "page_index": 670, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:48+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Give regular expression for the following finite automata. BK b TP.HCM a a Contents qo q2 Motivation Alphabets, words and languages a Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.26" }, { "page_index": 671, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_039.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_039.png", "page_index": 671, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:50+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Give regular expression for the following finite automata. BK b TP.HCM a b a Contents qo 12 Motivation b Alphabets,words and languages a Regular expression or rationnal expression Non-deterministic and this one. finite automata Deterministic finite automata b b Recognized languages Determinisation Minimization qo q1 DFAs combination Some applications a 4.26" }, { "page_index": 672, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_040.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_040.png", "page_index": 672, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:53+07:00" }, "raw_text": "Nondeterministic finite automata Automata TVHoai, HTNguyen, NAKhuong, LHTrang Definition BK TP.HCM A nondeterministic finite automata (NFA, Automat hüu han ph don dinh) is mathematically represented by a 5-tuples Q,E,qo,,F) where Contents Motivation Q a finite set of states. Alphabets, words and languages I is the alphabet of the automata. Regular expression or qo E Q is the initial state rationnal expression Non-deterministic : Q x -> Q is a transition function. finite automata Deterministic finite F C Q is the set of final/accepting states automata Recognized languages Determinisation Remark Minimization According to an event, a state may go to one or more states. DFAs combination Some applications 4.27" }, { "page_index": 673, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_041.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_041.png", "page_index": 673, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:56+07:00" }, "raw_text": "NFA with empty symbol a Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Other definition of NFA Finite automaton with transitions defined by character x (in ) or Contents Motivation empty character e. Alphabets, words and languages b Regular expression or a a rationnal expression b Non-deterministic € finite automata Deterministic finite qo q1 q2 automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.28" }, { "page_index": 674, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_042.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_042.png", "page_index": 674, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T05:59:59+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong,LHTrang BK TP.HCM Contents Motivation Consider the set of strings on {a,b} in which every aa is followed Alphabets, words and immediately by b. languages For example aab, aaba, aabaabbaab are in the language, Regular expression or rationnal expression but aaab and aabaa are not. Non-deterministic Construct an accepting NFA finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.29" }, { "page_index": 675, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_043.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_043.png", "page_index": 675, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:01+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Let E={a,b,c} Contents Motivation Propose NFA presenting each of the following languages Alphabets, words and languages all strings with exactly one 'a'. Regular expression or all strings of even number of appearances of 'b'. rationnal expression Non-deterministic all strings which the number of appearances of 'b' is divisible by 3. finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.30" }, { "page_index": 676, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_044.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_044.png", "page_index": 676, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:04+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c} BK Construct an accepting finite automata for languages represented TP.HCM by the following regular expressions Ei =a*c+b*a, Contents E2 = b*ab+ a*aba*, Motivation Alphabets, words and E3 = aab + cab*ac, languages Regular expression or E4 = b(ca + ac)(aa)* + a*(a + b) rationnal expression E5 = (ab)2*c + bac, Non-deterministic finite automata E6 = bb*ac+ b*a, Deterministic finite automata E7 = (b + c)ab+ ba(c + ab)* Recognized languages E8 = (b + c)*ba + a(c + a)* Determinisation Minimization Eg=[a(b+c)*+bc*]* DFAs combination E1o = b*ac+ bb*a. Some applications 4.31" }, { "page_index": 677, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_045.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_045.png", "page_index": 677, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:08+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b} BK Give 3 valid strings & 5 invalid strings in language L2, with L TP.HCM represented by the following finite automata. Contents a Motivation Alphabets, words and languages B Regular expression or rationnal expression Non-deterministic a finite automata Deterministic finite 6 automata Recognized languages Determinisation 6 Minimization DFAs combination a Some applications 4.32" }, { "page_index": 678, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_046.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_046.png", "page_index": 678, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:10+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b} BK Give 3 valid strings & 5 invalid strings in language L2, with L TP.HCM represented by the following finite automata. Contents a Motivation Alphabets, words and languages a,E 4 B Regular expression or rationnal expression Non-deterministic a finite automata Deterministic finite 6 automata Recognized languages a Determinisation Minimization DFAs combination a Some applications 4.33" }, { "page_index": 679, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_047.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_047.png", "page_index": 679, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:13+07:00" }, "raw_text": "Deterministic finite automata Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK Definition TP.HCM A deterministic finite automata (DFA, Automat hüu han don dinh) is given by a 5-tuplet (Q,,9o,,F) with Contents Q a finite set of states. Motivation Alphabets, words and is the input alphabet of the automata. languages Qo E Q is the initial state. Regular expression or rationnal expression : Q x - Q is a transition function Non-deterministic finite automata F C Q is the set of final/accepting states. Deterministic finite automata Recognized languages Condition Determinisation Transition function is an application Minimization DFAs combination Some applications 4.34" }, { "page_index": 680, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_048.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_048.png", "page_index": 680, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:17+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b} Hereinafter, a deterministic and complete automata that BK TP.HCM recognizes the set of strings which contain an odd number of a Contents b b Motivation a Alphabets, words and languages qo q1 Regular expression or rationnal expression Non-deterministic a finite automata Deterministicfinite automata Recognized languages Determinisation Q={qo,q1} Minimization (qo,a) =q1, (qo,b) = qo, (q1,a) = qo, (q1,b) =q1 DFAs combination F={q1}. Some applications 4.35" }, { "page_index": 681, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_049.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_049.png", "page_index": 681, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:20+07:00" }, "raw_text": "Configurations and executions Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK Let A=(Q,E,qo,,F) TP.HCM A configuration (c&u hinh) of automata A is a couple (q,u) where Contents q E Q and u E E*. Motivation We define the relation -> of derivation between configurations : Alphabets,words and (q,a.u) -(q',u) iif (q,a) = q' languages Regular expression or rationnal expression Non-deterministic finite automata An execution (thuc thi) of automata A is a sequence of Deterministic finite configurations automata Recognized languages (qo,uo)...(qn,un) such that (qi,ui) - (qi+1, ui+1), for i=0,1, . .,n-1. Determinisation Minimization DFAs combination Some applications 4.36" }, { "page_index": 682, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_050.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_050.png", "page_index": 682, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:23+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen. NAKhuong, LHTrang BK TP.HCM Let E={0,1} Give a DFA that accepts all words that contain a number of 0 Contents Motivation multiple of 3. Alphabets, words and Give an execution of this automata on 1101010 languages Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.37" }, { "page_index": 683, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_051.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_051.png", "page_index": 683, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:25+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Let E={0,1} Contents Give a DFA that accepts all words that contain a number of 0 Motivation multiple of 3 Alphabets, words and Give an execution of this automata on 1101010 languages Regular expression or rationnal expression Let E={a,b} Non-deterministic finite automata Deterministic finite Give a DFA that accepts all strings containing 2 characters a. automata Give an execution of this automata on aabb, ababb and bbaa. Recognized languages Determinisation Minimization DFAs combination Some applications 4.37" }, { "page_index": 684, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_052.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_052.png", "page_index": 684, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:29+07:00" }, "raw_text": "Recognized languages Automata TVHoai,HTNguyen NAKhuong, LHTrang BK TP.HCM Definition A language L over an alphabet , defined as a sub-set of *, is Contents recognized if there exists a finite automata accepting all strings of Motivation L. Alphabets, words and languages Regular expression or Proposition rationnal expression Non-deterministic If L1 and L2 are two recognized languages, then finite automata o L1 U L2 and L1 N L2 are also recognized; Deterministic finite automata . L1.L2 and L* are also recognized. Recognized languages Determinisation Minimization DFAs combination Some applications 4.38" }, { "page_index": 685, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_053.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_053.png", "page_index": 685, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:31+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK Sub-string ab TP.HCM Construct a DFA that recognizes the language over the alphabet {a,b} containing the sub-string ab. Contents Motivation Alphabets, words and Regular expression languages (a + b)*ab(a+ b)* Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.39" }, { "page_index": 686, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_054.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_054.png", "page_index": 686, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:35+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK Sub-string ab TP.HCM Construct a DFA that recognizes the language over the alphabet {a,b} containing the sub-string ab. Contents Motivation Regular expression Alphabets, words and languages (a + b)*ab(a + b)* Regular expression or Automata rationnal expression b a, b Non-deterministic a finite automata Transition table a b Deterministic finite a automata > qo qo q1 q2 Recognized languages q1 qo Determinisation q1 q1 q2 Minimization q2* q2 q2 DFAs combination Some applications 4.39" }, { "page_index": 687, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_055.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_055.png", "page_index": 687, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:39+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK Let E={a,b,c} TP.HCM Propose DFA presenting each of the following languages Contents all strings which the number of appearances of 'aa' and the one of Motivation b' are the same. Alphabets, words and all strings which the number of appearances of 'a' is equal to the languages one of 'b' plus the one of 'c'. Regular expression or rationnal expression 3 all strings including at least one 'a' and whose the first appearance Non-deterministic finite automata of 'a' is not followed by a 'c'. Deterministic finite all strings which the difference between number of appearances of automata Recognized languages a' and the one of 'c' is less than 1. Determinisation @ all strings which there is at least 'b' or 'cb' after 'a' or 'aa'. Minimization DFAs combination Some applications 4.40" }, { "page_index": 688, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_056.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_056.png", "page_index": 688, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:42+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Let E={a,b,c} Construct DFAs that recognize the languages represented by the Contents following regular expressions Motivation Alphabets, words and . E1 =a*+b*a, languages E2 = b* + a*aba*. Regular expression or rationnal expressior E3 = aab + cab*ac, Non-deterministic finite automata E4 = bb*ac+ b*a, Deterministic finite automata E5 = b*ac + 66*a Recognized languages Determinisation Minimization DFAs combination Some applications 4.41" }, { "page_index": 689, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_057.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_057.png", "page_index": 689, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:45+07:00" }, "raw_text": "From NFA to DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK Given a NFA TP.HCM a Contents Motivation Alphabets, words and Transition table languages Regular expression or a b rationnal expression Non-deterministic finite automata a Deterministic finite automata € Recognized languages Determinisation 2 Minimization DFAs combination Some applications 4.42" }, { "page_index": 690, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_058.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_058.png", "page_index": 690, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:48+07:00" }, "raw_text": "From NFA to DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK Given a NFA TP.HCM b a Contents Motivation Transition table Alphabets, words and languages a b Regular expression or rationnal expression >{0} {1}{0} Non-deterministic finite automata a Deterministic finite automata € Recognized languages Determinisation 2 Minimization DFAs combination Some applications 4.42" }, { "page_index": 691, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_059.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_059.png", "page_index": 691, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:51+07:00" }, "raw_text": "From NFA to DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK Given a NFA TP.HCM b a Contents Transition table Motivation Alphabets,words and languages a Regular expression or >{0} {1} {0} rationnal expression {1} {0,2} {1} Non-deterministic finite automata a Deterministic finite automata Recognized languages Determinisation 2 Minimization DFAs combination Some applications 4.42" }, { "page_index": 692, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_060.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_060.png", "page_index": 692, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:54+07:00" }, "raw_text": "From NFA to DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK Given a NFA TP.HCM b a Contents Transition table Motivation Alphabets,words and 6 languages a Regular expression or ->{0} {1} {0} rationnal expression {1} {0,2} {1} Non-deterministic finite automata a {0,2}* {1} {0} Deterministic finite automata Recognized languages Determinisation 2 Minimization DFAs combination Some applications 4.42" }, { "page_index": 693, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_061.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_061.png", "page_index": 693, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:00:58+07:00" }, "raw_text": "From NFA to DFA Automata TVHoai, HTNguyen, Transition table NAKhuong, LHTrang a ->{0} {1} {0} BK {1} {0,2} {1} TP.HCM Given a NFA {0,2}* {1} {0} b b Contents a Corresponding DFA Motivation Alphabets,words and b b languages Regular expression or a rationnal expression Non-deterministic {0} {1} finite automata a Deterministic finite automata Recognized languages € Determinisation a a Minimization DFAs combination Some applications {0,2} 4.42" }, { "page_index": 694, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_062.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_062.png", "page_index": 694, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:02+07:00" }, "raw_text": "Other example of determinisation Automata TVHoai, HTNguyen, NAKhuong, LHTrang Given a NFA BK TP.HCM b b a,b Contents Motivation Alphabets,words and languages a Regular expression or rationnal expression Non-deterministic finite automata Transition table Deterministic finite automata a b Recognized languages ->{0} {1} {0,1} Determinisation {1}* {0} {1} Minimization {0,1}* {0,1} {0,1} DFAs combination Some applications 4.43" }, { "page_index": 695, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_063.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_063.png", "page_index": 695, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:06+07:00" }, "raw_text": "Other example of determinisation Automata TVHoai, HTNguyen, NAKhuong, LHTrang Given a NFA BK Corresponding DFA TP.HCM b b b a,b Contents a Motivation {0} {1} Alphabets,words and languages a Regular expression or rationnal expression a Non-deterministic finite automata Transition table Deterministic finite automata a b Recognized languages ->{0} {1} {0,1} Determinisation {1}* {0} {1} a,b {0,1} Minimization {0,1}* {0,1} {0,1} DFAs combination Some applications 4.43" }, { "page_index": 696, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_064.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_064.png", "page_index": 696, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:09+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, Let E={a,b,c} NAKhuong, LHTrang Determine DFAs which corresponds to the following NFAs: BK TP.HCM a a,E Motivation Alphabets, words and a C,€ languages Regular expression or rationnalexpression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.44" }, { "page_index": 697, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_065.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_065.png", "page_index": 697, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:12+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, Let E={a,b,c} NAKhuong, LHTrang Determine DFAs which corresponds to the following NFAs: BK TP.HCM b,c a b a. Votivatior Alphabets,words and a C,€ languages b,c Regular expression or b a rationnalexpressior a Non-deterministic finite automata 6 Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.44" }, { "page_index": 698, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_066.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_066.png", "page_index": 698, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:15+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c} BK TP.HCM Determine DFAs which corresponds to the following NFAs: b Contents Motivation a C Alphabets, words and languages Regular expression or rationnal expression Non-deterministic a finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.45" }, { "page_index": 699, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_067.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_067.png", "page_index": 699, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:18+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c} BK Determine DFAs which corresponds to the following NFAs: TP.HCM b b.c Contents a Motivation Alphabets, words and languages Regular expression or rationnalexpression b Non-deterministic finite automata Deterministic finite a automata Recognized languages 3 Determinisation Minimization DFAs combination a,c Some applications 4.46" }, { "page_index": 700, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_068.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_068.png", "page_index": 700, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:21+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Let E={a,b,c} Contents Determine finite automata, not necessarily deterministic, Motivation recognizing the following languages: Alphabets, words and languages Li ={a,ab,ca,cab,acc} Regular expression or rationnal expression L2 ={ set of words of even number of a} Non-deterministic . L3 ={ set of words containing ab and ending with b} finite automata Deterministic finite Then, determine the corresponding complete DFAs automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.47" }, { "page_index": 701, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_069.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_069.png", "page_index": 701, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:24+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c} Construct DFAs for languages represented by following expressions BK TP.HCM . Ei =a*+b*a, E2 =b* + a*aba*, Contents E3 = (aab + ab*)*, Motivation E4 = b(ca + ac)(aa)* + a*(a + b) Alphabets, words and languages E5 = ba*b + baa + baba, Regular expression or rationnal expressior E6 = (ba*b + baa + baaba)*, Non-deterministic finite automata Er = ba*b + baa + aba(a + b)*. Deterministic finite automata Eg = [a(b + c)* + bc*]* Recognized languages E1o = bb*ac+ ba*b Determinisation E11 = bb*ac + b*a, Minimization DFAs combination E12 =(b+ c)ab+ ba(c+ a6)* Some applications : E13 = (b+ c)*ba+a(c+a)*, 4.48" }, { "page_index": 702, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_070.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_070.png", "page_index": 702, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:26+07:00" }, "raw_text": "Example Automata TVHoai,HTNguyen NAKhuong, LHTrang Determine a DFA that recognizes the language over the alphabet BK {a, b} with an even number of a and an even number b. TP.HCM Contents Motivation Alphabets, words and languages Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.49" }, { "page_index": 703, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_071.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_071.png", "page_index": 703, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:30+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong,LHTrang Determine a DFA that recognizes the language over the alphabet BK {a,b} with an even number of a and an even number b TP.HCM Automata Contents Motivation Alphabets,words and languages Regular expression or qo q1 rationnal expression Non-deterministic finite automata b Deterministic finite automata a a a a Recognized languages Determinisation b Minimization DFAs combination 92 q3 Some applications b 4.49" }, { "page_index": 704, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_072.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_072.png", "page_index": 704, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:34+07:00" }, "raw_text": "Example Automata TVHoai, HTNguyen, NAKhuong, LHTrang Determine a DFA that recognizes the language over the alphabet BK {a,b} with an even number of a and an even number b. TP.HCM Automata Contents Motivation Alphabets,words and Transition table languages Regular expression or qo q1 a b rationnal expression Non-deterministic 92 q1 finite automata 6 q1 q3 qo Deterministic finite automata 92 qo 93 a a a a Recognized languages q3 q1 q2 Determinisation b :: start state Minimization *: final state(s) DFAs combination q2 q3 Some applications b 4.49" }, { "page_index": 705, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_073.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_073.png", "page_index": 705, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:38+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b BK 2 TP.HCM Contents b a a Motivation 1. Alphabets,words and languages Regular expression or a b rationnal expression 3 5 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages 2 3 4 5 Determinisation s 0 1 Minimization cl(s) DFAs combination Some applications 4.50" }, { "page_index": 706, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_074.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_074.png", "page_index": 706, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:42+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b BK 2 TP.HCM Contents b a a a,b Motivation Alphabets,words and languages Regular expression or a b rationnal expression 3 5 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages 2 3 4 5 Determinisation s 0 1 Minimization cl(s) 1 1 1 1 DFAs combination Some applications 4.50" }, { "page_index": 707, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_075.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_075.png", "page_index": 707, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:46+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b BK 2 TP.HCM Contents b a a Motivation Alphabets,words and languages Regular expression or a b rationnal expression 3 5 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages 0 1 2 3 4 5 Determinisation s Minimization cl(s) 1 I1 1 11 1 1 DFAs combination Some applications 4.50" }, { "page_index": 708, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_076.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_076.png", "page_index": 708, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:50+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation cl(s) 1 II II - - 1 Minimization cl(s.a) DFAs combination cl(s.b) Some applications 4.50" }, { "page_index": 709, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_077.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_077.png", "page_index": 709, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:54+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation cl(s) - 1 1I 1 Minimization cl(s.a) 11 DFAs combination cl(s.b) I Some applications 4.50" }, { "page_index": 710, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_078.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_078.png", "page_index": 710, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:01:59+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation cl(s) 11 1 1I 1 Minimization cl(s.a) 11 DFAs combination cl(s.b) I Some applications 4.50" }, { "page_index": 711, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_079.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_079.png", "page_index": 711, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:03+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation cl(s) 1 II 1I 1 Minimization cl(s.a) I1 I 1 DFAs combination cl(s.b) I 1 1 Some applications 4.50" }, { "page_index": 712, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_080.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_080.png", "page_index": 712, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:08+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation cl(s) 1 II II 1 Minimization cl(s.a) I1 - 1 1 DFAs combination cl(s.b) I 1 1 1 Some applications 4.50" }, { "page_index": 713, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_081.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_081.png", "page_index": 713, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:13+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation cl(s) 1 II 1 II - 1 Minimization cl(s.a) I1 I 1 1 I1 DFAs combination cl(s.b) I 1 1 1 1 Some applications 4.50" }, { "page_index": 714, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_082.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_082.png", "page_index": 714, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:19+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation cl(s) 1 II 1 II - 1 Minimization cl(s.a) I1 I 1 1 I1 1 DFAs combination cl(s.b) I 1 1 1 1 Some applications 4.50" }, { "page_index": 715, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_083.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_083.png", "page_index": 715, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:24+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 5 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation 0 1 2 3 4 5 cl(s) 1 II 1 II 1 1 Minimization 1 I1 I1 cl(s.a) 1 - 1 I1 1 DFAs combination cl(s.b) 1 1 1 1 1 1 Some applications 4.50" }, { "page_index": 716, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_084.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_084.png", "page_index": 716, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:30+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a 6 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 5 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation 0 1 2 3 4 5 cl(s) 1 II 1 II 1 1 Minimization 1 I1 I11 I1 cl(s.a) 1 - 1 I1 1 DFAs combination cl(s.b) 1 1 1 1 1 1 Some applications 4.50" }, { "page_index": 717, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_085.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_085.png", "page_index": 717, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:35+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 5 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation 0 1 2 3 4 5 cl(s) 1 II 1 II 1 1 Minimization 1 I1 I11 I1 IV cl(s.a) 1 - 1 I1 1 DFAs combination cl(s.b) 1 1 1 1 1 1 Some applications 4.50" }, { "page_index": 718, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_086.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_086.png", "page_index": 718, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:42+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages b Regular expression or a 3 5 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 Determinisation 0 1 2 3 4 5 cl(s) 1 II 1 II 1 1 Minimization 1 I1 I11 I1 IV I1I cl(s.a) 1 - 1 I1 1 DFAs combination cl(s.b) 1 1 1 1 1 1 Some applications 4.50" }, { "page_index": 719, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_087.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_087.png", "page_index": 719, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:48+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang a 6 2 BK TP.HCM Contents b a a a,b a,b Motivation Alphabets, words and languages Regular expression or a 5 rationnal expression 4 Non-deterministic finite automata Deterministic finite automata equivalence relationships Recognized languages s 0 1 2 3 4 5 1 2 3 4 Determinisation cl(s) II 1 II 1 1 - II III II IV III Minimization cl(s.a) II - - II II IV III IV 11 I1I DFAs combination cl(s.b) II 1 1 1 1 1 II III II1 1 II1 II Some applications 4.50" }, { "page_index": 720, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_088.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_088.png", "page_index": 720, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:02:55+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai,HTNguyen, a NAKhuong,LHTrang 0 2 BK TP.HCM a a a,b a,b Contents Motivation a 3 Alphabets, words and languages Regular expression or rationnal expression equivalence relationships Non-deterministic finite automata s 0 1 2 3 4 5 1 2 3 4 5 Deterministic finite cl(s) II II 1 - II III II IV III automata cl(s.a) II - - - II - II IV III IV II III Recognized languages cl(s.b) I 1 - 1 1 1 II III III 1 II1 III Determinisation Minimization 0 1 2 3 4 5 DFAs combination S cl(s) 1 III IV III Some applications cl(s.a) I IV III II III cl(s.b) III III III II1 4.50" }, { "page_index": 721, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_089.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_089.png", "page_index": 721, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:03+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, a NAKhuong, LHTrang 0 2 BK TP.HCM a a a,b a,b Contents Motivation a 3 Alphabets, words and languages Regular expression or rationnal expression equivalence relationships Non-deterministic finite automata s 0 1 2 3 4 5 1 2 3 4 5 Deterministic finite cl(s) II II 1 - II III II IV III automata cl(s.a) II - - - II - II IV III IV II III Recognized languages cl(s.b) I 1 - 1 1 1 II III III 1 II1 III Determinisation Minimization 1 2 3 4 DFAs combination S cl(s) 1 II III V IV III Some applications cl(s.a) IV III II III cl(s.b) III III III III 4.50" }, { "page_index": 722, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_090.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_090.png", "page_index": 722, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:12+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, a NAKhuong, LHTrang 0 2 BK TP.HCM a a a,b a,b Contents Motivation a 3 Alphabets, words and languages Regular expression or rationnal expression equivalence relationships Non-deterministic finite automata s 0 1 2 3 4 5 1 2 3 4 5 Deterministic finite cl(s) II 1 II 1 - II III II IV III automata cl(s.a) II - - - II II IV III IV II III Recognized languages cl(s.b) I 1 - 1 1 1 II III III 1 II1 III Determinisation Minimization 1 2 3 4 5 DFAs combination S cl(s) 1 II III V IV III Some applications cl(s.a) II IV III II III cl(s.b) V III III III III 4.50" }, { "page_index": 723, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_091.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_091.png", "page_index": 723, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:21+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, a NAKhuong,LHTrang 0 2 BK TP.HCM a a a,b a,b Contents Motivation a 3 Alphabets, words and languages Regular expression or rationnal expression equivalence relationships Non-deterministic finite automata s 0 1 2 3 4 5 1 2 3 4 5 Deterministic finite cl(s) II 1 II 1 - II III II IV III automata cl(s.a) II - - - II II IV III IV II III Recognized languages cl(s.b) I 1 - 1 1 1 II III III 1 II1 III Determinisation Minimization 0 1 2 3 4 5 DFAs combination S cl(s) 1 I1 III V IV II1 Some applications cl(s.a) I1 IV III IV II III cl(s.b) V III III - III III 4.50" }, { "page_index": 724, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_092.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_092.png", "page_index": 724, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:25+07:00" }, "raw_text": "From a DFA to a smaller DFA Automata TVHoai, HTNguyen, NAKhuong, LHTrang {0} a {1} BK a,b TP.HCM b b 6 a a Contents Motivation {2,5} b Alphabets,words and languages {3} a {4} Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite equivalence relationships automata Recognized languages s 0 1 2 3 4 5 Determinisation cl(s) 1 I III V IV 111 Minimization cl(s.a) 1I IV IV II 11I DFAs combination cl(s.b) V 111 I11 I11 I11 Some applications 1 4.51" }, { "page_index": 725, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_093.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_093.png", "page_index": 725, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:30+07:00" }, "raw_text": "Another example of minimization Automata TVHoai, HTNguyen, NAKhuong,LHTrang a a b BK 2 TP.HCM Contents a Motivation Alphabets, words and languages Regular expression or rationnal expression 3 5 Non-deterministic b a finite automata Deterministic finite a automata Recognized languages Determinisation Minimization equivalence relationships DFAs combination s 0 2 3 4 5 Some applications cl(s) 1 1 11 1 1 1 cl(s.a) 1 1 1 1 - 1 cl(s.b) - 1I 11 - - I1 4.52" }, { "page_index": 726, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_094.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_094.png", "page_index": 726, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:36+07:00" }, "raw_text": "Another example of minimization Automata TVHoai, HTNguyen, NAKhuong, LHTrang a a BK 2 TP.HCM Contents a Motivation Alphabets, words and languages Regular expression or rationnal expression 3 4 5 Non-deterministic a finite automata Deterministic finite a automata Recognized languages Determinisation Minimization equivalence relationships DFAs combination s 0 1 2 3 4 5 0 1 2 3 4 5 Some applications cl(s) 1 1 I1 1 - 1 1 III 11 1 1 I11 cl(s.a) 1 1 1 1 I1I 1 1 I1I 1 1 cl(s.b) II I1 - II 1 I1 II 1 - II 4.52" }, { "page_index": 727, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_095.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_095.png", "page_index": 727, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:40+07:00" }, "raw_text": "Another example of minimization Automata TVHoai, HTNguyen, NAKhuong, LHTrang a a BK 2 TP.HCM Contents Motivation a Alphabets, words and languages Regular expression or rationnal expression 3 4 5 Non-deterministic b a finite automata Deterministic finite a a automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.53" }, { "page_index": 728, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_096.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_096.png", "page_index": 728, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:45+07:00" }, "raw_text": "Another example of minimization Automata TVHoai, HTNguyen, NAKhuong, LHTrang a a b BK 2 TP.HCM Contents Motivation a Alphabets, words and languages Regular expression or rationnal expression 3 5 Non-deterministic b a finite automata Deterministic finite a automata Recognized languages Determinisation Minimization equivalence relationships DFAs combination s 0 2 3 4 5 Some applications cl(s) 1 I11 11 1 1 I11 cl(s.a) I11 1 1 I11 1 1 cl(s.b) - I1 I1 - - I1 4.53" }, { "page_index": 729, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_097.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_097.png", "page_index": 729, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:52+07:00" }, "raw_text": "Another example of minimization Automata TVHoai, HTNguyen, NAKhuong, LHTrang a a BK 2 TP.HCM Contents a Motivation Alphabets, words and languages Regular expression or rationnal expression 3 4 5 Non-deterministic a finite automata Deterministic finite a a automata Recognized languages Determinisation Minimization equivalence relationships DFAs combination s 0 1 2 3 4 5 0 1 2 3 4 5 Some applications cl(s) 1 I11 I1 1 I11 1 111 1I 1 IV III cl(s.a) III - 1 III - 1 III IV 1 III IV IV cl(s.b) - II II 1 - 1I IV I1 II IV - 4.53" }, { "page_index": 730, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_098.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_098.png", "page_index": 730, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:03:56+07:00" }, "raw_text": "Another example of minimization Automata TVHoai, HTNguyen, NAKhuong, LHTrang a b BK a {0,3} {1,5} 2 TP.HCM Contents b Motivation a Alphabets,words and languages Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation equivalence relationships Minimization 0 1 2 3 4 5 DFAs combination s III 11 IV I1I Some applications cl(s) 1 1 cl(s.a) I11 IV 1 IV IV cl(s.b) IV II I1 IV 1 II 4.54" }, { "page_index": 731, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_099.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_099.png", "page_index": 731, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:00+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, Let E={a,b} NAKhuong, LHTrang Determine minimal DFA which corresponds, to the following DFA: BK TP.HCM 8 6 Contents Motivation a a a a a Alphabets, words and languages Regular expression or rationnal expression 2 Non-deterministic finite automata Deterministic finite b automata Recognized languages a a b Determinisation Minimization a DFAs combination Some applications a 4.55" }, { "page_index": 732, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_100.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_100.png", "page_index": 732, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:02+07:00" }, "raw_text": "Exercise Automata TVHoai,HTNguyen NAKhuong, LHTrang BK TP.HCM 6 Contents Motivation a Alphabets, words and languages Regular expression or rationnal expression a Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.56" }, { "page_index": 733, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_101.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_101.png", "page_index": 733, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:04+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong,LHTrang BK TP.HCM 6 Contents Motivation a Alphabets, words and languages Regular expression or rationnal expression a Non-deterministic finite automata Deterministic finite Hint automata Recognized languages Two-steps approach: (NFA -> DFA); min (DFA) Determinisation Minimization DFAs combination Some applications 4.56" }, { "page_index": 734, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_102.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_102.png", "page_index": 734, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:06+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong,LHTrang BK TP.HCM o={a,b} Contents Determine minimimal DFA regconized the languages represented Motivation by the following regular expressions: Alphabets, words and languages E1 = (a+b)*b(a+b)* Regular expression or rationnal expressior Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.57" }, { "page_index": 735, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_103.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_103.png", "page_index": 735, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:09+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong,LHTrang BK TP.HCM o={a,b} Contents Determine minimimal DFA regconized the languages represented Motivation by the following regular expressions: Alphabets, words and languages 0 E1 =(a+b)*b(a+b)* Regular expression or rationnal expressior @ E2=((a+b)2)*+((a+b)3)* Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combinatior Some applications 4.57" }, { "page_index": 736, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_104.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_104.png", "page_index": 736, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:12+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM o={a,b} Contents Determine minimimal DFA regconized the languages represented Motivation by the following regular expressions: Alphabets, words and languages E1 =(a+b)*b(a+ b)* Regular expression or 2 E2 =((a+b)2)*+ ((a+b)3)* rationnal expression Non-deterministic @ E3 =((a+b)2)++ ((a+b)3)+ finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.57" }, { "page_index": 737, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_105.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_105.png", "page_index": 737, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:15+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM o={a,b} Contents Determine minimimal DFA regconized the languages represented Motivation by the following regular expressions: Alphabets, words and languages E1 =(a+b)*b(a+ b)* Regular expression or rationnal expression 2 E2 =((a+b)2)*+((a+b)3)* Non-deterministic 3 E3 =((a+b)2)+ +((a+b)3)+ finite automata Deterministic finite 4 E4 = baa* + ab+(a +b)ab*. automata Recognized languages Determinisation Minimization DFAs combinatior Some applications 4.57" }, { "page_index": 738, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_106.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_106.png", "page_index": 738, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:19+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM o={a,b,c,d} Determine minimimal complete DFA regconized the languages Contents Motivation consisting of all strings where all 'a' is followed by a 'b' and all 'c' is Alphabets, words and followed by a 'b'. languages Then, deduce the corresponding regular expressions Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.58" }, { "page_index": 739, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_107.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_107.png", "page_index": 739, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:21+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM o={a,b,c,d} Contents Determine minimimal complete DFA regconized the languages Motivation consisting of all strings where all 'a' is followed by a 'b' and all 'c' is Alphabets, words and followed by a 'b'. languages Then, deduce the corresponding regular expressions Regular expression or rationnal expression Non-deterministic o={a,b} finite automata Deterministic finite Give a NFA (as simple as possible) for the language defined by the automata regular expression ab* + a(ba)*. Then determine the equivalent DFA Recognized languages Determinisation Minimization DFAs combination Some applications 4.58" }, { "page_index": 740, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_108.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_108.png", "page_index": 740, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:24+07:00" }, "raw_text": "Exercise Automata TVHoai, HTNguyen, NAKhuong, LHTrang Let E={a,b,c} BK Determine minimimal DFA regconized the languages represented by the TP.HCM following regular expressions: Contents O a*+b*. Motivation a*b+ b*a, Alphabets, words and languages b(ca+ ac)(aa)*+ a*(a + b Regular expression or (a*b+b*a)*: rationnal expression 4 Non-deterministic a*bc + bca*. finite automata Deterministic finite 6 b(c+ c)(aa)*+ (a+ c)a*, automata Recognized languages aab + cab*ac, Determinisation b(ca + ac)(a)* + a(a + b)*, Minimization 9 ab(b+ c)ab+ ba(c+b)* +(b+ c)ab(b+ c) DFAs combinatior Some applications 4.59" }, { "page_index": 741, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_109.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_109.png", "page_index": 741, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:28+07:00" }, "raw_text": "Equivalent automatons Automata TVHoai, HTNguyen, NAKhuong, LHTrang Two following automatas are eguivalent? BK TP.HCM qo Po P3 Contents a Motivation Alphabets,words and languages a a b b b Regular expression or a a a rationnal expression Non-deterministic a finite automata Deterministic finite a p2 automata q1 92 P1 Recognized languages Determinisation a Minimization 6 DFAs combination Some applications 4.60" }, { "page_index": 742, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_110.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_110.png", "page_index": 742, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:31+07:00" }, "raw_text": "Equivalent automatons Automata TVHoai, HTNguyen, NAKhuong, LHTrang Two following DFAs are equivalent? BK TP.HCM qo Po P3 Contents b Motivation Alphabets,words and languages a a b b b b Regular expression or a a rationnal expression Non-deterministic a finite automata Deterministic finite a p2 automata q1 92 P1 Recognized languages Determinisation a Minimization 6 DFAs combination Some applications 4.61" }, { "page_index": 743, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_111.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_111.png", "page_index": 743, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:34+07:00" }, "raw_text": "Combination of two automata Automata TVHoai, HTNguyen. NAKhuong, LHTrang BK TP.HCM E={a,b} @ Given two languages La, L defined by regular expressions Contents Motivation Ea = a(a + b)* and Eb = a*(ba)* Alphabets, words and Give a DFA for the language La, Lb. languages Regular expression or c) Then, determine a (minimized) DFA for the following languages. rationnal expression Non-deterministic @ L1 =La oLb finite automata @ L2=LaNLb Deterministic finite 3 L3 =LaULb automata 4 L4=LaLb Recognized languages Determinisation Minimization DFAs combination Some applications 4.62" }, { "page_index": 744, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_112.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_112.png", "page_index": 744, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:37+07:00" }, "raw_text": "Combination of two automata Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM E={a,b} @ Given two languages La, L defined by regular expressions Contents Ea =(a*b+b*a)+ and Eb =(a+b)*b(a+b)*a Motivation Alphabets, words and Give a DFA for the language La, Lb. languages Regular expression or c) Then, determine a (minimized) DFA for the following languages. rationnal expression Non-deterministic @ L1=La oLb finite automata @ L2=LaNLb Deterministic finite 3 L3 =LaULb automata 4 L4=LaLb Recognized languages Determinisation Minimization DFAs combination Some applications 4.62" }, { "page_index": 745, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_113.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_113.png", "page_index": 745, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:40+07:00" }, "raw_text": "Combination of two automata Automata TVHoai, HTNguyen. NAKhuong, LHTrang BK TP.HCM E={a,b} @) Given two languages La, L defined by regular expressions Contents Ea = ab* + a(ba)* and Eb = baa* + ab+ (a + b)ab* Motivation Alphabets, words and Give a DFA for the language La, Lb. languages Regular expression or c Then, determine a (minimized) DFA for the following languages rationnal expression Non-deterministic @ L1 =La oLb finite automata @ L2 =LaOLb Deterministic finite 3 L3 =LaULb automata 4 L4=LaLb Recognized languages Determinisation Minimization DFAs combination Some applications 4.62" }, { "page_index": 746, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_114.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_114.png", "page_index": 746, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:42+07:00" }, "raw_text": "Odd Parity Detector Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Contents Motivation Alphabets,words and Describe DFA for Odd Parity Detector languages Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.63" }, { "page_index": 747, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_115.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_115.png", "page_index": 747, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:43+07:00" }, "raw_text": "TCP/IP protocol Automata TVHoai, HTNguyen, NAKhuong, LHTrang BK TP.HCM Contents Motivation Alphabets,words and Describe DFA for a demonstration of TCP/lP protocol languages Regular expression or rationnal expression Non-deterministic finite automata Deterministic finite automata Recognized languages Determinisation Minimization DFAs combination Some applications 4.64" }, { "page_index": 748, "chapter_num": 4, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_116.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_4/slide_116.png", "page_index": 748, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:47+07:00" }, "raw_text": "Application Automata TVHoai, HTNguyen, NAKhuong,LHTrang BK TP.HCM Propose an automata to describe a vehicular multi-information Contents Motivation display system with a given number of buttons. Alphabets, words and languages For example, digital speedo meter of Honda Lead motor with only Regular expression or rationnal expressior one button can display information about: petroleum level, speed Non-deterministic trip, date, time, engine oil life. finite automata Hint: we distinguish two different actions: quickly press the Deterministic finite automata button; press the button and hold-down over two seconds.) Recognized languages Determinisation Minimization DFAs combination Some applications 4.65" }, { "page_index": 749, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_001.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_001.png", "page_index": 749, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:49+07:00" }, "raw_text": "Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen Chapter 6 Dynamical Systems BK TP.HCM Discrete Mathematics Il/Mathematical Modelling Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs Homeworks Nguyen An Khuong, Huynh Tuong Nguyen Faculty of Computer Science and Engineering University of Technology, VNU-HCM 6.1" }, { "page_index": 750, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_002.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_002.png", "page_index": 750, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:51+07:00" }, "raw_text": "Contents Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen BK Introduction TP.HCM Contents Malthusian Growth Model Introduction Malthusian Growth Model Properties of the 3 Properties of the systems systems System of ODEs Homeworks System of ODEs Homeworks 6.2" }, { "page_index": 751, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_003.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_003.png", "page_index": 751, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:54+07:00" }, "raw_text": "Change?! Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen : Dynamical systems: Tools for constructing and manipulating BK models TP.HCM . So we often have to model dynamic systems. Discrete -> difference equations (\"linear\" vs \"nonlinear\" Contents \"single variable\" vs \"multivariate\" Introduction Continuous > differential equations (\"ordinary\" vs \"partial\": - Malthusian Growth Model \"linear\" vs \"nonlinear\") Properties of the We will formulate the equations, analyze their properties and systems System of ODEs Iearn how to solve them. Homeworks To start there are many good references on this subject including: F.R. Giordano, W.P. Fox & S.B. Horton, A First Course in Mathematical Modeling, 5th ed., Cengage, 2014 A Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd. Cambridge University Press, 2008 6.3" }, { "page_index": 752, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_004.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_004.png", "page_index": 752, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:56+07:00" }, "raw_text": "Single Species Equations: Growth Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM Basic concept that individuals divide to increase a population Contents can be modeled mathematically using a differential equation Introduction Can loosely be applied to populations that don't divide to Malthusian Growth Model populate Properties of the systems Attributed to Malthus, who in 1798 found small group of System of ODEs organisms obeyed growth law Homeworks The solution to the equation concerned him greatly 6.4" }, { "page_index": 753, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_005.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_005.png", "page_index": 753, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:04:59+07:00" }, "raw_text": "Exponential Growth Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen As an example, consider the classic example of bacteria on a BK petri dish. TP.HCM Let's say the number on the plate grows by 10% each hour and the initial population is o = 1000. Contents : After the first hour: 1 = 1000 + (0.1) * 1000 = 1000 * (1.1) Introduction Malthusian Growth After the second hour: Model x2 = 1000 * (1.1) + (0.1) * (1000 * 1.1) = Properties of the systems 1000* (1.1) * (1.1) = 1000* (1.1)2 System of ODEs : After the third hour: x3 = 1000 * (1.1)3 etc. Homeworks In general, xt = o(1+ r)t which can be written as t = xoat where a = (1 + r) The solution for the Bacteria is thus an exponential function of base 1 + r. The (discrete) dynamical system: t = axt-1. (recurrent relation/difference equation) 6.5" }, { "page_index": 754, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_006.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_006.png", "page_index": 754, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:02+07:00" }, "raw_text": "Instantaneous Exponential Growth Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen In the last slide we examined growth at some kind of finite increment using an average growth rate r over that increment. What happens if the growth process is continuous? BK TP.HCM . Divide the growth process over the time increment into nt stages with the growth rate for each stage being r where r is Contents the average growth over the increment. Introduction Malthusian Growth Model x(t) = xo(1+ -)) nt (1) Properties of the systems : Look at the limit as we divide our interval into pieces System of ODEs Homeworks lim xo[(1+ 2)n/rrt (2) n-7x We can pull εo out of the limit. In brackets we have the irrational number e: (3) 6.6" }, { "page_index": 755, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_007.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_007.png", "page_index": 755, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:05+07:00" }, "raw_text": "Instantaneous Exponential Growth (cont'd) Dynamical Systems Nguyen An Khuong. Our time-dependent population equation satisfies the Huynh Tuong Nguyen fundamental differential equation: dx BK x : rx (4) TP.HCM dt where r is the (in this situation) constant growth rate. Contents See how to derive the equation above in page 462, [Giordano et al.], eqn(11.5).) Introduction Malthusian Growth The solution to these equation can be found by separating Model variables and integrating. Properties of the systems System of ODEs lnx(t)=rt +c (5) Homeworks and applying the initial condition that x(0) = , at t = 0 to get lnx(t)= rt +lnxo (6) Take the exponential of both sides to give the exact solution to the instantaneous growth equation x(t) = xoert (7) 6.7" }, { "page_index": 756, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_008.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_008.png", "page_index": 756, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:09+07:00" }, "raw_text": "Exponential Growth: Solution properties Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen We can calculate the time to double the population (known as the \"rule of 70\" in financial circles) BK TP.HCM ln(2) .70 tdouble (8) r ? Contents What does the solution look like? On a log plot, it is a Introduction straight line of slope r. Malthusian Growth Model P Properties of the systems System of ODEs 4 Homeworks + ls this realistic? 6.8" }, { "page_index": 757, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_009.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_009.png", "page_index": 757, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:11+07:00" }, "raw_text": "What if our growth rate was negative?: Mortality Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen x = -mx (9) BK We have exponential decay. Solution will tend toward zero, no TP.HCM matter what the initia/ condition was Contents Introduction p Malthusian Growth Model Properties of the systems System of ODEs Homeworks t 70 Most systems have both growth and decay terms Example, our phytoplankton equation will have terms for growth as a function of light, temperature, and nutrients, and decay (mortality). 6.9" }, { "page_index": 758, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_010.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_010.png", "page_index": 758, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:14+07:00" }, "raw_text": "More realistic model: Finite Resources Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen We now know that growth cannot continue forever because of BK finite resources and in fact in simplified scenarios will reach a TP.HCM given constant level K know as the carrying capacity of the system. Contents . Verhulst noticed that simple populations appear to be capped Introduction and added an additional term to remove excess capacity Malthusian Growth Model Properties of the 1 systems x =rx(1 (10) System of ODEs K Homeworks We can always nondimensionalize this by the carrying capacity to give x=rx(1 -x) (11) This is commonly known as /ogistic growth 6.10" }, { "page_index": 759, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_011.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_011.png", "page_index": 759, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:17+07:00" }, "raw_text": "Logistic Growth: Solution Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM This equation has solutions that tend toward K for all initial conditions except for x = o for which there is no growth Contents (assuming there is no such thing as a negative population Introduction Malthusian Growth So, this is mathematically equivalent to a density dependent Model growth rate r' = r(1 - x). Properties of the systems Solution to this equation is System of ODEs Homeworks x.K (12) 6.11" }, { "page_index": 760, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_012.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_012.png", "page_index": 760, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:20+07:00" }, "raw_text": "Arbitrary Growth: Polynomial Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen These equations are of the general form BK x = g(x) (13) TP.HCM where g(x) is a polynomial: g(x) = ao + a1x + a2x Contents . Malthus, all coefficients are zero except for a1 = r. Growth Introduction rate constant Malthusian Growth Verhults (logistic), growth rate decreases monotonically Model ao = 0 by argument that a population can't spontaneously Properties of the systems exist System of ODEs One can fit any population, but will not elucidate any Homeworks dynamics Allee effect: Growth rate might be maximal somewhere in between x = 0 and the carrying capacity. Low numbers, can't find mates, high numbers, competition/resources. (14) If a1 > 0 and az < 0 we have the Allee effect 6.12" }, { "page_index": 761, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_013.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_013.png", "page_index": 761, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:23+07:00" }, "raw_text": "Critical Points (Fixed Points) Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen Fixed points are steady state solutions of ordinary differentia BK equations. TP.HCM For example, consider the general form of our single species population equation ε = f(x): Contents Introduction Malthusian Growth x=0=f(x) =0 (15) Model Properties of the In ecosystem dynamics, these fixed points are also known as systems System of ODEs critical points. For the Malthus system (exponential growth) Homeworks we have: x = ax = 0 = x = 0 (16) There is only one critical point (x = 0). How could we reach this solution? 6.13" }, { "page_index": 762, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_014.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_014.png", "page_index": 762, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:27+07:00" }, "raw_text": "Stability of Critical Points Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen What is the nature of the solution near the critical point? Consider. BK two possibilities for a frog pond starting from a steady state: TP.HCM Five frogs are killed in a freak accident. The frog population returns to the steady state. stab/e Contents Introduction The frog population crashes: unstab/e Malthusian Growth Model Let's look at the stability of our exponential growth : = aa Properties of the critical point x = 0. systems System of ODEs If we perturb x to ε = 5, what will happen? Homeworks x(t) = xoeat = 5eat (17) System will grow exponentially away from the critical point: critical point is unstable 6.14" }, { "page_index": 763, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_015.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_015.png", "page_index": 763, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:29+07:00" }, "raw_text": "Stability of Critical Points: cont'd Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM B Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs A Homeworks B: Unstable : A: Stable This example relates to states of potential energy, but the analogy holds for energy in an ecosystem. 6.15" }, { "page_index": 764, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_016.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_016.png", "page_index": 764, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:32+07:00" }, "raw_text": "Stability of Critical Points: Logistic Eguatior Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen BK Logistic equation : TP.HCM x = ax(1 (18) K Solve for fixed points with x = f(x) = 0. Two fixed points: Contents Introduction 0 x1 = 0 Nothing exists Malthusian Growth Model @ 2 = K Population is at carrying capacity Properties of the Though experiment: Are these stable: systems System of ODEs x = 0: Perturb system from nothing, what will happen: Homeworks growth! (Unstable) x = K: Perturb system from carrying capacity, what will happen return to K (Stable) We will analyze this more rigorously 6.16" }, { "page_index": 765, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_017.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_017.png", "page_index": 765, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:34+07:00" }, "raw_text": "Stability of Linear Systems Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen BK For linear ODEs of the form: : = ac, the stability can be TP.HCM determined simply from the sign of a: Contents . a > 0: Unstable (growth) Introduction : a < 0: Stable (decay) Malthusian Growth Model This is obvious from the solution: x(t) = oeat.What about Properties of the nonlinear ODEs: systems x = ax(1 (19) System of ODEs Homeworks Trick is to linearize about the critical point. What is linearization: We approximate a nonlinear function with a linear function that is quite accurate at a given point 6.17" }, { "page_index": 766, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_018.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_018.png", "page_index": 766, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:38+07:00" }, "raw_text": "Linearization Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen We linearize a function near a given point using a Taylor Series. BK For example, take a nonlinear function g(x TP.HCM With approximation theory, we can show that the following series f(x) converges to the exact function g(x) at x = a Contents Introduction Malthusian Growth f(x) = g(a) +g (a)(x - a) +H.O.T. (20) Model Properties of the . If we retain only the first two terms (zeroth order and linear) systems we have the linearization C(x) of the function q(x) about the System of ODEs Homeworks point a. L(x)= g(a)+g(a)(x-a) g(x) (21) What do we need to compute this linearization? 1 The first derivative of the function g(x) evaluated at the point x = a The value of the function at a (g(a) 6.18" }, { "page_index": 767, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_019.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_019.png", "page_index": 767, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:40+07:00" }, "raw_text": "Linearization example: Sin Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen The sin function is a complex, irrational function However, if we are working around small angles, we can often BK replace it with a more compact function using the TP.HCM linearization. - For example, the governing equation for a pendulum is: Contents Introduction IG + mgl sin @ = 0 (22) Malthusian Growth Model where O is the angle of perturbation from the vertical. Properties of the systems We can simplify the solution if we linearize sin about @ = 0: System of ODEs Homeworks L(sin ) = 0 + cos(O)Theta=0(6 - 0) = (23) Using the so-called small angle approximation: we can simplify the D.E. to be: I6+ mglO = 0 (24) 6.19" }, { "page_index": 768, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_020.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_020.png", "page_index": 768, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:43+07:00" }, "raw_text": "Stability of the Logistic Equation Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen BK Let's analyze the stability of the critical points of the Logistic TP.HCM Equation. First, let's evaluate the first derivative Contents Introduction 2x f'(x)=a (25) Malthusian Growth K Model Properties of the At the point x = 0, we have, for the Linearization: systems System of ODEs Homeworks L(x) = 0+ a(x - 0) = ax (26) So, near the critical point x = 0, our logistic equation behaves as L(x) = ax (why is this not surprising) Thus it is an unstable critical point. 6.20" }, { "page_index": 769, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_021.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_021.png", "page_index": 769, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:45+07:00" }, "raw_text": "Stability of the Logistic Equation, cont'd Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen At the point x = K, we have, for the Linearization BK TP.HCM (27) C Contents 2 Introduction Thus, the linearized differential equation is: Malthusian Growth Model 1 1 (28) Properties of the systems 2 2 System of ODEs This is a stable differential equation with a single steady state Homeworks value: x = K. So, the fixed point x = K is stable. This is not surprising, given an understanding of the solution x(t) of the differential equation near c = K. 6.21" }, { "page_index": 770, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_022.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_022.png", "page_index": 770, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:49+07:00" }, "raw_text": "System of ODEs: Species Interaction Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen : We will now expand our analysis to more realistic system which include multiple, interacting species. BK . We will study two fundamental models, the predator-prey and TP.HCM the competing species. Later, when we begin studying marine ecosystems, you should Contents Introduction see the analogy with these fundamental models. Malthusian Growth For example, a classic NP model is a predator-prey model Model where the nutrient is the prey and phytoplankton is the Properties of the systems predator. System of ODEs . What do we hope to gain from the analysis: Homeworks - What are the steady states of my system . Are my steady states stable or unstable How do the interaction parameterizations influence the system stability How does the system parameterization influence the dynamics of the system response (time scales, oscillation rates, relative magnitude of populations, etc) 6.22" }, { "page_index": 771, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_023.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_023.png", "page_index": 771, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:52+07:00" }, "raw_text": "System of ODEs: Predator-Prey Models Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen We will start with a fundamental model of interaction in an BK ecosystem: The predator-prey model. History: TP.HCM Fur traders noted remarkable cycles in numbers of lynx and hare furs in the 1800s Contents Introduction Later, Volterra noted similar fluctuations in fish populations Malthusian Growth and derived a fundamental set of eguations to describe them Model Properties of the : Lotka derived simultaneously a similar set of equations systems Predator Prey (Lotka-Volterra) Equations System of ODEs Homeworks dx (ax - bxy) (29) dt dy (-cy+ dxy) (30) dt 6.23" }, { "page_index": 772, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_024.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_024.png", "page_index": 772, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:55+07:00" }, "raw_text": "System of ODEs: Predator-Prey Models Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen Parameters: : a: growth rate of prey BK TP.HCM b: consumption of x by y : c: natural mortality of y Contents d: consumption of x by y Introduction Note: b and d are different because there is an efficiency to Malthusian Growth Model consumption Properties of the In this model, they were looking to answer two questions systems System of ODEs Can we explain the cycles in a typical predator-prey system? Homeworks . Why is it that only in some systems the predation limits the prey density? . What happens if the state or parameters are changed. For example, disease could alter the mortality rate, or a change in habitat could modify the ability of the prey to hid and affect the rate of predation 6.24" }, { "page_index": 773, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_025.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_025.png", "page_index": 773, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:05:57+07:00" }, "raw_text": "System of ODEs: Predator-Prey Models Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen The model relates prey x and predator y and these variables can BK TP.HCM represent biomass or population densities.Here we have used the following assumptions: Contents There are no time lags, predators respond instantaneously to Introduction a change in the prey and vice versa Malthusian Growth Model Prey grow exponentially without predators to control their Properties of the population. systems System of ODEs . Predators depend on prey to survive, otherwise, natural Homeworks mortality will wipe them out c : Predation rate depends on the likelihood that a predator comes across prey, thus is proportional to prey population growth rate of the predator is proportional to food intake 6.25" }, { "page_index": 774, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_026.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_026.png", "page_index": 774, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:00+07:00" }, "raw_text": "System of ODEs: Consumption Responses Dynamical Systems Nguyen An Khuong. - Is a linear dependence of predation rate on prey realistic? Huynh Tuong Nguyen No, realistically, should consider additional ecosystem dynamics: BK Saturation Above a certain density of prey, the consumption TP.HCM will level out (can only eat so much). We will use this when we parameterize grazing of Z on P Contents Introduction Malthusian Growth Model Type I Properties of the systems System of ODEs Homeworks Prey Density (Number Area-1) Hinh: Type 1 Predation with Saturation (From Uldaho WLF 448 6.26" }, { "page_index": 775, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_027.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_027.png", "page_index": 775, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:02+07:00" }, "raw_text": "System of ODEs: Consumption Responses, cont'd Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen Refuge : below a density, prey have places to hide: BK TP.HCM Type ll w/ prey refuge Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs Homeworks Prey Density (Number Area-1) Hinh: Type 2 Predation with Refuge Effects (From UIdaho WLF 448 6.27" }, { "page_index": 776, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_028.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_028.png", "page_index": 776, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:04+07:00" }, "raw_text": "Analysis of L-V model Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen identify critical points in the system and think about their physical meaning BK plot the solution in phase (x,y) space using directional vectors TP.HCM examine Nullclines Contents solve the full nonlinear system using Matlab and explore the Introduction results of phase plane trajectories by specifying a whole slew Malthusian Growth of initial conditions. Model Properties of the But first, we need to take a step backward and work on some systems quantitative tools. First let's look at the generic representation of System of ODEs a coupled set of two ordinary differential equations.Let's look at Homeworks the generic representation first. dx (31) dy g(x,y) (32) dt 6.28" }, { "page_index": 777, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_029.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_029.png", "page_index": 777, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:06+07:00" }, "raw_text": "Phase Space Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Contents Introduction Phase space is the space of state variables. If we plot, for example Malthusian Growth the time-dependent solution of two state variables together, we are Model working in the phase p/ane. For example: Properties of the systems System of ODEs Homeworks 6.29" }, { "page_index": 778, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_030.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_030.png", "page_index": 778, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:09+07:00" }, "raw_text": "Directional Gradients Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen BK Consider the governing equations in generic form TP.HCM f(x,y) (33) Contents y g(x,y) (34) Introduction Malthusian Growth We may not have an explicit solution for x(t) or y(t). But we do Model have direct information on the rate of change of our two species Properties of the systems For example, at the position in the phase plane 1, y1, we have: System of ODEs Homeworks xx1,Y1 f(x1,y1) (35) yx1,Y1 g(x1,y1) (36) (37) 6.30" }, { "page_index": 779, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_031.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_031.png", "page_index": 779, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:12+07:00" }, "raw_text": "Directional Gradients, cont'd Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen We may not know the solution x(t),y(t), but the differential BK equations tell as at any point in the phase plane (x,y) what the TP.HCM rate of change of the solution with time is. We can approximate the derivative using a forward Euler step: Contents Introduction dx x : (38) Malthusian Growth Model dt t1 - to Properties of the systems or given a solution (o,yo) at to, we can determine the System of ODEs approximate solution at t1 : Homeworks x = x2 - x1 (t2 - t1)f(x1,y1 (39) and y = y2 - y1 (t2 - t1)g(x1,y1 (40) 6.31" }, { "page_index": 780, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_032.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_032.png", "page_index": 780, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:15+07:00" }, "raw_text": "Directional Gradients, cont'd Dynamical Systems So we can compute the vector at each point. If we only wish to Nguyen An Khuong, Huynh Tuong Nguyen piece together the gradients, then all we care about is the relative size of q and f and their signs. We can normalize the vectors so they are all the same length (magnitude) but give different BK TP.HCM directions. If we evaluate the gradients at enough points, we can get a sense of the behavior of the system. Contents Introduction X2 Malthusian Growth Model Properties of the systems Ay System of ODEs Homeworks x x Hinh: Phase nlane Model stat 6.32" }, { "page_index": 781, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_033.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_033.png", "page_index": 781, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:19+07:00" }, "raw_text": "Directional Gradients, cont'd Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen The general procedure is as follows: BK TP.HCM We plot our phase plane with species on the x-axis and species y on the y-axis Contents @ We turn this into a regular grid of points, that is we divide up Introduction x into segments and y into segments so that we have a mesh Malthusian Growth Model of locations x, y. At each of these locations we evaluate our Properties of the functions as above to find out what ε, y are at each point systems @ We then draw an arrow with the tail located at the point , y. System of ODEs Homeworks The head of the vector is located at the point x + ε,y + y where ε = f(x,y) and x : g(x,y) are related the rates V f2+g2 Vf2+g2 of changes of prey ε and predator y. This arrow is pointing in the direction that ecosystem is going at the moment. 6.33" }, { "page_index": 782, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_034.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_034.png", "page_index": 782, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:22+07:00" }, "raw_text": "Nullclines (a.k.a. isoclines) Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Nul/cline: Curve along which one of the variables is a steady state Contents . Along the x isocline, we have f = 0 Introduction Along the x isocl/ine the directional gradients must be parallel Malthusian Growth Model to the y - axis, but can be oriented up or down Properties of the How do we determine the direction? We look at the equation systems System of ODEs for y and check the sign. Homeworks . If y = g(x,y) > 0, y is increasing . We can also look at the y - nullcline along which y = 0 and the directional gradients are parallel to the x - axis. 6.34" }, { "page_index": 783, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_035.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_035.png", "page_index": 783, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:24+07:00" }, "raw_text": "Work at Example: Pred-Prey Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM dx (ax - bxy) (41) Contents dt Introduction dy Malthusian Growth = (-cy+dxy (42) Model dt Properties of the systems Identify the critical points: System of ODEs 1,41 = 0,0: Extinction of both species Homeworks are coupled. 6.35" }, { "page_index": 784, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_036.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_036.png", "page_index": 784, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:27+07:00" }, "raw_text": "Nullclines of Pred-Prey Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM x nul/c/ine If we set : = 0 we find that we have two straight lines, y = and x = 0. For this system, to the right of x = Contents we have that y > 0 and to the left we have that y < 0. Introduction Malthusian Growth y nul/cline If we set y = 0 we find that we two straight lines Model given by x = and y = 0. Now, if we examine what happens Properties of the systems to along the y nullcline, we see that when y > , we have System of ODEs that ε < 0 and below that line, it is positive. Homeworks Note: Where the nullclines intersect is, by definition, a critical point. 6.36" }, { "page_index": 785, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_037.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_037.png", "page_index": 785, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:29+07:00" }, "raw_text": "Sketch of pred-prey analysis Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen y=0 BK TP.HCM Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs Homeworks d What is happening in the ecosystem? 6.37" }, { "page_index": 786, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_038.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_038.png", "page_index": 786, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:32+07:00" }, "raw_text": "Analysis of Critical Points Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen In our one-dimensional logistic equation we were able to analyze BK the stability of the critical points by linearizing the system at the TP.HCM critical point. Essentially, we transformed ε = f(x) = x a For a coupled set of ODEs Contents dx Introduction f(x,y) (43) Malthusian Growth dt Model Properties of the dy systems g(x,y) (44) dt System of ODEs Homeworks The linear system is: c a11x + a12y (45) y a21x + a22y (46) How do we linearize? 6.38" }, { "page_index": 787, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_039.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_039.png", "page_index": 787, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:34+07:00" }, "raw_text": "Solving nonlinear ODEs Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen BK TP.HCM There are a few analytical techniques for solving nonlinear ODEs (e.g. separation of variables). Contents Introduction However, for the complex form typical of dynamical systems, Malthusian Growth these do not generally work. Model Properties of the : We will have to use numerical techiques there (and Matlab systems will help greatly.) System of ODEs Homeworks A well known solver we should re in detail is called Runge-Kutta 6.39" }, { "page_index": 788, "chapter_num": 6, "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_040.png", "metadata": { "doc_type": "slide", "course_id": "CO2011", "source_file": "/workspace/data/converted/CO2011_MaThematical Modeling/Chapter_6/slide_040.png", "page_index": 788, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-31T06:06:36+07:00" }, "raw_text": "Homeworks and next week plan Dynamical Systems Nguyen An Khuong. Huynh Tuong Nguyen BK TP.HCM Do ALL exercises in both: Contents Section 1.4 (pages 52-56) and Introduction Section 11.1 (pages 468-470) in: Malthusian Growth Model F.R. Giordano, W.P. Fox & S.B. Horton, A First Course in Properties of the systems Mathematical Modeling, 5th ed., Cengage, 2014. System of ODEs . Due: a week from today lecture. Homeworks Next week plan: Exercise Session 6.40" } ] }