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For parallel computing, using a programming model instead of a language is common. The reason is that details of the parallel hardware leak into the abstractions used to program the hardware. This causes the programmer to have to map patterns in the algorithm onto patterns in the execution model . As a consequence, no one parallel programming language maps well to all computation problems. Thus, it is more convenient to use a base sequential language and insert API calls to parallel execution models via a programming model. Such parallel programming models can be classified according to abstractions that reflect the hardware, such as shared memory, distributed memory with message passing, notions of place visible in the code, and so forth. These can be considered flavors of programming paradigm that apply to only parallel languages and programming models.
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Some programming language researchers criticise the notion of paradigms as a classification of programming languages, e.g. Harper, and Krishnamurthi. They argue that many programming languages cannot be strictly classified into one paradigm, but rather include features from several paradigms. See Comparison of multi-paradigm programming languages.
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Different approaches to programming have developed over time. Classification of each approach was either described at the time the approach was first developed, but often not until some time later, retrospectively. An early approach consciously identified as such is structured programming, advocated since the mid 1960s. The concept of a programming paradigm as such dates at least to 1978, in the Turing Award lecture of Robert W. Floyd, entitled The Paradigms of Programming, which cites the notion of paradigm as used by Thomas Kuhn in his The Structure of Scientific Revolutions . Early programming languages did not have clearly defined programming paradigms and sometimes programs made extensive use of goto statements. Liberal use of which lead to spaghetti code which is difficult to understand and maintain. This led to the development of structured programming paradigms that disallowed the use of goto statements; only allowing the use of more structured programming constructs.
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Machine code is the lowest-level of computer programming as it is machine instructions that define behavior at the lowest level of abstract possible for a computer. As it is the most prescriptive way to code it is classified as imperative.
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It is sometimes called the first-generation programming language.
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Assembly language introduced mnemonics for machine instructions and memory addresses. Assembly is classified as imperative and is sometimes called the second-generation programming language.
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In the 1960s, assembly languages were developed to support library COPY and quite sophisticated conditional macro generation and preprocessing abilities, CALL to subroutine, external variables and common sections , enabling significant code re-use and isolation from hardware specifics via the use of logical operators such as READ/WRITE/GET/PUT. Assembly was, and still is, used for time-critical systems and often in embedded systems as it gives the most control of what the machine does.
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Procedural languages, also called the third-generation programming languages are the first described as high-level languages. They support vocabulary related to the problem being solved. For example,
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These languages are classified as procedural paradigm. They directly control the step by step process that a computer program follows. The efficacy and efficiency of such a program is therefore highly dependent on the programmer's skill.
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In attempt to improve on procedural languages, object-oriented programming languages were created, such as Simula, Smalltalk, C++, Eiffel, Python, PHP, Java, and C#. In these languages, data and methods to manipulate the data are in the same code unit called an object. This encapsulation ensures that the only way that an object can access data is via methods of the object that contains the data. Thus, an object's inner workings may be changed without affecting code that uses the object.
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There is controversy raised by Alexander Stepanov, Richard Stallman and other programmers, concerning the efficacy of the OOP paradigm versus the procedural paradigm. The need for every object to have associative methods leads some skeptics to associate OOP with software bloat; an attempt to resolve this dilemma came through polymorphism.
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Although most OOP languages are third-generation, it is possible to create an object-oriented assembler language. High Level Assembly is an example of this that fully supports advanced data types and object-oriented assembly language programming – despite its early origins. Thus, differing programming paradigms can be seen rather like motivational memes of their advocates, rather than necessarily representing progress from one level to the next. Precise comparisons of competing paradigms' efficacy are frequently made more difficult because of new and differing terminology applied to similar entities and processes together with numerous implementation distinctions across languages.
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A declarative programming program describes what the problem is, not how to solve it. The program is structured as a set of properties to find in the expected result, not as a procedure to follow. Given a database or a set of rules, the computer tries to find a solution matching all the desired properties. An archetype of a declarative language is the fourth generation language SQL, and the family of functional languages and logic programming.
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Functional programming is a subset of declarative programming. Programs written using this paradigm use functions, blocks of code intended to behave like mathematical functions. Functional languages discourage changes in the value of variables through assignment, making a great deal of use of recursion instead.
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The logic programming paradigm views computation as automated reasoning over a body of knowledge. Facts about the problem domain are expressed as logic formulas, and programs are executed by applying inference rules over them until an answer to the problem is found, or the set of formulas is proved inconsistent.
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Symbolic programming is a paradigm that describes programs able to manipulate formulas and program components as data. Programs can thus effectively modify themselves, and appear to "learn", making them suited for applications such as artificial intelligence, expert systems, natural-language processing and computer games. Languages that support this paradigm include Lisp and Prolog.
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Differentiable programming structures programs so that they can be differentiated throughout, usually via automatic differentiation.
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Literate programming, as a form of imperative programming, structures programs as a human-centered web, as in a hypertext essay: documentation is integral to the program, and the program is structured following the logic of prose exposition, rather than compiler convenience.
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Symbolic techniques such as reflection, which allow the program to refer to itself, might also be considered as a programming paradigm. However, this is compatible with the major paradigms and thus is not a real paradigm in its own right.
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Classification of the principal programming paradigms
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How programming paradigms evolve and get adopted?
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The structured chart subject to these constraints, particularly the loop constraint implying a single exit , may however use additional variables in the form of bits in order to keep track of information that the original program represents by the program location. The construction was based on Böhm's programming language P′′.
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The theorem forms the basis of structured programming, a programming paradigm which eschews goto commands and exclusively uses subroutines, sequences, selection and iteration.
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The theorem is typically credited: 381 to a 1966 paper by Corrado Böhm and Giuseppe Jacopini. David Harel wrote in 1980 that the Böhm–Jacopini paper enjoyed "universal popularity",: 381 particularly with proponents of structured programming. Harel also noted that "due to its rather technical style is apparently more often cited than read in detail": 381 and, after reviewing a large number of papers published up to 1980, Harel argued that the contents of the Böhm–Jacopini proof were usually misrepresented as a folk theorem that essentially contains a simpler result, a result which itself can be traced to the inception of modern computing theory in the papers of von Neumann and Kleene.: 383
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Harel also writes that the more generic name was proposed by H.D. Mills as "The Structure Theorem" in the early 1970s.: 381
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This version of the theorem replaces all the original program's control flow with a single global while loop that simulates a program counter going over all possible labels in the original non-structured program. Harel traced the origin of this folk theorem to two papers marking the beginning of computing. One is the 1946 description of the von Neumann architecture, which explains how a program counter operates in terms of a while loop. Harel notes that the single loop used by the folk version of the structured programming theorem basically just provides operational semantics for the execution of a flowchart on a von Neumann computer.: 383 Another, even older source that Harel traced the folk version of the theorem is Stephen Kleene's normal form theorem from 1936.: 383
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Donald Knuth criticized this form of the proof, which results in pseudocode like the one below, by pointing out that the structure of the original program is completely lost in this transformation.: 274 Similarly, Bruce Ian Mills wrote about this approach that "The spirit of block structure is a style, not a language. By simulating a Von Neumann machine, we can produce the behavior of any spaghetti code within the confines of a block-structured language. This does not prevent it from being spaghetti."
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The proof in Böhm and Jacopini's paper proceeds by induction on the structure of the flow chart.: 381 Because it employed pattern matching in graphs, the proof of Böhm and Jacopini's was not really practical as a program transformation algorithm, and thus opened the door for additional research in this direction.
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The Reversible Structured Program Theorem is an important concept in the field of reversible computing. It posits that any computation achievable by a reversible program can also be accomplished through a reversible program using only a structured combination of control flow constructs such as sequences, selections, and iterations. Any computation achievable by a traditional, irreversible program can also be accomplished through a reversible program, but with the additional constraint that each step must be reversible and some extra output. Furthermore, any reversible unstructured program can also be accomplished through a structured reversible program with only one iteration without any extra output. This theorem lays the foundational principles for constructing reversible algorithms within a structured programming framework.
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For the Structured Program Theorem, both local and global methods of proof are known. However, for its reversible version, while a global method of proof is recognized, a local approach similar to that undertaken by Böhm and Jacopini is not yet known. This distinction is an example that underscores the challenges and nuances in establishing the foundations of reversible computing compared to traditional computing paradigms.
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The Böhm–Jacopini proof did not settle the question of whether to adopt structured programming for software development, partly because the construction was more likely to obscure a program than to improve it. On the contrary, it signalled the beginning of the debate. Edsger Dijkstra's famous letter, "Go To Statement Considered Harmful," followed in 1968.
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Some academics took a purist approach to the Böhm–Jacopini result and argued that even instructions like break and return from the middle of loops are bad practice as they are not needed in the Böhm–Jacopini proof, and thus they advocated that all loops should have a single exit point. This purist approach is embodied in the Pascal programming language , which up to the mid-1990s was the preferred tool for teaching introductory programming classes in academia.
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Edward Yourdon notes that in the 1970s there was even philosophical opposition to transforming unstructured programs into structured ones by automated means, based on the argument that one needed to think in structured programming fashion from the get go. The pragmatic counterpoint was that such transformations benefited a large body of existing programs. Among the first proposals for an automated transformation was a 1971 paper by Edward Ashcroft and Zohar Manna.
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The direct application of the Böhm–Jacopini theorem may result in additional local variables being introduced in the structured chart, and may also result in some code duplication. The latter issue is called the loop and a half problem in this context. Pascal is affected by both of these problems and according to empirical studies cited by Eric S. Roberts, student programmers had difficulty formulating correct solutions in Pascal for several simple problems, including writing a function for searching an element in an array. A 1980 study by Henry Shapiro cited by Roberts found that using only the Pascal-provided control structures, the correct solution was given by only 20% of the subjects, while no subject wrote incorrect code for this problem if allowed to write a return from the middle of a loop.
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In 1973, S. Rao Kosaraju proved that it's possible to avoid adding additional variables in structured programming, as long as arbitrary-depth, multi-level breaks from loops are allowed. Furthermore, Kosaraju proved that a strict hierarchy of programs exists, nowadays called the Kosaraju hierarchy, in that for every integer n, there exists a program containing a multi-level break of depth n that cannot be rewritten as program with multi-level breaks of depth less than n . Kosaraju cites the multi-level break construct to the BLISS programming language. The multi-level breaks, in the form a leave label keyword were actually introduced in the BLISS-11 version of that language; the original BLISS only had single-level breaks. The BLISS family of languages didn't provide an unrestricted goto. The Java programming language would later follow this approach as well.: 960–965
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A simpler result from Kosaraju's paper is that a program is reducible to a structured program if and only if it does not contain a loop with two distinct exits. Reducibility was defined by Kosaraju, loosely speaking, as computing the same function and using the same "primitive actions" and predicates as the original program, but possibly using different control flow structures. Inspired by this result, in section VI of his highly-cited paper that introduced the notion of cyclomatic complexity, Thomas J. McCabe described an analogue of Kuratowski's theorem for the control-flow graphs of non-structured programs, which is to say, the minimal subgraphs that make the CFG of a program non-structured. These subgraphs have a very good description in natural language. They are:
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McCabe actually found that these four graphs are not independent when appearing as subgraphs, meaning that a necessary and sufficient condition for a program to be non-structured is for its CFG to have as subgraph one of any subset of three of these four graphs. He also found that if a non-structured program contains one of these four sub-graphs, it must contain another distinct one from the set of four. This latter result helps explain how the control flow of non-structured program becomes entangled in what is popularly called "spaghetti code". McCabe also devised a numerical measure that, given an arbitrary program, quantifies how far off it is from the ideal of being a structured program; McCabe called his measure essential complexity.
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McCabe's characterization of the forbidden graphs for structured programming can be considered incomplete, at least if the Dijkstra's D structures are considered the building blocks.: 274–275
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Up to 1990 there were quite a few proposed methods for eliminating gotos from existing programs, while preserving most of their structure. The various approaches to this problem also proposed several notions of equivalence, which are stricter than simply Turing equivalence, in order to avoid output like the folk theorem discussed above. The strictness of the chosen notion of equivalence dictates the minimal set of control flow structures needed. The 1988 JACM paper by Lyle Ramshaw surveys the field up to that point, as well proposing its own method. Ramshaw's algorithm was used for example in some Java decompilers because the Java virtual machine code has branch instructions with targets expressed as offsets, but the high-level Java language only has multi-level break and continue statements. Ammarguellat proposed a transformation method that goes back to enforcing single-exit.
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In the 1980s IBM researcher Harlan Mills oversaw the development of the COBOL Structuring Facility, which applied a structuring algorithm to COBOL code. Mills's transformation involved the following steps for each procedure.
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This construction can be improved by converting some cases of the selection statement into subprocedures.
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Material not yet covered above:
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The machine operates on an infinite memory tape divided into discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write, which direction to move the head, and whether to halt is based on a finite table that specifies what to do for each combination of the current state and the symbol that is read. Like a real computer program, it is possible for a Turing machine to go into an infinite loop which will never halt.
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The Turing machine was invented in 1936 by Alan Turing, who called it an "a-machine" . It was Turing's doctoral advisor, Alonzo Church, who later coined the term "Turing machine" in a review. With this model, Turing was able to answer two questions in the negative:
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Thus by providing a mathematical description of a very simple device capable of arbitrary computations, he was able to prove properties of computation in general—and in particular, the uncomputability of the Entscheidungsproblem .
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Turing machines proved the existence of fundamental limitations on the power of mechanical computation. While they can express arbitrary computations, their minimalist design makes them too slow for computation in practice: real-world computers are based on different designs that, unlike Turing machines, use random-access memory.
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Turing completeness is the ability for a computational model or a system of instructions to simulate a Turing machine. A programming language that is Turing complete is theoretically capable of expressing all tasks accomplishable by computers; nearly all programming languages are Turing complete if the limitations of finite memory are ignored.
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A Turing machine is an idealised model of a central processing unit that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. Typically, the sequential memory is represented as a tape of infinite length on which the machine can perform read and write operations.
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In the context of formal language theory, a Turing machine is capable of enumerating some arbitrary subset of valid strings of an alphabet. A set of strings which can be enumerated in this manner is called a recursively enumerable language. The Turing machine can equivalently be defined as a model that recognises valid input strings, rather than enumerating output strings.
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Given a Turing machine M and an arbitrary string s, it is generally not possible to decide whether M will eventually produce s. This is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing.
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The Turing machine is capable of processing an unrestricted grammar, which further implies that it is capable of robustly evaluating first-order logic in an infinite number of ways. This is famously demonstrated through lambda calculus.
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A Turing machine that is able to simulate any other Turing machine is called a universal Turing machine . Another mathematical formalism, lambda calculus, with a similar "universal" nature was introduced by Alonzo Church. Church's work intertwined with Turing's to form the basis for the Church–Turing thesis. This thesis states that Turing machines, lambda calculus, and other similar formalisms of computation do indeed capture the informal notion of effective methods in logic and mathematics and thus provide a model through which one can reason about an algorithm or "mechanical procedure" in a mathematically precise way without being tied to any particular formalism. Studying the abstract properties of Turing machines has yielded many insights into computer science, computability theory, and complexity theory.
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In his 1948 essay, "Intelligent Machinery", Turing wrote that his machine consisted of:
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...an unlimited memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed. At any moment there is one symbol in the machine; it is called the scanned symbol. The machine can alter the scanned symbol, and its behavior is in part determined by that symbol, but the symbols on the tape elsewhere do not affect the behavior of the machine. However, the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings.
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The Turing machine mathematically models a machine that mechanically operates on a tape. On this tape are symbols, which the machine can read and write, one at a time, using a tape head. Operation is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; if the symbol seen is 1, change into state 17; in state 17, if the symbol seen is 0, write a 1 and change to state 6;" etc. In the original article , Turing imagines not a mechanism, but a person whom he calls the "computer", who executes these deterministic mechanical rules slavishly .
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More explicitly, a Turing machine consists of:
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In the 4-tuple models, erasing or writing a symbol and moving the head left or right are specified as separate instructions. The table tells the machine to erase or write a symbol or move the head left or right, and then assume the same or a new state as prescribed, but not both actions and in the same instruction. In some models, if there is no entry in the table for the current combination of symbol and state, then the machine will halt; other models require all entries to be filled.
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Every part of the machine and its actions is finite, discrete and distinguishable; it is the unlimited amount of tape and runtime that gives it an unbounded amount of storage space.
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The 7-tuple for the 3-state busy beaver looks like this :
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In the words of van Emde Boas , p. 6: "The set-theoretical object provides only partial information on how the machine will behave and what its computations will look like."
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For instance,
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The most common convention represents each "Turing instruction" in a "Turing table" by one of nine 5-tuples, per the convention of Turing/Davis in The Undecidable, p. 126–127 and Davis p. 152):
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Other authors p. 119, Hopcroft and Ullman p. 158, Stone p. 9) adopt a different convention, with new state qm listed immediately after the scanned symbol Sj:
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For the remainder of this article "definition 1" will be used.
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In the following table, Turing's original model allowed only the first three lines that he called N1, N2, N3 . He allowed for erasure of the "scanned square" by naming a 0th symbol S0 = "erase" or "blank", etc. However, he did not allow for non-printing, so every instruction-line includes "print symbol Sk" or "erase" , The Undecidable, p. 300). The abbreviations are Turing's . Subsequent to Turing's original paper in 1936–1937, machine-models have allowed all nine possible types of five-tuples:
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Any Turing table can be constructed from the above nine 5-tuples. For technical reasons, the three non-printing or "N" instructions can usually be dispensed with. For examples see Turing machine examples.
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Less frequently the use of 4-tuples are encountered: these represent a further atomization of the Turing instructions , Boolos & Jeffrey , Davis-Sigal-Weyuker ); also see more at Post–Turing machine.
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The word "state" used in context of Turing machines can be a source of confusion, as it can mean two things. Most commentators after Turing have used "state" to mean the name/designator of the current instruction to be performed—i.e. the contents of the state register. But Turing made a strong distinction between a record of what he called the machine's "m-configuration", and the machine's "state of progress" through the computation—the current state of the total system. What Turing called "the state formula" includes both the current instruction and all the symbols on the tape:
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Thus the state of progress of the computation at any stage is completely determined by the note of instructions and the symbols on the tape. That is, the state of the system may be described by a single expression consisting of the symbols on the tape followed by Δ and then by the note of instructions. This expression is called the "state formula".
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Earlier in his paper Turing carried this even further: he gives an example where he placed a symbol of the current "m-configuration"—the instruction's label—beneath the scanned square, together with all the symbols on the tape ; this he calls "the complete configuration" . To print the "complete configuration" on one line, he places the state-label/m-configuration to the left of the scanned symbol.
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A variant of this is seen in Kleene where Kleene shows how to write the Gödel number of a machine's "situation": he places the "m-configuration" symbol q4 over the scanned square in roughly the center of the 6 non-blank squares on the tape and puts it to the right of the scanned square. But Kleene refers to "q4" itself as "the machine state" . Hopcroft and Ullman call this composite the "instantaneous description" and follow the Turing convention of putting the "current state" to the left of the scanned symbol , that is, the instantaneous description is the composite of non-blank symbols to the left, state of the machine, the current symbol scanned by the head, and the non-blank symbols to the right.
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Example: total state of 3-state 2-symbol busy beaver after 3 "moves" :
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This means: after three moves the tape has ... 000110000 ... on it, the head is scanning the right-most 1, and the state is A. Blanks can be part of the total state as shown here: B01; the tape has a single 1 on it, but the head is scanning the 0 to its left and the state is B.
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"State" in the context of Turing machines should be clarified as to which is being described: the current instruction, or the list of symbols on the tape together with the current instruction, or the list of symbols on the tape together with the current instruction placed to the left of the scanned symbol or to the right of the scanned symbol.
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Turing's biographer Andrew Hodges has noted and discussed this confusion.
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To the right: the above table as expressed as a "state transition" diagram.
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Usually large tables are better left as tables . They are more readily simulated by computer in tabular form . However, certain concepts—e.g. machines with "reset" states and machines with repeating patterns —can be more readily seen when viewed as a drawing.
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Whether a drawing represents an improvement on its table must be decided by the reader for the particular context.
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The reader should again be cautioned that such diagrams represent a snapshot of their table frozen in time, not the course of a computation through time and space. While every time the busy beaver machine "runs" it will always follow the same state-trajectory, this is not true for the "copy" machine that can be provided with variable input "parameters".
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The diagram "progress of the computation" shows the three-state busy beaver's "state" progress through its computation from start to finish. On the far right is the Turing "complete configuration" at each step. If the machine were to be stopped and cleared to blank both the "state register" and entire tape, these "configurations" could be used to rekindle a computation anywhere in its progress The Undecidable, pp. 139–140).
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Many machines that might be thought to have more computational capability than a simple universal Turing machine can be shown to have no more power ). They might compute faster, perhaps, or use less memory, or their instruction set might be smaller, but they cannot compute more powerfully .
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A Turing machine is equivalent to a single-stack pushdown automaton that has been made more flexible and concise by relaxing the last-in-first-out requirement of its stack. In addition, a Turing machine is also equivalent to a two-stack PDA with standard LIFO semantics, by using one stack to model the tape left of the head and the other stack for the tape to the right.
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At the other extreme, some very simple models turn out to be Turing-equivalent, i.e. to have the same computational power as the Turing machine model.
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Common equivalent models are the multi-tape Turing machine, multi-track Turing machine, machines with input and output, and the non-deterministic Turing machine as opposed to the deterministic Turing machine for which the action table has at most one entry for each combination of symbol and state.
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Read-only, right-moving Turing machines are equivalent to DFAs .
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For practical and didactical intentions the equivalent register machine can be used as a usual assembly programming language.
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A relevant question is whether or not the computation model represented by concrete programming languages is Turing equivalent. While the computation of a real computer is based on finite states and thus not capable to simulate a Turing machine, programming languages themselves do not necessarily have this limitation. Kirner et al., 2009 have shown that among the general-purpose programming languages some are Turing complete while others are not. For example, ANSI C is not Turing-equivalent, as all instantiations of ANSI C imply a finite-space memory. This is because the size of memory reference data types, called pointers, is accessible inside the language. However, other programming languages like Pascal do not have this feature, which allows them to be Turing complete in principle.
It is just Turing complete in principle, as memory allocation in a programming language is allowed to fail, which means the programming language can be Turing complete when ignoring failed memory allocations, but the compiled programs executable on a real computer cannot.
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Early in his paper Turing makes a distinction between an "automatic machine"—its "motion ... completely determined by the configuration" and a "choice machine":
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...whose motion is only partially determined by the configuration ... When such a machine reaches one of these ambiguous configurations, it cannot go on until some arbitrary choice has been made by an external operator. This would be the case if we were using machines to deal with axiomatic systems.
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Turing does not elaborate further except in a footnote in which he describes how to use an a-machine to "find all the provable formulae of the calculus" rather than use a choice machine. He "suppose that the choices are always between two possibilities 0 and 1. Each proof will then be determined by a sequence of choices i1, i2, ..., in , and hence the number 2n + i12n-1 + i22n-2 + ... +in completely determines the proof. The automatic machine carries out successively proof 1, proof 2, proof 3, ..."
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This is indeed the technique by which a deterministic Turing machine can be used to mimic the action of a nondeterministic Turing machine; Turing solved the matter in a footnote and appears to dismiss it from further consideration.
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An oracle machine or o-machine is a Turing a-machine that pauses its computation at state "o" while, to complete its calculation, it "awaits the decision" of "the oracle"—an entity unspecified by Turing "apart from saying that it cannot be a machine" , The Undecidable, p. 166–168).
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As Turing wrote in The Undecidable, p. 128 :
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It is possible to invent a single machine which can be used to compute any computable sequence. If this machine U is supplied with the tape on the beginning of which is written the string of quintuples separated by semicolons of some computing machine M, then U will compute the same sequence as M.
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This finding is now taken for granted, but at the time it was considered astonishing. The model of computation that Turing called his "universal machine"—"U" for short—is considered by some ) to have been the fundamental theoretical breakthrough that led to the notion of the stored-program computer.
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Turing's paper ... contains, in essence, the invention of the modern computer and some of the programming techniques that accompanied it.
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In terms of computational complexity, a multi-tape universal Turing machine need only be slower by logarithmic factor compared to the machines it simulates. This result was obtained in 1966 by F. C. Hennie and R. E. Stearns.
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It is often believed that Turing machines, unlike simpler automata, are as powerful as real machines, and are able to execute any operation that a real program can. What is neglected in this statement is that, because a real machine can only have a finite number of configurations, it is nothing but a finite-state machine, whereas a Turing machine has an unlimited amount of storage space available for its computations.
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There are a number of ways to explain why Turing machines are useful models of real computers:
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A limitation of Turing machines is that they do not model the strengths of a particular arrangement well. For instance, modern stored-program computers are actually instances of a more specific form of abstract machine known as the random-access stored-program machine or RASP machine model. Like the universal Turing machine, the RASP stores its "program" in "memory" external to its finite-state machine's "instructions". Unlike the universal Turing machine, the RASP has an infinite number of distinguishable, numbered but unbounded "registers"—memory "cells" that can contain any integer , Hartmanis , and in particular Cook-Rechow ; references at random-access machine). The RASP's finite-state machine is equipped with the capability for indirect addressing ; thus the RASP's "program" can address any register in the register-sequence. The upshot of this distinction is that there are computational optimizations that can be performed based on the memory indices, which are not possible in a general Turing machine; thus when Turing machines are used as the basis for bounding running times, a "false lower bound" can be proven on certain algorithms' running times . An example of this is binary search, an algorithm that can be shown to perform more quickly when using the RASP model of computation rather than the Turing machine model.
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