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@@ -17,11 +17,11 @@
 {"homework": "hw3", "exercise_number": "2", "content": "Give a regular expression for language\n\n$$L = \\{w \\in \\{0, 1\\}^* : \\text{ the number of 1's in } w \\text{ is even, and the number of 0's is even}\\}.$$", "full_id": "hw3_ex2"}
 {"homework": "hw3", "exercise_number": "3", "content": "Construct NFAs corresponding to the following regular expressions:\n\n* $0^+ \\cup (01)^*$;\n* $(0 \\cup 1^+)0^*1^+$.", "full_id": "hw3_ex3"}
 {"homework": "hw5", "exercise_number": "1", "content": "The input is two binary numbers, $x$ and $y$ (with the most significant bit on the left), separated by a symbol #, for example, 100#100⊔. (Assume there are no leading zeros unless the number is zero.) Accept if $x$ and $y$ are equal.\n\nDraw the state diagram of the corresponding Turing machine and submit its source code on eLearning.", "full_id": "hw5_ex1"}
-{"homework": "hw5", "exercise_number": "2", "content": "Let language\n\n$$L = \\{0^n : n \\text{ is a Fibonacci number}\\}.$$\n\nDesign a Turing machine that decides L. Draw the state diagram of the corresponding Turing machine and submit its source code on eLearning.", "full_id": "hw5_ex2"}
-{"homework": "hw5", "exercise_number": "3", "content": "In the definition of the Turing machine, the head can move left or right. A Turing machine whose head can move left, right, or stay in place is called a *Turing machine with a stationary move*, or more formally:\n\n$$\\delta : Q \\times \\Gamma \\to Q \\times \\Gamma \\times \\{L, R, S\\}.$$\n\nProve that if a language $L$ can be decided by a Turing machine with stationary moves, then $L$ can also be decided by a standard Turing machine.", "full_id": "hw5_ex3"}
+{"homework": "hw5", "exercise_number": "2", "content": "(30') Let language\n\n$$L = \\{0^n : n \\text{ is a Fibonacci number}\\}.$$\n\nDesign a Turing machine that decides L. Draw the state diagram of the corresponding Turing machine and submit its source code on eLearning.", "full_id": "hw5_ex2"}
+{"homework": "hw5", "exercise_number": "3", "content": "(40') In the definition of the Turing machine, the head can move left or right. A Turing machine whose head can move left, right, or stay in place is called a *Turing machine with a stationary move*, or more formally:\n\n$$\\delta : Q \\times \\Gamma \\to Q \\times \\Gamma \\times \\{L, R, S\\}.$$\n\nProve that if a language $L$ can be decided by a Turing machine with stationary moves, then $L$ can also be decided by a standard Turing machine.", "full_id": "hw5_ex3"}
 {"homework": "hw6", "exercise_number": "1", "content": "Define a *two-dimensional* Turing machine to be a TM where each of its tapes is an infinite grid (and the machine can move not only Left and Right but also Up and Down).\n\nShow that if language L can be decided in time T(n) by a two-dimensional TM then L ∈ DTIME(T(n)<sup>2</sup>).", "full_id": "hw6_ex1"}
 {"homework": "hw6", "exercise_number": "2", "content": "Define a *one-tape two-head Turing machine* as a Turing machine equipped with a single tape but two independent heads operating on it. Each head can read, write, and move simultaneously and independently of the other. Both heads may read the same tape cell at the same time. Conflicts are resolved as follows:\n\n* If both heads attempt to write to the same cell simultaneously, the first head goes first, and the second head overwrites any previous content.\n\n* If the first head tries to write while the second head tries to read the same cell, the first head goes first, and the second head reads the updated content.\n\n* If the first head tries to read while the second head tries to write the same cell, the first head goes first, reading the old content, and then the second head updates the content.\n\nShow that if language L can be decided in time T(n) by a one-tape two-head Turing machine, then L ∈ DTIME(T(n)<sup>2</sup>).", "full_id": "hw6_ex2"}
-{"homework": "hw6", "exercise_number": "3", "content": "Prove the number of multitape Turing machines is countable.", "full_id": "hw6_ex3"}
+{"homework": "hw6", "exercise_number": "3", "content": "(30') Prove the number of multitape Turing machines is countable.", "full_id": "hw6_ex3"}
 {"homework": "hw9", "exercise_number": "1", "content": "Prove that the Post Correspondence Problem is decidable over the unary alphabet Σ = {1}.", "full_id": "hw9_ex1"}
 {"homework": "hw9", "exercise_number": "2", "content": "Prove that the Post Correspondence Problem is undecidable over the binary alphabet Σ = {0, 1}, assuming that the PCP is undecidable over an arbitrary finite alphabet.", "full_id": "hw9_ex2"}
 {"homework": "hw9", "exercise_number": "3", "content": "Express the *Twin Prime Conjecture* in first-order Peano Arithmetic using only the symbols 0, S, +, ×, = and quantifiers ∀, ∃.\n\n(a) Define a formula Prime(x) expressing that x is a prime number.\n\n(b) Define a formula TwinPrime(p) expressing that p is the smaller number of a twin prime pair.\n\n(c) Write a single first-order PA formula that states: *for every number x, there exists a larger twin prime p > x*.", "full_id": "hw9_ex3"}