or P = NP = NP-complete?

Transcription

1 P = NP? Research Seminar TBW Southwestphalia University of Applied Sciences Campus Hagen Summer Term 2010 NP-complete NP P or P = NP = NP-complete? 1 / 64

2 Contents 1 Bibliography 2 Decision problems languages Definitions 3 Turing Machines Definition Church-Turing Thesis: Algorithm = Turing Machine Universal Turing Machine Decidability Nondeterministic Turing Machines 2 / 64

3 Bibliography Sanjeev Arora and Boaz Barak. Computational Complexity. A Modern Approach. Cambridge University Press, Cambridge, Ronald Lewis Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics. Addison-Wesley, Upper Saddle River, NJ, 2nd edition, Dirk W. Hoffmann. Theoretische Informatik. Carl Hanser Verlag, München, C. M. Papadimitriou. Computational Complexity. Addison-Wesley, Reading, Massachusetts, Michael Sipser. Introduction to the Theory of Computation. Thomson Course Technology, Boston, / 64

4 Bibliography Web Sources Web Sources Wo] gwoegi/p-versus-np.htm Gerhard J. Woeginger (TU Eindhoven): The P-versus-NP page. A list of links to a number of purported solutions to the problem. Zoo] Complexity Zoo Wiki at Stanford University. last visited ] 4 / 64

5 Decision problems languages Definitions Übersicht 1 Bibliography 2 Decision problems languages Definitions 3 Turing Machines Definition Church-Turing Thesis: Algorithm = Turing Machine Universal Turing Machine Decidability Nondeterministic Turing Machines 5 / 64

6 Decision problems languages Definitions Strings Let Σ be a given alphabet. Then a string over Σ is a finite tuple of symbols x k Σ. x 1 x 2... x n 6 / 64

7 Decision problems languages Definitions Strings Let Σ be a given alphabet. Then a string over Σ is a finite tuple of symbols x k Σ. x 1 x 2... x n The length x of a string x is the number of its symbols. 6 / 64

8 Decision problems languages Definitions Strings Let Σ be a given alphabet. Then a string over Σ is a finite tuple of symbols x k Σ. x 1 x 2... x n The length x of a string x is the number of its symbols. By Σ n we denote the set of strings of length n, and by Σ the set of strings of arbitrary length. 6 / 64

9 Decision problems languages Definitions Strings Let Σ be a given alphabet. Then a string over Σ is a finite tuple of symbols x k Σ. x 1 x 2... x n The length x of a string x is the number of its symbols. By Σ n we denote the set of strings of length n, and by Σ the set of strings of arbitrary length. Especially, Σ 0 is the set consisting of the empty string (""), and Σ = n 0 Σ n. (1) Note that { } {""}. 6 / 64

10 Decision problems languages Definitions Languages and decision problems Often, a string is also called a word. 7 / 64

11 Decision problems languages Definitions Languages and decision problems Often, a string is also called a word. A language is a set of words Pap94, p. 24], Sip06, p. 14]. 7 / 64

12 Decision problems languages Definitions Languages and decision problems Often, a string is also called a word. A language is a set of words Pap94, p. 24], Sip06, p. 14]. A decision problem for a given language L is defined by the word problem Hof09, pp 164, 176, 182, 184] to decide whether, for a given word x Σ, we have x L, i.e., whether the word is contained in the language. 7 / 64

13 Decision problems languages Definitions Languages and decision problems Often, a string is also called a word. A language is a set of words Pap94, p. 24], Sip06, p. 14]. A decision problem for a given language L is defined by the word problem Hof09, pp 164, 176, 182, 184] to decide whether, for a given word x Σ, we have x L, i.e., whether the word is contained in the language. In the literature on complexity theory, language often refers to the special meaning decision problem AB09, p. 3]. 7 / 64

14 Decision problems languages Definitions Encodings Often a string of a given language is intended to represent certain objects other than strings, such as graphs, polynomials, or grammars. 8 / 64

15 Decision problems languages Definitions Encodings Often a string of a given language is intended to represent certain objects other than strings, such as graphs, polynomials, or grammars. Encoding an object A into its representation as a string is A. Encoding of several objects A 1, A 2,..., A k : A 1, A 2,..., A k. 8 / 64

16 Decision problems languages Definitions Encodings Often a string of a given language is intended to represent certain objects other than strings, such as graphs, polynomials, or grammars. Encoding an object A into its representation as a string is A. Encoding of several objects A 1, A 2,..., A k : A 1, A 2,..., A k. For a graph G and alphabet Σ = {1, 2, 3, 4, (, ),,} Sip06, 3.23]: G 1 4 G (1,2,3,4)((1,2),(2,3),(3,1),(1,4)) / 64

17 Decision problems languages Definitions Encodings Often a string of a given language is intended to represent certain objects other than strings, such as graphs, polynomials, or grammars. Encoding an object A into its representation as a string is A. Encoding of several objects A 1, A 2,..., A k : A 1, A 2,..., A k. For a graph G and alphabet Σ = {1, 2, 3, 4, (, ),,} Sip06, 3.23]: G 1 4 G (1,2,3,4)((1,2),(2,3),(3,1),(1,4)) 3 2 Strings of different encodings can be translated into each other in a unique way. 8 / 64

18 Decision problems languages Übersicht 1 Bibliography 2 Decision problems languages Definitions 3 Turing Machines Definition Church-Turing Thesis: Algorithm = Turing Machine Universal Turing Machine Decidability Nondeterministic Turing Machines 9 / 64

19 Decision problems languages Palindromes A palindrome (παλιν again, δρoµoς direction) is a string which can be read the same way in either direction, for instance / 64

20 Decision problems languages Palindromes A palindrome (παλιν again, δρoµoς direction) is a string which can be read the same way in either direction, for instance... stets, Regallager, Rentner, Reliefpfeiler, Ein Esel lese nie 10 / 64

21 Decision problems languages Palindromes A palindrome (παλιν again, δρoµoς direction) is a string which can be read the same way in either direction, for instance... stets, Regallager, Rentner, Reliefpfeiler, Ein Esel lese nie step on no pets, rats live on no evil star, Able was I ere I saw Elba 10 / 64

22 Decision problems languages Palindromes A palindrome (παλιν again, δρoµoς direction) is a string which can be read the same way in either direction, for instance... stets, Regallager, Rentner, Reliefpfeiler, Ein Esel lese nie step on no pets, rats live on no evil star, Able was I ere I saw Elba Historical example: Pompeii 79 AD ( SATOR square ) R O T A S O P E R A T E N E T A R E P O S A T O R Wheels and] works holds Arepo the sower last vis ] (Reading with alternate directions and beginning with the S: Sator opera tenet = The sower holds the works in the possible meaning The creator preserves his creation ). 10 / 64

23 Decision problems languages Notation: CAPITAL LETTERS Then we have the language PALINDROME = {x Σ : x 1 x 2... x x = x x x x 1... x 1 } It induces the corresponding word problem, i.e., the decision problem whether a word x Σ satisfies x PALINDROME. 11 / 64

24 Decision problems languages Notation: CAPITAL LETTERS Then we have the language PALINDROME = {x Σ : x 1 x 2... x x = x x x x 1... x 1 } It induces the corresponding word problem, i.e., the decision problem whether a word x Σ satisfies x PALINDROME. We consider a decision problem as a mathematical object, and not just a thing to solve. When problems are treated as mathematical objects, we denote it with capital letters, e.g., PALINDROME. 11 / 64

25 Decision problems languages Notation: CAPITAL LETTERS Then we have the language PALINDROME = {x Σ : x 1 x 2... x x = x x x x 1... x 1 } It induces the corresponding word problem, i.e., the decision problem whether a word x Σ satisfies x PALINDROME. We consider a decision problem as a mathematical object, and not just a thing to solve. When problems are treated as mathematical objects, we denote it with capital letters, e.g., PALINDROME. Usually, the problem is characterized by the language as a given mathematical set where the underlying alphabet is not explicitly given AB09, p 3], Sip06, p 143, 166],... and sometimes the corresponding word problem is simply identified with the language itself Pap94, pp 3, 8]. 11 / 64

26 Decision problems languages EVENS and SEVENS The set of even numbers is given by EVENS = { n : n N, n mod 2 = 0}. 12 / 64

27 Decision problems languages EVENS and SEVENS The set of even numbers is given by EVENS = { n : n N, n mod 2 = 0}. It is decidable by the simple algorithm of merely looking at the last digit in the decimal system, or any other number system to an even base b = 2k for some k N. 12 / 64

28 Decision problems languages EVENS and SEVENS The set of even numbers is given by EVENS = { n : n N, n mod 2 = 0}. It is decidable by the simple algorithm of merely looking at the last digit in the decimal system, or any other number system to an even base b = 2k for some k N. Its running time: T (n) = O(1). 12 / 64

29 Decision problems languages EVENS and SEVENS The set of even numbers is given by EVENS = { n : n N, n mod 2 = 0}. It is decidable by the simple algorithm of merely looking at the last digit in the decimal system, or any other number system to an even base b = 2k for some k N. Its running time: T (n) = O(1). Let SEVENS = { n : n N, n mod 7 = 0}. 12 / 64

30 Decision problems languages EVENS and SEVENS The set of even numbers is given by EVENS = { n : n N, n mod 2 = 0}. It is decidable by the simple algorithm of merely looking at the last digit in the decimal system, or any other number system to an even base b = 2k for some k N. Its running time: T (n) = O(1). Let SEVENS = { n : n N, n mod 7 = 0}. It is decidable by the algorithm of dividing the input n by 7 and determining the remainder. 12 / 64

31 Decision problems languages EVENS and SEVENS The set of even numbers is given by EVENS = { n : n N, n mod 2 = 0}. It is decidable by the simple algorithm of merely looking at the last digit in the decimal system, or any other number system to an even base b = 2k for some k N. Its running time: T (n) = O(1). Let SEVENS = { n : n N, n mod 7 = 0}. It is decidable by the algorithm of dividing the input n by 7 and determining the remainder. Running time T (n) = O(log n). 12 / 64

32 Definition Übersicht 1 Bibliography 2 Decision problems languages Definitions 3 Turing Machines Definition Church-Turing Thesis: Algorithm = Turing Machine Universal Turing Machine Decidability Nondeterministic Turing Machines 13 / 64

33 Definition Definition A Turing machine M with m tapes, m 1, is a tuple (Q, Γ, δ) where Q is a finite set of possible states in which M s register can be, containing a start symbol q s and a halting state q h ; Γ is a finite set of the symbols, the alphabet of M, which M s tapes can attain, containing a blank symbol, a start symbol, and the numbers 0 and 1; a function δ : Q Γ m Q Γ m {,, } m describes the rules the machine M uses in performing each step; this function is called the transition function of M. Γ is also called the tape alphabet of M. It has to be distinguished from the alphabet Σ = Γ \ {, } which underlies a language which is written on one of the tapes as input. 14 / 64

34 Definition Turing machine with binary alphabet and single tape (m = 1) Program δ Finite state Control Q Read/Write Head Tape L R 15 / 64

35 Übersicht 1 Bibliography 2 Decision problems languages Definitions 3 Turing Machines Definition Church-Turing Thesis: Algorithm = Turing Machine Universal Turing Machine Decidability Nondeterministic Turing Machines 16 / 64

36 Example: Binary successor q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) Consider the single-tape Turing machine M = (Q, Γ, δ) (m = 1) with the states the tape alphabet Q = {q s, q 1, q h }, Γ = {0, 1,, }, and the transition function δ as given by the table on the left. 17 / 64

37 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, ?? 18 / 64

38 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, / 64

39 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, q s, / 64

40 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, q s, q s, / 64

41 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, q s, q s, q s, / 64

42 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, q s, q s, q s, q s, / 64

43 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, q s, q s, q s, q s, q s, / 64

44 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, q s, q s, q s, q s, q s, q 1, / 64

45 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, q s, q s, q s, q s, q s, q 1, q 1, / 64

46 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, q s, q s, q s, q s, q s, q 1, q 1, q 1, / 64

47 Transition function and its computation for input q Q σ Γ δ(q, σ) q s 0 (q s, 0, ) q s 1 (q s, 1, ) q s (q 1,, ) q s (q s,, ) q 1 0 (q h, 1, ) q 1 1 (q 1, 0, ) q 1 (q h,, ) 0. q s, q s, q s, q s, q s, q s, q s, q 1, q 1, q 1, q h, / 64

48 Example: Palindrome checker q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) Let M = (Q, Γ, δ) be a single-tape Turing machine where Q = {q s, q 0, q 1, q 0, q 1, q w, q y, q n } with two halting states q y, q n ( yes, no ), Γ = {0, 1,, }, and δ is given by the tables on the left. 29 / 64

49 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, ?? 30 / 64

50 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, / 64

51 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, / 64

52 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, / 64

53 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, / 64

54 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, q 1, / 64

55 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, q 1, q w, 0 36 / 64

56 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, q 1, q w, 0 7. q w, 0 37 / 64

57 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, q 1, q w, 0 7. q w, 0 8. q s, 0 38 / 64

58 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, q 1, q w, 0 7. q w, 0 8. q s, 0 9. q 0, 39 / 64

59 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, q 1, q w, 0 7. q w, 0 8. q s, 0 9. q 0, 10. q 0, 40 / 64

60 Transition function and computation for input 101 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, q 1, q w, 0 7. q w, 0 8. q s, 0 9. q 0, 10. q 0, 11. q y, 41 / 64

61 Functioning of the palindrome checker 1 In state q s, the Turing machine M searches its string for the first input symbol x (after the ) 2 M remembers the symbol x by changing its corresponding state q x (This capability of a Turing machine to remember finite information in its state is crucial) 3 The symbol x is deleted by 4 M changes to its walking state q w and moves to the right until the first is met 5 M moves one place to the left and changes to the state q x, still remembering the first symbol x 6 If the last symbol x of the string equals the remembered symbol x, it is replaced by; otherwise, the machine halts with qn ( no ). 7 If the remaining string is empty, M halts with q y ( yes ); M is in state q w and moves to the left until a symbol is reached. 8 Repeat step 1 42 / 64

62 Transition function and computation for input 100 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, ?? 43 / 64

63 Transition function and computation for input 100 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, / 64

64 Transition function and computation for input 100 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, / 64

65 Transition function and computation for input 100 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, / 64

66 Transition function and computation for input 100 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, / 64

67 Transition function and computation for input 100 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, q 1, / 64

68 Transition function and computation for input 100 q σ δ(q, σ) q s 0 (q s,, ) q s 1 (q s,, ) q s (q s,, ) q s (q y,, ) q 0 0 (q s, 0, ) q 0 1 (q s, 1, ) q 0 ( q 0,, ) q 1 0 (q 1, 0, ) q 1 1 (q 1, 1, ) q 1 ( q 1,, ) q σ δ(q, σ) q 0 0 (q w,, ) q 0 1 (q n, 1, ) q 0 (q y,, ) q 1 0 (q n, 0, ) q 1 1 (q w,, ) q 1 (q y,, ) q w 0 (q w, 0, ) q w 1 (q w, 1, ) q w (q s,, ) 0. q s, q s, q 1, q 1, q 1, q 1, q n, 0 49 / 64

69 Church-Turing Thesis: Algorithm = Turing Machine Übersicht 1 Bibliography 2 Decision problems languages Definitions 3 Turing Machines Definition Church-Turing Thesis: Algorithm = Turing Machine Universal Turing Machine Decidability Nondeterministic Turing Machines 50 / 64

70 Church-Turing Thesis: Algorithm = Turing Machine Church-Turing Thesis Every physically realizable algorithm can be simulated by a Turing machine. 51 / 64

71 Universal Turing Machine Übersicht 1 Bibliography 2 Decision problems languages Definitions 3 Turing Machines Definition Church-Turing Thesis: Algorithm = Turing Machine Universal Turing Machine Decidability Nondeterministic Turing Machines 52 / 64

72 Universal Turing Machine Universal Turing Machine A universal Turing machine is a Turing machine which can simulate the execution of any other Turing machine. 53 / 64

73 Universal Turing Machine Universal Turing Machine A universal Turing machine is a Turing machine which can simulate the execution of any other Turing machine. Computers (desktop, laptop, smart phones,...) are universal Turing machines, since any program can run on them. 53 / 64

74 Universal Turing Machine Universal Turing Machine A universal Turing machine is a Turing machine which can simulate the execution of any other Turing machine. Computers (desktop, laptop, smart phones,...) are universal Turing machines, since any program can run on them. However, it was a surprising observation in the 1930 s that general-purpose computers are possible at all. After all, the parameters of a universal Turing machine alphabet size, number of states, and number of tapes are fixed. 53 / 64

75 Universal Turing Machine Universal Turing Machine A universal Turing machine is a Turing machine which can simulate the execution of any other Turing machine. Computers (desktop, laptop, smart phones,...) are universal Turing machines, since any program can run on them. However, it was a surprising observation in the 1930 s that general-purpose computers are possible at all. After all, the parameters of a universal Turing machine alphabet size, number of states, and number of tapes are fixed. Alan Turing ( ) was the first to show in the 1930 s that this is not a hurdle because of the ability to use encodings. 53 / 64

76 Universal Turing Machine Efficient universal Turing machine Theorem. Efficient universal Turing machine] There exists a Turing machine U such that for every with x, α {0, 1} U(x, α) = M α (x), where M α denotes the Turing machine represented by α. Moreover, if M α halts on input x within T steps, then U(x, α) halts within kt ln T steps, where k is a number independent of n = x and depending only on M α s alphabet size, number of tapes, and number of states. 54 / 64

77 Decidability Übersicht 1 Bibliography 2 Decision problems languages Definitions 3 Turing Machines Definition Church-Turing Thesis: Algorithm = Turing Machine Universal Turing Machine Decidability Nondeterministic Turing Machines 55 / 64

78 Decidability Decidability A language L is called Turing-recognizable, if there exists a Turing machine which halts accepting on every input x L. (However, there possibly might exist no Turing machine which ever rejects an input x / L.) 56 / 64

79 Decidability Decidability A language L is called Turing-recognizable, if there exists a Turing machine which halts accepting on every input x L. (However, there possibly might exist no Turing machine which ever rejects an input x / L.) A language is called decidable, if there exists a Turing machine which, on every input, halts, either accepting or rejecting. 56 / 64

80 Decidability Decidability A language L is called Turing-recognizable, if there exists a Turing machine which halts accepting on every input x L. (However, there possibly might exist no Turing machine which ever rejects an input x / L.) A language is called decidable, if there exists a Turing machine which, on every input, halts, either accepting or rejecting. Every decidable language is Turing-recognizable, but not vice versa. 56 / 64

81 Decidability Decidability, computability, enumerability Since deciding a language is equivalent to computing a function, a decidable language is also called computable AB09, p 23], Sip06, 5.17]. 57 / 64

82 Decidability Decidability, computability, enumerability Since deciding a language is equivalent to computing a function, a decidable language is also called computable AB09, p 23], Sip06, 5.17]. The Turing-recognizable languages are exactly the recursively enumerable languages Hof09, 4.6]. 57 / 64

83 Decidability Decidability, computability, enumerability Since deciding a language is equivalent to computing a function, a decidable language is also called computable AB09, p 23], Sip06, 5.17]. The Turing-recognizable languages are exactly the recursively enumerable languages Hof09, 4.6]. The decidable languages are also called recursive languages Pap94, 2.2]. (Recursive languages are not contained in the Chomsky hierarchy, but all regular, context-free, and context-sensitive languages are recursive Hof09, p 302], Sip06, p 172].) 57 / 64

84 Decidability Excursion: Diophantine sets A set X N n 0 is Diophantine if it satisfies X = {(x 1,..., x n ) : y N m 0 s.t. f (x 1,..., x n, y 1,..., y m ) = 0} where f : Z m+n Z is a polynomial with integer coefficients 58 / 64

85 Decidability Excursion: Diophantine sets A set X N n 0 is Diophantine if it satisfies X = {(x 1,..., x n ) : y N m 0 s.t. f (x 1,..., x n, y 1,..., y m ) = 0} where f : Z m+n Z is a polynomial with integer coefficients Matiyasevich s theorem (1970): A set X N n 0 is Turing-recognizable if and only if it is Diophantine. 58 / 64

86 Decidability Excursion: Diophantine sets A set X N n 0 is Diophantine if it satisfies X = {(x 1,..., x n ) : y N m 0 s.t. f (x 1,..., x n, y 1,..., y m ) = 0} where f : Z m+n Z is a polynomial with integer coefficients Matiyasevich s theorem (1970): A set X N n 0 is Turing-recognizable if and only if it is Diophantine. Matiyasevich s theorem resolved the tenth Hilbert problem (1900): There is no general algorithm to decide a Diophantine equation. 58 / 64

87 Decidability Excursion: Diophantine sets A set X N n 0 is Diophantine if it satisfies X = {(x 1,..., x n ) : y N m 0 s.t. f (x 1,..., x n, y 1,..., y m ) = 0} where f : Z m+n Z is a polynomial with integer coefficients Matiyasevich s theorem (1970): A set X N n 0 is Turing-recognizable if and only if it is Diophantine. Matiyasevich s theorem resolved the tenth Hilbert problem (1900): There is no general algorithm to decide a Diophantine equation. Moreover: Any Turing-recognizable language uniquely corresponds to a Diophantine equation: Computability means number theory 58 / 64

88 Decidability Excursion: 1 of Diophantine sets The set of all even non-negative integers is defined by the Diophantine equation x = 2y, i.e., f (x, y) = x 2y / 64

89 Decidability Excursion: 1 of Diophantine sets The set of all even non-negative integers is defined by the Diophantine equation x = 2y, i.e., f (x, y) = x 2y. The set of all squares is defined by the Diophantine equation x = y 2, i.e., f (x, y) = x y / 64

90 Decidability Excursion: 1 of Diophantine sets The set of all even non-negative integers is defined by the Diophantine equation x = 2y, i.e., f (x, y) = x 2y. The set of all squares is defined by the Diophantine equation x = y 2, i.e., f (x, y) = x y 2. The set of all non-negative integers that are not squares is defined by Pell s equation (y 1 + 1) 2 x(y 2 + 1) 2 = 1, i.e., f (x, y 1, y 2 ) = (y 1 + 1) 2 x(y 2 + 1) 2 1, provided that the unknowns y 1 and y 2 range over non-negative integers / 64

91 Decidability Excursion: Fibonacci numbers 2 as Diophantine sets The set of all Fibonacci numbers is defined by f (x, y) = (x 2 + xy y 2 ) / 64

92 Decidability Excursion: Fibonacci numbers 2 as Diophantine sets The set of all Fibonacci numbers is defined by f (x, y) = (x 2 + xy y 2 ) 2 1. E.g., (x, y) = (0, 1), (1, 1), (1,2), (2, 3), (3, 5), (5, 8), solve f = / 64

93 Decidability Excursion: Fibonacci numbers 2 as Diophantine sets The set of all Fibonacci numbers is defined by f (x, y) = (x 2 + xy y 2 ) 2 1. E.g., (x, y) = (0, 1), (1, 1), (1,2), (2, 3), (3, 5), (5, 8), solve f = 0. In fact, the Diophantine equation follows from Cassini s identity F 2 n+1 F n+1f n F 2 n = ( 1) n, where F n denotes the n-th Fibonacci number GKP94, (6.106)] / 64

94 Decidability Excursion: Fibonacci numbers 2 as Diophantine sets The set of all Fibonacci numbers is defined by f (x, y) = (x 2 + xy y 2 ) 2 1. E.g., (x, y) = (0, 1), (1, 1), (1,2), (2, 3), (3, 5), (5, 8), solve f = 0. In fact, the Diophantine equation follows from Cassini s identity F 2 n+1 F n+1f n F 2 n = ( 1) n, where F n denotes the n-th Fibonacci number GKP94, (6.106)]. The Fibonacci numbers play a important role in Matiyasevich s proof of his theorem GKP94, pp 294] / 64

95 Decidability Excursion: Fibonacci numbers 2 as Diophantine sets The set of all Fibonacci numbers is defined by f (x, y) = (x 2 + xy y 2 ) 2 1. E.g., (x, y) = (0, 1), (1, 1), (1,2), (2, 3), (3, 5), (5, 8), solve f = 0. In fact, the Diophantine equation follows from Cassini s identity F 2 n+1 F n+1f n F 2 n = ( 1) n, where F n denotes the n-th Fibonacci number GKP94, (6.106)]. The Fibonacci numbers play a important role in Matiyasevich s proof of his theorem GKP94, pp 294]. F n can be computed in a closed form: F n = φn ˆφ n 5 where φ = (1 + 5)/ is the golden ratio and ˆφ = 1/φ, both satisfying the equation x 2 = x + 1 GKP94, pp 299] / 64

96 Nondeterministic Turing Machines Übersicht 1 Bibliography 2 Decision problems languages Definitions 3 Turing Machines Definition Church-Turing Thesis: Algorithm = Turing Machine Universal Turing Machine Decidability Nondeterministic Turing Machines 61 / 64

97 Nondeterministic Turing Machines Nondeterministic Turing Machines A non-deterministic Turing machine M is a tuple (Q, Γ, δ) where the set Q of machine states and the tape alphabet Γ are given as for a deterministic Turing machine, but the transition function is δ : Q Γ m P(Q Γ m {,, } m ), Here P(A) for an arbitrary set A denotes the power set of A, i.e., the set of all possible subsets of A. 62 / 64

98 Nondeterministic Turing Machines Nondeterministic Turing Machines A non-deterministic Turing machine M is a tuple (Q, Γ, δ) where the set Q of machine states and the tape alphabet Γ are given as for a deterministic Turing machine, but the transition function is δ : Q Γ m P(Q Γ m {,, } m ), Here P(A) for an arbitrary set A denotes the power set of A, i.e., the set of all possible subsets of A. In particular,, A P(A), and P(A) = 2 A. 62 / 64

99 Nondeterministic Turing Machines Nondeterministic Turing Machines A non-deterministic Turing machine M is a tuple (Q, Γ, δ) where the set Q of machine states and the tape alphabet Γ are given as for a deterministic Turing machine, but the transition function is δ : Q Γ m P(Q Γ m {,, } m ), Here P(A) for an arbitrary set A denotes the power set of A, i.e., the set of all possible subsets of A. In particular,, A P(A), and P(A) = 2 A. For instance, for A = {a, b, c} we have / 64

100 Nondeterministic Turing Machines Deterministic versus nondeterministic Turing Machines time The succession of computational steps in a deterministic Turing machine (left), where for each state-symbol combination there is at most one next step, and in a nondeterministic Turing-machine (right), where for each state-symbol combination there may be more than one next step. 63 / 64

101 Nondeterministic Turing Machines Deterministic versus nondeterministic Turing Machines time The succession of computational steps in a deterministic Turing machine (left), where for each state-symbol combination there is at most one next step, and in a nondeterministic Turing-machine (right), where for each state-symbol combination there may be more than one next step. In the deterministic case, a problem is accepted if the Turing machine halts with the final state yes, whereas in the nondeterministic case it is solved if there is at least one final state yes. 63 / 64

102 Nondeterministic Turing Machines Discussion Any questions? 64 / 64