VI. Church-Turing Thesis

Transcription

1 VI. Church-Turing Thesis Yuxi Fu BASICS, Shanghai Jiao Tong University

2 Fundamental Question How do computation models characterize the informal notion of effective computability? Computability Theory, by Y. Fu VI. Church-Turing Thesis 1 / 41

3 Fundamental Result Theorem. The set of functions definable in λ-calculus (Turing Machine Model, Unlimited Random Access Machine Model) is precisely the set of recursive functions. Proof. We have already showed that µ-definable λ-definable Turing definable URM-definable. We have to show that URM-definable µ-definable. Computability Theory, by Y. Fu VI. Church-Turing Thesis 2 / 41

4 Synopsis 1. Gödel Encoding 2. Kleene s Proof 3. Church-Turing Thesis Computability Theory, by Y. Fu VI. Church-Turing Thesis 3 / 41

5 Gödel Encoding Computability Theory, by Y. Fu VI. Church-Turing Thesis 4 / 41

6 Godel s Insight The set of syntactical objects of a formal system is denumerable. More importantly, every syntactical object can be coded up effectively by a number in such a way that a unique syntactical object can be recovered from the number. This is the crucial technique Gödel used in his proof of the Incompleteness Theorem. Computability Theory, by Y. Fu VI. Church-Turing Thesis 5 / 41

7 Enumeration An enumeration of a set X is a surjection g : ω X ; this is often represented by writing {x 0, x 1, x 2,...}. It is an enumeration without repetition if g is injective. Computability Theory, by Y. Fu VI. Church-Turing Thesis 6 / 41

8 Denumeration A set X is denumerable if there is a bijection f : X ω. (denumerate = denote + enumerate) Let X be a set of finite objects. Then X is effectively denumerable if there is a bijection f : X ω such that both f and f 1 are computable. Computability Theory, by Y. Fu VI. Church-Turing Thesis 7 / 41

9 Encoding Pair Fact. ω ω is effectively denumerable. Proof. A bijection π : ω ω ω is defined by π(m, n) π 1 (l) def = 2 m (2n + 1) 1, def = (π 1 (l), π 2 (l)), where π 1 (x) π 2 (x) def = (x + 1) 1, def = ((x + 1)/2 π 1(x) 1)/2. Computability Theory, by Y. Fu VI. Church-Turing Thesis 8 / 41

10 Encoding Tuple Fact. ω + ω + ω + is effectively denumerable. Proof. A bijection ζ : ω + ω + ω + ω is defined by ζ(m, n, q) ζ 1 (l) def = π(π(m 1, n 1), q 1), def = (π 1 (π 1 (l)) + 1, π 2 (π 1 (l)) + 1, π 2 (l) + 1). Computability Theory, by Y. Fu VI. Church-Turing Thesis 9 / 41

11 Encoding Finite String Fact. k>0 ωk is effectively denumerable. Proof. A bijection τ : k>0 ωk ω is defined by τ(a 1,..., a k ) def = 2 a a 1+a a 1+a 2 +a a 1+a 2 +a ,a k +k 1 1. Now given x it is easy to find b 1 < b 2 <... < b k such that 2 b b b b k = x + 1. It is then clear how to calculate a 1, a 2, a 3,..., a k. Details are next. Computability Theory, by Y. Fu VI. Church-Turing Thesis 10 / 41

12 Encoding Finite String A number x ω has a unique expression as x = α i 2 i, i=0 where α i is either 0 or 1 for all i The function α(i, x) = α i is primitive recursive: α(i, x) = rm(2, qt(2 i, x)). 2. The function l(x) = if x > 0 then k else 0 is primitive recursive: l(x) = i<x α(i, x). Computability Theory, by Y. Fu VI. Church-Turing Thesis 11 / 41

13 Encoding Finite String 3. If x > 0 then it has a unique expression as x = 2 b b b k, where 1 k and 0 b 1 < b 2 <... < b k. The function b(i, x) = if (x > 0) (1 i l(x)) then b i else 0 is primitive recursive: { ( ) µy<x b(i, x) = k y α(k, x) = i, if (x > 0) (1 i l(x)); 0, otherwise. Computability Theory, by Y. Fu VI. Church-Turing Thesis 12 / 41

14 Encoding Finite String 4. If x > 0 then it has a unique expression as x = 2 a a 1+a a l +a a k +k 1. The function a(i, x) = a i is primitive recursive: a(i, x) = b(i, x), if i = 0 or i = 1, a(i + 1, x) = (b(i + 1, x) b(i, x)) 1, if i 1. We conclude that a 1, a 2, a 3,..., a k can be calculated by primitive recursive functions. Computability Theory, by Y. Fu VI. Church-Turing Thesis 13 / 41

15 Encoding Programme Let I be the set of all instructions. Let P be the set of all programs. The objects in I, and P as well, are finite objects. Computability Theory, by Y. Fu VI. Church-Turing Thesis 14 / 41

16 Encoding Programme Theorem. I is effectively denumerable. Proof. The bijection β : I ω is defined as follows: β(z(n)) = 4(n 1), β(s(n)) = 4(n 1) + 1, β(t (m, n)) = 4π(m 1, n 1) + 2, β(j(m, n, q)) = 4ζ(m, n, q) + 3. The converse β 1 is easy. Computability Theory, by Y. Fu VI. Church-Turing Thesis 15 / 41

17 Encoding Programme Theorem. P is effectively denumerable. Proof. The bijection γ : P ω is defined as follows: assuming P = I 1,..., I s. The converse γ 1 is obvious. γ(p) = τ(β(i 1 ),..., β(i s )), Computability Theory, by Y. Fu VI. Church-Turing Thesis 16 / 41

18 Gödel Number of Programme The value γ(p) is called the Gödel number of P. P n = the programme with Godel index n = γ 1 (n) We shall fix this particular encoding function γ throughout. Computability Theory, by Y. Fu VI. Church-Turing Thesis 17 / 41

19 Example Let P be the program T (1, 3), S(4), Z(6). β(t (1, 3)) = 18, β(s(4)) = 13, β(z(6)) = 20. γ(p) = Computability Theory, by Y. Fu VI. Church-Turing Thesis 18 / 41

20 Example Consider P = β(i 1 ) = 4 + 1, β(i 2 ) = 4π(1, 0) + 2. So P 4127 is S(2); T (2, 1). Computability Theory, by Y. Fu VI. Church-Turing Thesis 19 / 41

21 Kleene s Proof Computability Theory, by Y. Fu VI. Church-Turing Thesis 20 / 41

22 Kleene demonstrated how to prove that machine computable functions are recursive functions. Computability Theory, by Y. Fu VI. Church-Turing Thesis 21 / 41

23 Proof in Detail The state of the computation of the program P e ( x) can be described by a configuration and an instruction number. A state can be coded up by the number σ = π(c, j), where c is the configuration that codes up the current values in the registers c = 2 r 1 3 r 2... = i 1 p r i i, and j is the next instruction number. Computability Theory, by Y. Fu VI. Church-Turing Thesis 22 / 41

24 Proof in Detail To describe the changes of the states of P e ( x), we introduce three (n + 2)-ary functions: c n (e, x, t) = the configuration after t steps of P e ( x), j n (e, x, t) = the number of the next instruction after t steps of P e ( x) (it is 0 if P e ( x) stops in t or less steps), σ n (e, x, t) = π(c n (e, x, t), j n (e, x, t)). If σ n is primitive recursive, then c n, j n are primitive recursive. Computability Theory, by Y. Fu VI. Church-Turing Thesis 23 / 41

25 Proof in Detail If the computation of P e ( x) stops, it does so in µt(j n (e, x, t) = 0) steps. Then the final configuration is c n (e, x, µt(j n (e, x, t) = 0)). We conclude that the value of the computation P e ( x) is (c n (e, x, µt(j n (e, x, t) = 0))) 1. Computability Theory, by Y. Fu VI. Church-Turing Thesis 24 / 41

26 Proof in Detail The function σ n can be defined as follows: σ n (e, x, 0) = π(2 x 1 3 x 2... pn xn, 1), σ n (e, x, t + 1) = π(config(e, σ n (e, x, t)), next(e, σ n (e, x, t))), where config(e, π(c, j)) is the configuration after t + 1 steps; next(e, π(c, j)) is the new number after t + 1 steps. Computability Theory, by Y. Fu VI. Church-Turing Thesis 25 / 41

27 Proof in Detail ln(e) = the number of instructions in P e ; gn(e, j) = { the code of Ij in P e, if 1 j ln(e), 0, otherwise. Both functions are primitive recursive since ln(e) = l(e + 1), gn(e, j) = a(j, e + 1). Computability Theory, by Y. Fu VI. Church-Turing Thesis 26 / 41

28 Proof in Detail u(z) = m whenever z = β(z(m)) or z = β(s(m)): u(z) = qt(4, z) + 1. u 1 (z) = m 1 and u 2 (z) = m 2 whenever z = β(t (m 1, m 2 )): u 1 (z) = π 1 (qt(4, z)) + 1, u 2 (z) = π 2 (qt(4, z)) + 1. v 1 (z) = m 1 and v 2 (z) = m 2 and v 3 (z) = q if z = β(j(m 1, m 2, q)): v 1 (z) = π 1 (π 1 (qt(4, z))) + 1, v 2 (z) = π 2 (π 1 (qt(4, z))) + 1, v 3 (z) = π 2 (qt(4, z)) + 1. Computability Theory, by Y. Fu VI. Church-Turing Thesis 27 / 41

29 Proof in Detail The change in the configuration c effected by instruction Z(m): zero(c, m) = qt(p m (c)m, c). The change in the configuration c effected by instruction S(m): succ(c, m) = p m c. The change in the configuration c effected by instruction T (m, n): tran(c, m, n) = qt(p (c)n n, p n (c)m c). Computability Theory, by Y. Fu VI. Church-Turing Thesis 28 / 41

30 Proof in Detail The following function ch(c, z) = the resulting configuration when the configuration c is operated on by the instruction with code number z. is primitive recursive since zero(c, u(z)), if rm(4, z) = 0, succ(c, u(z)), if rm(4, z) = 1, ch(c, z) = tran(c, u 1 (z), u 2 (z)), if rm(4, z) = 2, c, if rm(4, z) = 3. Computability Theory, by Y. Fu VI. Church-Turing Thesis 29 / 41

31 Proof in Detail The following function the number j of the next instruction when the configuration c is operated if j > 0, v(c, j, z) = on by the jth instruction with code z, 0, if j = 0. is primitive recursive since j + 1, if rm(4, z) 3, v(c, j, z) = j + 1, v 3 (z), if rm(4, z) = 3 (c) v1 (z) (c) v2 (z), if rm(4, z) = 3 (c) v1 (z) = (c) v2 (z). Computability Theory, by Y. Fu VI. Church-Turing Thesis 30 / 41

32 Proof in Detail config(e, σ) = { ch(π1 (σ), gn(e, π 2 (σ))), if 1 π 2 (σ) ln(e), π 1 (σ), otherwise. next(e, σ) = { v(π1 (σ), π 2 (σ), gn(e, π 2 (σ))), if 1 π 2 (σ) ln(e), 0, otherwise. Computability Theory, by Y. Fu VI. Church-Turing Thesis 31 / 41

33 Proof in Detail We conclude that the functions c n, j n, σ n are primitive recursive. Computability Theory, by Y. Fu VI. Church-Turing Thesis 32 / 41

34 Further Constructions For each n 1, the following predicates are primitive recursive: 1. S n (e, x, y, t) def = P e ( x) y in t or fewer steps. 2. H n (e, x, t) def = P e ( x) in t or fewer steps. They are defined by S n (e, x, y, t) H n (e, x, t) def = j n (e, x, t) = 0 (c n (e, x, t)) 1 = y, def = j n (e, x, t) = 0. Computability Theory, by Y. Fu VI. Church-Turing Thesis 33 / 41

35 Kleene s Normal Form Theorem Let φ (n) e denote the n-ary function computed by P e. Theorem. (Kleene) There is a primitive recursive function U(x) and, for each n 1, a primitive recursive predicate T n (e, x, z) such that 1. φ (n) e ( x) is defined if and only if z.t n (e, x, z). 2. φ (n) e ( x) U(µzT n (e, x, z)). Proof. (1) T n (e, x, z) = S n (e, x, π 1 (z), π 2 (z)). (2) Let U(x) = π 1 (x). Then φ (n) e ( x) U(µz.T n (e, x, z)). Computability Theory, by Y. Fu VI. Church-Turing Thesis 34 / 41

36 Every computable function can be obtained from a primitive recursive function by using at most one application of the µ-operator in a standard manner. Computability Theory, by Y. Fu VI. Church-Turing Thesis 35 / 41

37 Church-Turing Thesis Computability Theory, by Y. Fu VI. Church-Turing Thesis 36 / 41

38 Church-Turing Thesis. The functions definable in all computation models are the same. They are precisely the computable functions. Church believed that all computable functions are λ-definable. Kleene termed it Church Thesis. Gödel accepted it only after he saw Turing s equivalence proof. Church-Turing Thesis is now universally accepted. Computability Theory, by Y. Fu VI. Church-Turing Thesis 37 / 41

39 Computable Function Let C be the set of all computable functions. Let C n be the set of all n-ary computable functions. Computability Theory, by Y. Fu VI. Church-Turing Thesis 38 / 41

40 Power of Church-Turing Thesis Noone has come up with a computable function that is not in C. When you are convincing people of your model of computation, you are constructing an effective translation from your model to a well-known computation model. Computability Theory, by Y. Fu VI. Church-Turing Thesis 39 / 41

41 Making Use of Church-Turing Thesis Church-Turing Thesis allows us to give an informal argument for the computability of a function. We will make use of CTT in this way without explicitly defining it. Computability Theory, by Y. Fu VI. Church-Turing Thesis 40 / 41

42 Comment on Church-Turing Thesis CTT and Physical Implementation Deterministic Turing Machines are physically implementable. This is the well-known von Neumann Architecture. Are quantum computers physically implementable? Can a quantum computer compute more or more efficiently? CTT, is it a Law of Nature or a Wisdom of Human? Computability Theory, by Y. Fu VI. Church-Turing Thesis 41 / 41