VII. Problem Index Yuxi Fu BASICS, Shanghai Jiao Tong University - - PowerPoint PPT Presentation

• VII. Problem Index

Yuxi Fu

BASICS, Shanghai Jiao Tong University

Motivation

By Church-Turing Thesis one may study computability theory using any of the computation models. It is much more instructive however to carry out the study in a model independent manner. The first step is to assign index to computable function.

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Agenda

We shall study three fundamental theorems about indexing.

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Synopsis

• 1. G¨
• del Index
• 2. S-m-n Theorem
• 3. Enumeration Theorem
• 4. Recursion Theorem

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• 1. G¨
• del Index

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Basic Idea

We see a number as an index for a problem/function if it is the G¨

• del number of a programme that solves/calculates the

problem/function.

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Definition

Suppose a ∈ ω and n ≥ 1. φ(n)

a

= the n ary function computed by Pa = f (n)

Pn ,

W (n)

a

= the domain of φ(n)

a

= {(x1, . . . , xn) | Pa(x1, . . . , xn) ↓}, E (n)

a

= the range of φ(n)

a .

The super script (n) is omitted when n = 1.

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Example

Let a = 4127. Then P4127 = S(2); T(2, 1). If the program is seen to calculate a unary function, then φ4127(x) = 1, W4127 = ω, E4127 = {1}. If the program is seen to calculate an n-ary function, then φ(n)

4127(x1, . . . , xn)

= x2 + 1, W n

4127

= ωn, E n

4127

= ω+.

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G¨

• del Index for Computable Function

Suppose f is an n-ary computable function. A number a is an index for f if f = φ(n)

a .

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Padding Lemma

Padding Lemma. Every computable function has infinite indices. Moreover for each x we can effectively find an infinite recursive set Ax of indices for φx.

Proof.

Systematically add useless instructions to Px.

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Computable Functions are Enumerable

We may list for example all the elements of C as φ0, φ1, φ2, . . . .

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Diagonal Method

• Fact. There is a total unary function that is not computable.

Proof.

Suppose φ0, φ1, φ2, . . . is an enumeration of C. Define f (n) = φn(n) + 1, if φn(n) is defined, 0, if φn(n) is undefined. By Church-Turing Thesis the function f (n) is not computable.

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• 2. S-m-n Theorem

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How do different indexing systems relate?

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S-m-n Theorem, the Unary Case

Given a binary function f (x, y), we get a unary computable function f (a, y) by fixing a value a for x. Let e be an index for f (a, y). Then f (a, y) ≃ φe(y). S-m-n Theorem states that the index e can be computed from a.

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S-m-n Theorem, the Unary Case

• Fact. Suppose that f (x, y) is a computable function. There is a

primitive recursive function k(x) such that f (x, y) ≃ φk(x)(y).

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S-m-n Theorem, the Unary Case

Let F be a program that computes f . Consider the following T(1, 2) Z(1) S(1) . . . S(1)      a times F The above program can be effectively constructed from a. Let k(a) be the G¨

• del number of the above program.

It can be effectively computed from the above program.

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Example

• 1. Let f (x, y) = yx. Then φk(x)(y) = yx. For each fixed n, k(n)

is an index for yn.

• 2. Let f (x, y) ≃

y, if y is a multiple of x, ↑,

• therwise.

. Then φk(n)(y) is defined if and only if y is a multiple of n.

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S-m-n Theorem. For m, n, there is an injective primitive recursive (m + 1)-function sm

n (x,

x) such that for all e the following holds: φm+n

e

( x, y) ≃ φn

sm

n (e,

x)(

y). S-m-n Theorem is also called Parameter Theorem.

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Proof of S-m-n Theorem

• Proof. Given e, x1, . . . , xm, we can effectively construct the

following program and its index T(n, m + n) . . . T(1, m + 1) Q(1, x1) . . . Q(m, xm) Pe where Q(i, x) is the program Z(i), S(i), . . . , S(i)

• x times

. The injectivity is achieved by padding enough useless instructions.

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• 3. Enumeration Theorem

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What makes it possible that every C program can be executed in a computer?

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General Remark

There are universal programs that embody all the programs. A program is universal if upon receiving the G¨

• del number of a

program it simulates the program indexed by the number.

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Intuition

Consider the function ψ(x, y) defined as follows ψ(x, y) ≃ φx(y). In an obvious sense ψ(x, ) is a universal function for the unary functions φ0, φ1, φ2, φ3, . . . .

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Universal Function

The universal function for n-ary computable functions is the (n + 1)-ary function ψ(n)

U

defined by ψ(n)

U (e, x1, . . . , xn) ≃ φ(n) e (x1, . . . , xn).

We write ψU for ψ(1)

U .

Question: Is ψ(n)

U

computable?

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Enumeration Theorem. For each n, the universal function ψ(n)

U

is computable.

Proof.

Given a number e, decode the number to get the program Pe; and then simulate the program Pe. If the simulation ever terminates, then return the number in R1. By Church-Turing Thesis, ψ(n)

U

is computable.

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Application: Undecidability

• Proposition. The problem ‘φx is total’ is undecidable.

Proof.

If ‘φx is total’ were decidable, then by Church’s Thesis f (x) = ψU(x, x) + 1, if φx is total, 0, if φx is not total. would be a total computable function that differs from every total computable function.

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Application: Effectiveness of Function Operation

• Proposition. There is a total computable function s(x, y) such

that φs(x,y) = φxφy for all x, y.

Proof.

Let f (x, y, z) ≃ φx(z)φy(z) ≃ ψU(x, z)ψU(y, z). By S-m-n Theorem there is a total function s(x, y) such that φs(x,y)(z) ≃ f (x, y, z).

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Application: Effectiveness of Set Operation

• Proposition. There is a total computable function s(x, y) such

that Ws(x,y) = Wx ∪ Wy.

Proof.

Let f (x, y, z) = 1, if z ∈ Wx or z ∈ Wy, undefined,

• therwise.

By S-m-n Theorem there is a total function s(x, y) such that φs(x,y)(z) ≃ f (x, y, z). Clearly Ws(x,y) = Wx ∪ Wy.

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Application: Effectiveness of Recursion

Consider f defined by the following recursion f (e1, e2, x, 0) ≃ φ(n)

e1 (

x) ≃ ψ(n)

U (e1,

x), and f (e1, e2, x, y + 1) ≃ φ(n+2)

e2

( x, y, f (e1, e2, x, y)) ≃ ψ(n+2)

U

(e2, x, y, f (e1, e2, x, y)). By S-m-n Theorem, there is a total computable function r(e1, e2) such that φ(n+1)

r(e1,e2)(

x, y) ≃ f (e1, e2, x, y).

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Application: Non-Primitive Recursive Total Function

• Theorem. There is a total computable function that is not

primitive recursive.

Proof.

• 1. The primitive recursive functions have a universal function.
• 2. Such a function cannot be primitive recursive by

diagonalisation.

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• 4. Recursion Theorem

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Recursion Theorem (Kleene, 1938). Let f be a total unary computable function. Then there is a number n such that φf (n) = φn.

Proof.

By S-m-n Theorem there is an injective primitive recursive function s(x) such that for all x φs(x)(y) ≃ φφx(x)(y), if φx(x) ↓; ↑,

• therwise.

(1) Let v be such that φv = s; f . Obviously φv is total and φv(v) ↓. It follows from (1) that φs(v) = φφv(v) = φf (s(v)). We are done by letting n be s(v).

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More about Recursion Theorem

• Fact. If f is partial then φf (n) = φn whenever f (n) ↓.
• Fact. The fixpoint n can be computed from an index of f by an

injective primitive recursive function.

Proof.

Let v(z) be an injective primitive recursive function such that φv(z) ≃ s; φz. Then let n(z) ≃ s(v(z)).

• Fact. There is an infinite set of fixpoints for f .

Proof.

By Padding Lemma there is an infinite set of indices v such that φv ≃ s; f .

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• Fact. If f is a total computable function, there is a number n such

that Wf (n) = Wn and Ef (n) = En.

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Self Referential Definition

• Fact. Let f (x, y) be a computable function. Then there is an

index e such that φe(y) ≃ f (e, y).

Proof.

By S-m-n Theorem there is a total computable function s(x) such that φs(x)(y) ≃ f (x, y). We are done by applying Recursion Theorem.

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Self Referential Definition

The previous corollary makes it meaningful to define a computable function φe(y) by a computable function f (e, y).

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Self Referential Definition

There is a number n such that φn(x) = xn. There is a number n such that Wn = {n}. This number is

• btained by applying the above corollary to the function

f (x, y) = 0, if x = y, ↑,

• therwise.

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Self Printing Program

• Fact. There is a program P such that for all x, P(x) ↓ γ(P).

Proof.

It says that there is a number n such that φn(x) = n for all x. Simply apply one of the corollaries to f (z, x) = z.

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Applying Recursion Theorem

• Theorem. Suppose that f is a total increasing function such that

◮ if m = n then φf (m) = φf (n), ◮ f (n) is the least index of the function φf (n).

Then f is not computable.

Proof.

Suppose f satisfies the conditions of the theorem. By the first condition f (n) > n if n is large enough. By the second condition φf (n) = φn for all large enough n. This contradicts to Recursion Theorem.

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Diagonalisation Implicit in Recursion Theorem

We may think of φx as providing an effective enumeration of φφx(0), φφx(1), φφx(2), . . . , φφx(i), . . . . The diagonal function φx(x) enumerates φφ0(0), φφ1(1), φφ2(2), . . .. Let f be total computable. There is some total computable s(x), due to S-m-n Theorem, and some index m for s(x) such that φφm(x)(y) ≃ φs(x)(y) ≃ φf (φx(x))(y). Both φm and f (φx(x)) enumerate the following φφm(0), φφm(1), φφm(2), . . ., φφm(i), . . ..

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Diagonalisation Implicit in Recursion Theorem

φφ0(0), φφ0(1), φφ0(2), . . . , φφ0(i), . . . φφ1(0), φφ1(1), φφ1(2), . . . , φφ1(i), . . . φφ2(0), φφ2(1), φφ2(2), . . . , φφ2(i), . . . . . . . . . . . . . . . φφm(0), φφm(1), φφm(2), . . ., φφm(m), . . . . . . . . . . . . . . . The diagonal intersects the m-row at φφm(m) = φf (φm(m)). It is important that φm(m) must be defined.

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Diagonalisation offers an intuition about Recursion Theorem. It also explains the power of Recursion Theorem.

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Generalized Recursion Theorem. Suppose f (x, z) is a total computable function. There is an injective primitive recursive function n(z) such that φf (n(z),z) = φn(z) for all z.

Proof.

By S-m-n Theorem there is an injective primitive recursive function s(x, z) such that φf (φx(x),z) = φs(x,z). By the same theorem there is an injective primitive recursive function m(z) such that s(x, z) = φm(z)(x). So φf (φx(x),z) = φφm(z)(x). We are done by letting x = m(z) and n(z) = φm(z)(m(z)). It is clear that n is an injective primitive recursive function.

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Basic recursion theory can be developed from S-m-n Theorem and Enumeration Theorem. Advanced recursion theory makes frequent use of Recursion Theorem.

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