mathematical logic - introduction to theory of computability · mathematical logic introduction to...

Mathematical LogicIntroduction to Theory of Computability

Nikolaj Popov and Tudor Jebelean

Research Institute for Symbolic Computation, Linz

[email protected]

Outline

ComputabilityPreliminariesPrimitive Recursive FunctionsPartial FunctionsEnumeration of the Computable Functions

DecidabilityDecidable and Semidecidable Sets

The Halting Problem

Outline

The Halting Problem

Introduction

Various notions of computation developed by Church, Turingand Kleene

The three computational models (recursion, λ-calculus, and Turingmachine) were shown to be equivalent (1934).

Church-Turing thesisAny real-world computation can be translated into an equivalentcomputation involving a Turing machine (or a program in anyreasonable programming language).

The intuitive notion of effective computability for functions andalgorithms is formally expressed by Turing machines or the lambdacalculus.

A function is computable, in the intuitive sense, if and only if it isTuring-computable.

Introduction

Preliminaries

Domain of a functionDom[f ] = {x | f [x ] is defined}

Range of a functionRan[f ] = {y | ∃x ∈ Dom[f ] ∧ f [x ] = y}

Graph of a functionGraph[f ] = {〈x , y〉 | x ∈ Dom[f ] ∧ f [x ] = y}

Partial equality(∀x , y) (f [x ] ' y ⇔ 〈x , y〉 ∈ Graph[f ])

Preliminaries

f is defined on xf [x ] ↓ ⇔ ∃y (x , y) ∈ Graph[f ]

Partial equality '(∀x) (f [x ] ' g[x ])iff(∀x) (f [x ] ↓ ⇔ g[x ] ↓)(∀x) (f [x ] ↓ ⇒ f [x ] = g[x ])

f is computablef is computable function iffthere exists a program P which computes it

Preliminaries

Superposition

h is a superposition of f ,g1, . . . ,gk

h[x ] ' f [g1[x ], . . . ,gk [x ]]

TheoremGiven the computable functions f ,g1, . . . ,gk , thenh[x ] ' f [g1[x ], . . . ,gk [x ]] is computable function.

Superposition preserves computability.

Superposition

h[x ] ' f [g1[x ], . . . ,gk [x ]]

Superposition

h[x ] ' f [g1[x ], . . . ,gk [x ]]

Primitive Recursion

h is obtained by weak primitive recursion from g and a

h[x ] '{

a ⇐ x = 0g[x ,h[x − 1]] ⇐ o.w.

h is obtained by primitive recursion from f and g

h[x , y ] '{

f [x ] ⇐ y = 0g[x , y ,h[x , y − 1]] ⇐ o.w.

Primitive recursion preserves computability.

Primitive Recursion

h[x ] '{

a ⇐ x = 0g[x ,h[x − 1]] ⇐ o.w.

h[x , y ] '{

f [x ] ⇐ y = 0g[x , y ,h[x , y − 1]] ⇐ o.w.

Primitive Recursion

h[x ] '{

a ⇐ x = 0g[x ,h[x − 1]] ⇐ o.w.

h[x , y ] '{

f [x ] ⇐ y = 0g[x , y ,h[x , y − 1]] ⇐ o.w.

Primitive Recursive Functions

The basic functions are primitive recursiveO[x ] ' 0S[x ] ' x + 1Ini [x ] ' xi

The superposition is primitive recursiveIf f ,g1, . . . ,gk are primitive recursive, thenh[x ] ' f [g1[x ], . . . ,gk [x ]]

is primitive recursive.

The primitive recursion is primitive recursiveIf f and g are primitive recursive, then

h[x , y ] '{

f [x ] ⇐ y = 0g[x , y ,h[x , y − 1]] ⇐ o.w.

h[x , y ] '{

f [x ] ⇐ y = 0g[x , y ,h[x , y − 1]] ⇐ o.w.

h[x , y ] '{

f [x ] ⇐ y = 0g[x , y ,h[x , y − 1]] ⇐ o.w.

Primitive Recursion

TheoremAll the primitive recursive functions are computable.

TheoremAll the primitive recursive functions are total.

Primitive Recursion

TheoremAll the primitive recursive functions are computable.

TheoremAll the primitive recursive functions are total.

Examples

Addition is primitive recursivef1[x , y ] ' x + y

f1[x , y ] '{

x ⇐ y = 0S[f1[x , y − 1]] ⇐ o.w.

Multiplication is primitive recursivef2[x , y ] ' x .y

f2[x , y ] '{

0 ⇐ y = 0x + f2[x , y − 1] ⇐ o.w.

Examples

f1[x , y ] '{

x ⇐ y = 0S[f1[x , y − 1]] ⇐ o.w.

f2[x , y ] '{

0 ⇐ y = 0x + f2[x , y − 1] ⇐ o.w.

Examples

f1[x , y ] '{

x ⇐ y = 0S[f1[x , y − 1]] ⇐ o.w.

f2[x , y ] '{

0 ⇐ y = 0x + f2[x , y − 1] ⇐ o.w.

Examples

f1[x , y ] '{

x ⇐ y = 0S[f1[x , y − 1]] ⇐ o.w.

f2[x , y ] '{

0 ⇐ y = 0x + f2[x , y − 1] ⇐ o.w.

ExamplesPower is primitive recursivef3[x , y ] ' xy

f3[x , y ] '{

1 ⇐ y = 0x .f3[x , y − 1]] ⇐ o.w.

Subtraction-dot-one is primitive recursive

f4[x ] ' x−̇1 '{

0 ⇐ x = 0x − 1 ⇐ o.w.

f4[x ] '{

0 ⇐ x = 0f4[x − 1] + 1 ⇐ o.w.

f3[x , y ] '{

1 ⇐ y = 0x .f3[x , y − 1]] ⇐ o.w.

f4[x ] ' x−̇1 '{

0 ⇐ x = 0x − 1 ⇐ o.w.

f4[x ] '{

0 ⇐ x = 0f4[x − 1] + 1 ⇐ o.w.

f3[x , y ] '{

1 ⇐ y = 0x .f3[x , y − 1]] ⇐ o.w.

f4[x ] ' x−̇1 '{

0 ⇐ x = 0x − 1 ⇐ o.w.

f4[x ] '{

0 ⇐ x = 0f4[x − 1] + 1 ⇐ o.w.

f3[x , y ] '{

1 ⇐ y = 0x .f3[x , y − 1]] ⇐ o.w.

f4[x ] ' x−̇1 '{

0 ⇐ x = 0x − 1 ⇐ o.w.

f4[x ] '{

0 ⇐ x = 0f4[x − 1] + 1 ⇐ o.w.

ExamplesSubtraction-dot is primitive recursive

f5[x , y ] ' x−̇y '{

0 ⇐ x < yx − y ⇐ o.w.

f5[x , y ] '{

x ⇐ y = 0f5[x , y − 1]−̇1 ⇐ o.w.

Factorial is primitive recursivef6[x ] ' x!

f6[x ] '{

1 ⇐ x = 0x .f6[x − 1]] ⇐ o.w.

f5[x , y ] ' x−̇y '{

0 ⇐ x < yx − y ⇐ o.w.

f5[x , y ] '{

x ⇐ y = 0f5[x , y − 1]−̇1 ⇐ o.w.

f6[x ] '{

1 ⇐ x = 0x .f6[x − 1]] ⇐ o.w.

f5[x , y ] ' x−̇y '{

0 ⇐ x < yx − y ⇐ o.w.

f5[x , y ] '{

x ⇐ y = 0f5[x , y − 1]−̇1 ⇐ o.w.

f6[x ] '{

1 ⇐ x = 0x .f6[x − 1]] ⇐ o.w.

f5[x , y ] ' x−̇y '{

0 ⇐ x < yx − y ⇐ o.w.

f5[x , y ] '{

x ⇐ y = 0f5[x , y − 1]−̇1 ⇐ o.w.

f6[x ] '{

1 ⇐ x = 0x .f6[x − 1]] ⇐ o.w.

ExamplesSign is primitive recursive

sg[x ] '{

0 ⇐ x = 01 ⇐ o.w.

sg[x ] '{

0 ⇐ x = 0O[sg[x − 1]] + 1 ⇐ o.w.

Opposite-sign is primitive recursive

sg[x ] '{

1 ⇐ x = 00 ⇐ o.w.

sg[x ] '{

1 ⇐ x = 0O[sg[x − 1]] ⇐ o.w.

sg[x ] '{

0 ⇐ x = 01 ⇐ o.w.

sg[x ] '{

0 ⇐ x = 0O[sg[x − 1]] + 1 ⇐ o.w.

sg[x ] '{

1 ⇐ x = 00 ⇐ o.w.

sg[x ] '{

1 ⇐ x = 0O[sg[x − 1]] ⇐ o.w.

sg[x ] '{

0 ⇐ x = 01 ⇐ o.w.

sg[x ] '{

0 ⇐ x = 0O[sg[x − 1]] + 1 ⇐ o.w.

sg[x ] '{

1 ⇐ x = 00 ⇐ o.w.

sg[x ] '{

1 ⇐ x = 0O[sg[x − 1]] ⇐ o.w.

sg[x ] '{

0 ⇐ x = 01 ⇐ o.w.

sg[x ] '{

0 ⇐ x = 0O[sg[x − 1]] + 1 ⇐ o.w.

sg[x ] '{

1 ⇐ x = 00 ⇐ o.w.

sg[x ] '{

1 ⇐ x = 0O[sg[x − 1]] ⇐ o.w.

Examples

Absolute value is primitive recursivemod [x , y ] ' |x − y |

mod [x , y ] ' (x−̇y) + (y−̇x)

Minimum is primitive recursivemin[x , y ]

min[x , y ] ' x−̇(x−̇y)

Maximum is primitive recursivemax [x , y ]

max [x , y ] ' x + (y−̇x)

Examples

mod [x , y ] ' (x−̇y) + (y−̇x)

min[x , y ] ' x−̇(x−̇y)

max [x , y ] ' x + (y−̇x)

Examples

mod [x , y ] ' (x−̇y) + (y−̇x)

min[x , y ] ' x−̇(x−̇y)

max [x , y ] ' x + (y−̇x)

Examples

mod [x , y ] ' (x−̇y) + (y−̇x)

min[x , y ] ' x−̇(x−̇y)

max [x , y ] ' x + (y−̇x)

Examples

mod [x , y ] ' (x−̇y) + (y−̇x)

min[x , y ] ' x−̇(x−̇y)

max [x , y ] ' x + (y−̇x)

Examples

mod [x , y ] ' (x−̇y) + (y−̇x)

min[x , y ] ' x−̇(x−̇y)

max [x , y ] ' x + (y−̇x)

Primitive recursion. Properties

Theorem If then elseLet f0, f1,g be primitive recursive.Then

h[x ] '{

f0[x ] ⇐ g[x ] = 0f1[x ] ⇐ o.w.

proof:h[x ] ' sg[g[x ]].f0[x ] + sg[g[x ]].f1[x ]

Theorem If then elseLet f0, f1,g be primitive recursive.Then

h[x ] '{

f0[x ] ⇐ g[x ] = 0f1[x ] ⇐ o.w.

proof:h[x ] ' sg[g[x ]].f0[x ] + sg[g[x ]].f1[x ]

Theorem If then1 . . . thenk elseLet f0, . . . , fk ,g0, . . . ,gk−1 be primitive recursive.Then

h[x ] '

f0[x ] ⇐ g0[x ] = 0f1[x ] ⇐ g0[x ] 6= 0 ∧ g1[x ] = 0. . .. . .fk [x ] ⇐ o.w.

Partial Functions

While loopinput [x ]y := 0while f [x , y ] > 0 do y := y + 1return[y ]

g is obtained by minimization from fg[x ] ' yiff∀z < y(f [x , z] ↓ ∧ f [x , z] > 0)

f [x , y ] ' 0

g is obtained by minimization from fg[x ] ' µy [f [x , y ] = 0]

Partial Functions

f [x , y ] ' 0

Partial Functions

f [x , y ] ' 0

Partial Functions

The basic functions are partialO[x ] ' 0S[x ] ' x + 1Ini [x ] ' xi

The superposition is partialIf f ,g1, . . . ,gk are partial, thenh[x ] ' f [g1[x ], . . . ,gk [x ]]

is partial.

Partial Functions

The basic functions are partialO[x ] ' 0S[x ] ' x + 1Ini [x ] ' xi

The superposition is partialIf f ,g1, . . . ,gk are partial, thenh[x ] ' f [g1[x ], . . . ,gk [x ]]

is partial.

Partial Functions

The primitive recursion is partialIf f and g are partial, then

h[x , y ] '{

f [x ] ⇐ y = 0g[x , y ,h[x , y − 1]] ⇐ o.w.

is partial.

The minimization is partialIf f is partial, theng[x ] ' µy [f [x , y ] = 0]

is partial.

Partial Functions

The primitive recursion is partialIf f and g are partial, then

h[x , y ] '{

f [x ] ⇐ y = 0g[x , y ,h[x , y − 1]] ⇐ o.w.

is partial.

The minimization is partialIf f is partial, theng[x ] ' µy [f [x , y ] = 0]

is partial.

Partial Functions

TheoremAll the partial functions are computable.

Alternative DefinitionPartial functions = Computable functions.

Partial Functions

TheoremAll the partial functions are computable.

Alternative DefinitionPartial functions = Computable functions.

Examples

Subtraction is partial

f [x , y ] '{

x − y ⇐ x ≥ y↑ ⇐ o.w.

f [x , y ] ' µz[z + y = x ]

Division is partial

g[x , y ] '{

x/y ⇐ ∃k(y .k = x)↑ ⇐ o.w.

g[x , y ] ' µk [k .y = x ]

Examples

f [x , y ] '{

x − y ⇐ x ≥ y↑ ⇐ o.w.

f [x , y ] ' µz[z + y = x ]

Division is partial

g[x , y ] '{

x/y ⇐ ∃k(y .k = x)↑ ⇐ o.w.

g[x , y ] ' µk [k .y = x ]

Examples

f [x , y ] '{

x − y ⇐ x ≥ y↑ ⇐ o.w.

f [x , y ] ' µz[z + y = x ]

Division is partial

g[x , y ] '{

x/y ⇐ ∃k(y .k = x)↑ ⇐ o.w.

g[x , y ] ' µk [k .y = x ]

Examples

f [x , y ] '{

x − y ⇐ x ≥ y↑ ⇐ o.w.

f [x , y ] ' µz[z + y = x ]

Division is partial

g[x , y ] '{

x/y ⇐ ∃k(y .k = x)↑ ⇐ o.w.

g[x , y ] ' µk [k .y = x ]

Enumeration of the computable functions

Enumeration = Encoding = Effective coding

I Uniqueness: each object has a unique codeI Totality: each natural number is a code of an objectI Effectiveness: For each object one can find algorithmically its

code and for each code (number) one can find its object.

I Let P0,P1, . . . ,Pn, . . .be a list of all the programs (on one variable), and0,1, . . . ,n, . . . be an effective coding of these programs.

I Each program Pn corresponds to a computable function ϕ

I Let ϕ0, ϕ1, . . . , ϕn, . . .be a list of all the computable functions (on one variable), and0,1, . . . ,n, . . . be the effective encoding of these functions. Theencoding of the functions is such that for any n, the program Pnis an implementation of the computable function ϕn.

Example

Total function which is not computable

f [x ] '{ϕx [x ] + 1 ⇐ ϕx [x ] ↓0 ⇐ o.w.

Assume f is computable. Then f = ϕa for some a.If a ∈ Dom[ϕa] then ϕa[a] ↓. Hence, f [a] = ϕa[a] = ϕa[a] + 1If a 6∈ Dom[ϕa] then ϕa[a] ↑. Hence, f [a] = ϕa[a] = 0, but ϕa[a] ↑

Example

Total function which is not computable

f [x ] '{ϕx [x ] + 1 ⇐ ϕx [x ] ↓0 ⇐ o.w.

Assume f is computable. Then f = ϕa for some a.If a ∈ Dom[ϕa] then ϕa[a] ↓. Hence, f [a] = ϕa[a] = ϕa[a] + 1If a 6∈ Dom[ϕa] then ϕa[a] ↑. Hence, f [a] = ϕa[a] = 0, but ϕa[a] ↑

Kleene’s S-m-n Theorem

S-m-n Theorem

For any n,m exists a primitive recursive function Smn , such that

for any a, x , y

ϕ(m+n)a [x , y ] ' ϕ(n)

Smn [a,x ][y ]

Property

Let F be a computable function. Then there exists a number e, suchthat,

F [e, x ] ' ϕe[x ]

S-m-n Theorem

For any n,m exists a primitive recursive function Smn , such that

for any a, x , y

ϕ(m+n)a [x , y ] ' ϕ(n)

Smn [a,x ][y ]

Property

Let F be a computable function. Then there exists a number e, suchthat,

F [e, x ] ' ϕe[x ]

Property – Quines

There exists a number e, such that,

e ' ϕe[x ]

Quine in Lisp((lambda (x) (list x (list ′quote x)))′(lambda (x) (list x (list ′quote x))))

Property – Quines

There exists a number e, such that,

e ' ϕe[x ]

Quine in Lisp((lambda (x) (list x (list ′quote x)))′(lambda (x) (list x (list ′quote x))))

Universal FunctionUniversal Function Theorem

The universal function

Φn[a, x ] ' ϕ(n)a [x ]

is computable.

Property

The class of all the total functions on n-variables does not have acomputable universal function.

proofAssume Φ[a, x ] is an universal function for the class of all the totalfunctions on one variable.

Let ϕ[x ] ' Φ[x , x ] + 1.

Since Φ is total, ϕ is also total and hence, there exists a, such that

ϕ[x ] ' Φ[a, x ].

ϕ[a] ' Φ[a,a], and also ϕ[a] ' Φ[a,a] + 1.

Φn[a, x ] ' ϕ(n)a [x ]

is computable.

Property

Let ϕ[x ] ' Φ[x , x ] + 1.

ϕ[x ] ' Φ[a, x ].

Φn[a, x ] ' ϕ(n)a [x ]

is computable.

Property

Let ϕ[x ] ' Φ[x , x ] + 1.

ϕ[x ] ' Φ[a, x ].

Φn[a, x ] ' ϕ(n)a [x ]

is computable.

Property

Let ϕ[x ] ' Φ[x , x ] + 1.

ϕ[x ] ' Φ[a, x ].

Outline

The Halting Problem

Decidable and Semidecidable Sets A ⊆ Nn

Characteristic function of a set χA

χA[x ] '{

1 ⇐ x ∈ A0 ⇐ o.w.

Decidable SetA set A is decidable iff χA is computable.

χA[x ] '{

1 ⇐ x ∈ A0 ⇐ o.w.

χA[x ] '{

1 ⇐ x ∈ A0 ⇐ o.w.

Semicharacteristic function of a set CA

CA[x ] '{

1 ⇐ x ∈ A↑ ⇐ o.w.

Semidecidable SetA set A is semidecidable iff CA is computable.

Semicharacteristic function of a set CA

CA[x ] '{

1 ⇐ x ∈ A↑ ⇐ o.w.

Semidecidable SetA set A is semidecidable iff CA is computable.

TheoremIf A is decidable then it is also semidecidable.

TheoremIf A is decidable then A is also decidable.

TheoremIf A and B are decidable thenA ∪ B, A ∩ B and A\B are decidable.

TheoremIf A and B are semidecidable thenA ∪ B and A ∩ B are semidecidable.

TheoremA set A is semidecidable iff there exists a computable function ϕ,such that,A = Dom[ϕ]

Post TheoremA set A is decidable iff A and A are semidecidable.

Kleene Set KThe set K = {x | ϕx [x ] ↓} is called Kleene set.

TheoremK is semidecidable but not decidable.

Outline

The Halting Problem

The halting problem is a decision problem which can be stated asfollows:

Given a source code of a program, decide whether the programterminates or will run forever.

This is equivalent to the problem of deciding, given a program and aninput, whether the program will eventually halt when executed on thatinput, or will run forever.

The Halting Problem

The halting problem is a decision problem which can be stated asfollows:

Given a source code of a program, decide whether the programterminates or will run forever.

This is equivalent to the problem of deciding, given a program and aninput, whether the program will eventually halt when executed on thatinput, or will run forever.

The Halting Problem

Assume there is a program P (a computable function) with thefollowing property:

P[Q, x ] '{

1 ⇐ Q[x ] ↓0 ⇐ o.w.

P[a, x ] '{

1 ⇐ ϕa[x ] ↓0 ⇐ o.w.

a ∈ K ⇔ ϕa[a] ↓ ⇔ P[a,a] = 1

a ∈ K ⇔ ϕa[a] ↑ ⇔ P[a,a] = 0

Thus K is decidable. This is a contradiction!Therefore, the Halting problem is not decidable – it is undecidable.

The Halting Problem

P[Q, x ] '{

1 ⇐ Q[x ] ↓0 ⇐ o.w.

P[a, x ] '{

1 ⇐ ϕa[x ] ↓0 ⇐ o.w.

a ∈ K ⇔ ϕa[a] ↓ ⇔ P[a,a] = 1

a ∈ K ⇔ ϕa[a] ↑ ⇔ P[a,a] = 0

The Halting Problem

P[Q, x ] '{

1 ⇐ Q[x ] ↓0 ⇐ o.w.

P[a, x ] '{

1 ⇐ ϕa[x ] ↓0 ⇐ o.w.

a ∈ K ⇔ ϕa[a] ↓ ⇔ P[a,a] = 1

a ∈ K ⇔ ϕa[a] ↑ ⇔ P[a,a] = 0

The Halting Problem

P[Q, x ] '{

1 ⇐ Q[x ] ↓0 ⇐ o.w.

P[a, x ] '{

1 ⇐ ϕa[x ] ↓0 ⇐ o.w.

a ∈ K ⇔ ϕa[a] ↓ ⇔ P[a,a] = 1

a ∈ K ⇔ ϕa[a] ↑ ⇔ P[a,a] = 0

Questionnaire

When is a function computable?

When is a problem decidable?

When is a problem semidecidable?

When is a problem undecidable?

Questionnaire

Factorial

fact [x ] '{

1 ⇐ x = 0x .fact [x − 1] ⇐ o.w.

sum[x ] '{

0 ⇐ x = 0x + sum[x − 1] ⇐ o.w.

Is fact [0] ' sum[0]?

Is fact [−2] ' fact [−5]?

Questionnaire

Factorial

fact [x ] '{

1 ⇐ x = 0x .fact [x − 1] ⇐ o.w.

sum[x ] '{

0 ⇐ x = 0x + sum[x − 1] ⇐ o.w.

Questionnaire

Factorial

fact [x ] '{

1 ⇐ x = 0x .fact [x − 1] ⇐ o.w.

sum[x ] '{

0 ⇐ x = 0x + sum[x − 1] ⇐ o.w.

Questionnaire

Factorial

fact [x ] '{

1 ⇐ x = 0x .fact [x − 1] ⇐ o.w.

sum[x ] '{

0 ⇐ x = 0x + sum[x − 1] ⇐ o.w.

Questionnaire

Problem P: The function fact is defined on x .

Is the problem P decidable?

Problem Q: An arbitrary function f is defined on x .

Is the problem Q decidable?

Questionnaire

How many indexes are there for the factorial function?

How many indexes are there for the universal function for theclass of all the total functions?

How many indexes are there for the universal function for theclass of all the partial recursive functions?

Questionnaire

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