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Mathematical LogicIntroduction to Theory of Computability
Nikolaj Popov and Tudor Jebelean
Research Institute for Symbolic Computation, Linz
popov@risc.uni-linz.ac.at
Outline
ComputabilityPreliminariesPrimitive Recursive FunctionsPartial FunctionsEnumeration of the Computable Functions
DecidabilityDecidable and Semidecidable Sets
The Halting Problem
Outline
ComputabilityPreliminariesPrimitive Recursive FunctionsPartial FunctionsEnumeration of the Computable Functions
DecidabilityDecidable and Semidecidable Sets
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
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
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
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
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.
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
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
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
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
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
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
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 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 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.
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 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 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 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.
is primitive recursive.
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.
is primitive recursive.
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.
is primitive recursive.
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
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
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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
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
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
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)
Primitive recursion. Properties
Theorem If then elseLet f0, f1,g be primitive recursive.Then
h[x ] '{
f0[x ] ⇐ g[x ] = 0f1[x ] ⇐ o.w.
is primitive recursive.
proof:h[x ] ' sg[g[x ]].f0[x ] + sg[g[x ]].f1[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.
is primitive recursive.
proof:h[x ] ' sg[g[x ]].f0[x ] + sg[g[x ]].f1[x ]
Primitive recursion. Properties
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.
is primitive recursive.
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
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
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
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
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
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
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 ]
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.
Enumeration of the computable functions
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.
Enumeration of the computable functions
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.
Enumeration of the computable functions
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 ]
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 ]
Kleene’s S-m-n Theorem
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))))
Kleene’s S-m-n Theorem
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.
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.
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.
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.
Outline
ComputabilityPreliminariesPrimitive Recursive FunctionsPartial FunctionsEnumeration of the Computable Functions
DecidabilityDecidable and Semidecidable Sets
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.
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.
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.
Decidable and Semidecidable Sets A ⊆ Nn
Semicharacteristic function of a set CA
CA[x ] '{
1 ⇐ x ∈ A↑ ⇐ o.w.
Semidecidable SetA set A is semidecidable iff CA is computable.
Decidable and Semidecidable Sets A ⊆ Nn
Semicharacteristic function of a set CA
CA[x ] '{
1 ⇐ x ∈ A↑ ⇐ o.w.
Semidecidable SetA set A is semidecidable iff CA is computable.
Decidable and Semidecidable Sets A ⊆ Nn
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.
Decidable and Semidecidable Sets A ⊆ Nn
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.
Decidable and Semidecidable Sets A ⊆ Nn
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.
Decidable and Semidecidable Sets A ⊆ Nn
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.
Decidable and Semidecidable Sets A ⊆ Nn
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.
Decidable and Semidecidable Sets A ⊆ Nn
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.
Decidable and Semidecidable Sets A ⊆ Nn
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.
Decidable and Semidecidable Sets A ⊆ Nn
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
ComputabilityPreliminariesPrimitive Recursive FunctionsPartial FunctionsEnumeration of the Computable Functions
DecidabilityDecidable and Semidecidable Sets
The Halting Problem
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
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
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
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.
Questionnaire
When is a function computable?
When is a problem decidable?
When is a problem semidecidable?
When is a problem undecidable?
Questionnaire
When is a function computable?
When is a problem decidable?
When is a problem semidecidable?
When is a problem undecidable?
Questionnaire
When is a function computable?
When is a problem decidable?
When is a problem semidecidable?
When is a problem undecidable?
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
sum[x ] '{
0 ⇐ x = 0x + sum[x − 1] ⇐ o.w.
Is fact [0] ' sum[0]?
Is fact [2] ' sum[2]?
Is fact [−2] ' fact [−5]?
Questionnaire
Factorial
fact [x ] '{
1 ⇐ x = 0x .fact [x − 1] ⇐ o.w.
Sum
sum[x ] '{
0 ⇐ x = 0x + sum[x − 1] ⇐ o.w.
Is fact [0] ' sum[0]?
Is fact [2] ' sum[2]?
Is fact [−2] ' fact [−5]?
Questionnaire
Factorial
fact [x ] '{
1 ⇐ x = 0x .fact [x − 1] ⇐ o.w.
Sum
sum[x ] '{
0 ⇐ x = 0x + sum[x − 1] ⇐ o.w.
Is fact [0] ' sum[0]?
Is fact [2] ' sum[2]?
Is fact [−2] ' fact [−5]?
Questionnaire
Factorial
fact [x ] '{
1 ⇐ x = 0x .fact [x − 1] ⇐ o.w.
Sum
sum[x ] '{
0 ⇐ x = 0x + sum[x − 1] ⇐ o.w.
Is fact [0] ' sum[0]?
Is fact [2] ' sum[2]?
Is fact [−2] ' fact [−5]?
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
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
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
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
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
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?
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