Comput Lection 3

8/13/2019 Comput Lection 3

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Lection 3 Computability Theory

Church Thesis

Gdelization

Universal Turing machine

Padding Lemma

Models of computation: -calculus

VU Logic and Computation p. 1


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Church-Turing Thesis

The functions computed by an algorithm coincide withthe class of functions computable by Turing machines.

Recall thatf :N N isTuring computableiff there exists a TMA

starting withq0, x,Ahalts ifff(x)

in that casew0qay, wheref(x) =y.



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Computable Functions

To prove that a functionf is computable:

write a Turing machine that computes the function

give an informal algorithm to computefand use Churchsthesis



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Universal Turing Machine

A generic TM that can beprogrammedto solve any problem thatcan be solved by a TM.

The program" that makes the generic machine simulated a

specific machineM is adescriptionofM. We need toencodeTMs so that their description can be

used as input to the generic TM.

(Enumeration ofGdelization)



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Godelization

T :T M N

A possible encoding (there are many other alternatives):

Given a TM we consider its functional matrix

we order all the strings identifying functional matrices of TM

according to their length lexicographically



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Theuniversal Turing machine,Uon inputT(M), w, whereMis a TM andwa string, behaves as follows:

1. ExtractM fromT(M)and check thatwis a correct string

2. SimulatesMon inputw

3. IfMon inputw enters its accept state,U accept, and ifMon inputwever enters its reject state,U reject.



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Theuniversal Turing machine,Uon inputT(M), w, whereMis a TM andwa string, behaves as follows:

1. ExtractM fromT(M)and check thatwis a correct string

2. SimulatesMon inputw

3. IfMon inputw enters its accept state,U accept, and ifMon inputwever enters its reject state,U reject.

Notice that ifMon inputw enters an infinite loop, so doesU oninputM, w.



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1. Informal argument (via Church thesis and enumeration ofTM)

2. Formal proof



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Universal Turing Machine informal argument

LetZx be thex-th Turing machine (w.r.t. the enumerationT)

Take the functiong:N N Nsuch that

g(x, y)Zx(y)



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Universal Turing Machine informal argument

LetZx be thex-th Turing machine (w.r.t. the enumerationT)

Take the functiong:N N Nsuch that

g(x, y)Zx(y)

g is computable (Churchs thesis)

Algorithm

fromxtoZx viaT1 :N T M (T1(x) =Zx)

Zx(y)



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Universal Turing Machine 2

A step ofMis simulated as follows:

Uscans tape 2 until it finds the matching transition (the firstcomponent matches the state written on tape 3 and the

second component matches the current symbol on tape 1). If it is found, change the state on tape 3 and update tape 1

accordingly.

If not found, the final state has been reached.



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Universal Turing Machine 3

The universal machineUobviously has afixed numberofstates.

Despite this, it can simulate machinesMwithanynumber

of states.Universal machines inspired the development ofstored-program computers in the 40s and 50s.



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Padding Lemma

0 Z001 Z112 Z22

3 Z33... is an enumeration of computable functions



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Padding Lemma

0 Z001 Z112 Z22

3 Z33... is an enumeration of computable functions

Padding LemmaEvery computable function admits an infinite number of indexes.



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Not all functions are (Turing) computable



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A not computable function

1, 2, . . . (Turing) computable functions

Example: The diagonal function:

g(x) =

ifx(x)0 otherwise



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Other Models of Computations

Formal notionsappeared, starting in 1936:

-calculusof Alonzo Church

Turing machinesof Alan Turing

Recursive functionsof Gdel and Kleene

Countermachines of Minsky

Normal algorithmsof Markov

Unrestrictedgrammars

Two stackautomata

Random accessmachines (RAMs)

...VU Logic and Computation p. 16


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These definitions look very different, but areprovably equivalent.

The Church-Turing Thesis:

The intuitive notion of reasonable models ofcomputation equals Turing machine algorithms.



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Why is this a thesis and not atheorem?

Because the notion of reasonable model of computing isnot well defined.

In 2008, Nachum Dershowitz (TAU) and Yu. Gurevitch(MSFT) gave an axiomatization of this notion.

This axiomatization seems to reasonably capture what a

reasonable model of computing is.

Under this axiomatization, DG gave a proof of theChurch-Turing thesis.



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calculus



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Why calculus?

A more abstract, machine-independent description of thecollection of computable functions than that provided by Turingmachines.

Historical Useful for formal semantics of programming languages

Its at the core of functional programming languages (LISP,

ML..)



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-Terms

are built from a given, countable collection of

variablesx, y, z . . .

by two operations for forming-terms:

abstraction: (x.M)(wherexis a variable andM a-term)

application: (M M)

(whereM andM are-terms)



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-Terms

Notational conventions:

(x1x2 . . . xn.M)means(x1.(x2. . . . (xn.M) . . . )

(M1M2 . . . M n)means(. . . (M1M2) . . . M n), i.e. application

is left-associative

drop outermost parentheses and those enclosing the bodyof a-abstraction



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Free and Bound Variables

Inx.Mwe callxthebound variableandM thebodyof theabstraction.



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Inx.Mwe callxthebound variableandM thebodyof theabstraction.

An occurrence ofxin a-termM is called

binding, if in betweenand .

boundif in the body of a binding occurrence of x

freeif neither binding not bound



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Sets offreeandboundvariables

F V(x) ={x}

F V(x.M) =F V(M)\ {x}

F V(M N) =F V(M) F V(N)

BV(x) =

BV(x.M) =BV(M) {x}BV(M N) =BV(M) BV(N)

IfF V(M) =,Mis called aclosed term, or combinator



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Rules for transforming terms



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-Equivalence M=M

x.Mis intended to represent the functionfsuch that

f(x) =M, for allx.



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-Equivalence M=M


f(x) =M, for allx.

Hence ifM =M[x/x]is the result of takingMand changingall occurrences ofxto some variablex that does not appearfree inM, thenx.Mandx.M both represent the samefunction.



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-Equivalence M=M


f(x) =M, for allx.

Hence ifM =M[x/x]is the result of takingMand changingall occurrences ofxto some variablex that does not appearfree inM, thenx.Mandx.M both represent the samefunction.

x.M x.M[x/x]



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-Equivalence

is the smallest relation onterms satisfying the following rules:

M=M

(refl) N=M

M=N

(symm)

M=N N=PM=P

(trans) M=Nx.M=x.N

()

M=M N=N

M N=MN (cong)

y /M

x.M=y.(M{y/x}) ()



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-Reduction

The process of evaluating lambda terms by "plugging argumentsinto functions".

x.Mcan be seen as a function on -terms via substitution:

map eachN toM[N/x].



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-Reduction

The process of evaluating lambda terms by "plugging argumentsinto functions".

x.Mcan be seen as a function on -terms via substitution:

map eachN toM[N/x].So the natural notion of computation for -terms is given bystepping from a

-redex(x.M)Nto the corresponding

-reductM[N/x]



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Substitution M[N/x]

x[N/x] =N

y[N/x] =y (ifx=y)

y.M[N/x] =y.(M[N/x])ifx=y andyF V(N x)

(M1M2)[N/x] =M1[N/x]M2[N/x]

The side conditionyF V(N x)makes substitution "capture

avoiding".

y.M[N/x] =y.(M{y

/y}[N/x]) y fresh



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Single-step -reduction

Is the smallest relationon terms satisfying

(x.M)NM[N/x] ()

MM

M NMN

(cong1)NN

M NMN

(cong2)

MN

x.Mx.N ()



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-reduction and -equivalence

is the reflexive and transitive closure of .MM ifMreduces toM in zero or more steps.



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-reduction and -equivalence

is the reflexive and transitive closure of .MM ifMreduces toM in zero or more steps.

M= Nholds ifNcan be obtained fromMperforming zero or

more steps of-reductions,-reductions and/or inversereduction steps (i.e. makingsymmetric). Formally,=

is

defined to be the reflexive, symmetric and transitive closure of, i.e. the smallest equivalence relation containing .



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Church-Rosser

Theoremisconfluent, that is, if

M1MM2

then there existsM such that

M1M M2


N l F


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Normal Form

Def.A-termN is in-normal form (nf) if it contains no-redex (i.e.no sub-terms of the form(x.M)M).

M has-nfN ifM=

N withN a-nf.


N l F


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Normal Form


M has

-nf

N if

M=

N with

N a

-nf.

Note that ifN is in-nf andNN, thenN= N.


N l F


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Normal Form


M has

-nf

N if

M=

N with

N a

-nf.

Note that ifN is in-nf andNN, thenN= N.

Corollary:The-nf ofMis unique up to-equivalence, if it

exists.


Non termination


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Non-termination

Someterms have no-nf


Non termination


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Non-termination

Someterms have no-nf

Ex.

:= (x.xx)(x.xx)


Encoding data in calculus


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Encoding data in -calculus

Computation incalculus is given by-reduction. To relate thisto Turing-machine computation, or computation in other modelswe have to see how to encode numbers, pairs, lists... as-terms.


Encoding data in calculus


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Encoding data in -calculus

Computation incalculus is given by-reduction. To relate thisto Turing-machine computation, or computation in other modelswe have to see how to encode numbers, pairs, lists... as-terms.

Example

T :=xy.x F :=xy.y

and:=ab.abF

if-then-else:=a.a


definable functions


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-definable functions

f :Nn N is-definable if there is a closed-termF thatrepresents it: for all(x1, . . . xn) Nn andy N

iff(x1, . . . xn) =ythenF x1 . . . xn= y

iff(x1, . . . xn), thenF x1 . . . xn has no-nf

Computable =-definable


Bibliography


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Bibliography

Sipser, Introduction to the Theory of Computation, PWSPublishing Co., 1997

Hindley, Selding, Introduction to Combinators and

-Calculus, London Mathematical Society, 1 990 Odifreddi, Classical Recursion Theory, North-Holland, 1989

http://arxiv.org/pdf/0804.3434v1.pdf(Ch. 1-4.2)Selinger, Lecture Notes on the Lambda Calculus

...

http://arxiv.org/pdf/0804.3434v1.pdfhttp://arxiv.org/pdf/0804.3434v1.pdf

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