slides church-turing thesis (moc)

TheChurch-TuringThesis

Criticism on andinterpretationsof theChurch-TuringThesis

Variations ontheChurch-TuringThesis

Physical/StrongChurch-Turing Thesis

FeasibilityThesis/ExtendedChurch-Turing Thesis

QuantumComplexity-TheoreticChurch-Turing Thesis

Concludingremarks

Church-Turing Thesis – revisited

Bas van Gijzel

June 7, 2010

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Outline

The Church-Turing Thesis

Criticism on and interpretations of the Church-Turing Thesis

Variations on the Church-Turing ThesisPhysical/Strong Church-Turing ThesisFeasibility Thesis/Extended Church-Turing ThesisQuantum Complexity-Theoretic Church-Turing Thesis

Concluding remarks

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So first...

A reintroduction to the Church-Turing Thesis and someconcepts from Sélinde’s talk.

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Alonzo Church (1903-1995)

Church wanted to define the notion of an effectivelycalculable function.

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An effectively calculable function: a smallmotivation

I What is an effectively calculable function?• An informal concept.• Explained as: “calculable by a finite (mechanical)

procedure”.

I Some possible intuitive notions:• Functions humans can solve in finite time.• Functions that can be constructively defined.

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An effectively calculable function: a smallmotivation

I What is an effectively calculable function?• An informal concept.• Explained as: “calculable by a finite (mechanical)

procedure”.

I Some possible intuitive notions:• Functions humans can solve in finite time.• Functions that can be constructively defined.

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The Church Thesis

I Church wanted to define this notion of effectivelycalculable by using λ-definable functions.

I After encountering Gödel this was deemedunsatisfactory and the ’thesis’ was changed.

New solution:

I Identify effectively calculable functions with therecursive functions.

I One identification is done in the context of calculabilityin a logic.

I Problem: steps of the proof were not proved recursive(or finite and mechanical).

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The Church Thesis

New solution:

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The Church Thesis

New solution:

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The Church Thesis

New solution:

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The Church Thesis

New solution:

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The Church Thesis by Church

Church’s formulation (Church1936):

We now define the notion, already discussed, of aneffectively calculable function of positive integersby identifying it with the notion of a recursivefunction of positive integers (or a λ-definablefunction of positive integers). This definition isthought to be justified by the considerations whichfollow, so far as positive justification can ever beobtained for the selection of formal definition tocorrespond to an intuitive notion.

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The Church Thesis in short (Sieg)

Due to Sieg (Sieg1997):

The steps of any effective procedure (governingproofs in a system of symbolic logic) must berecursive.

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Alan Turing (1912-1954)

Turing wanted to define the notion of effective calculabilityor informal computability in terms of an idealizedhuman/computor.

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The Turing Thesis

I The processes or steps taken by this idealized humanare formalised and restrained. Conditions as by Sieg(Sieg2001):

• (B) (Boundedness) There is a fixed bound on thenumber of configurations a computer can immediatelyrecognize.

• (L) (Locality) A computor can change only immediatelyrecognizable (sub-)configurations.

• (D) (Determinacy) The immediately recognizable(sub-)configuration determines uniquely the nextcomputation step (and id).

I Turing then argued that number theoretic functionscalculable by such a computor are also computable bya Turing machine.

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The Turing Thesis

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The Turing Thesis

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The Turing Thesis in short

Adaptation of Shagrir (Shagrir2002) which he calls theGandy-Sieg account:

Any function that can be computed by a computor,with the previous restrictions (B, L and D), is Turingcomputable (in the sense of a Turing machine).

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The two short formulations combined:Any effectively calculable function is Turingcomputable.

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Interpretation of Turing’s Thesis(1)

Again the interpretation of Gandy-Sieg:

Any function that can be computed by a computorwith the previous restrictions is Turing computable(in the sense of a Turing machine).

I This formulation uses computors, and thus humans, toformulate the concept effective calculability.

I Church, as can be seen in his review (Church1937), alsoprobably understood Turing’s Thesis as a thesis aboutcomputors, not about computing machines.(Hodges2006 and Shagrir2002)

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I This is in contrast to common modern opinions andinterpretations of the Church-Turing Thesis.

I The Church-Turing Thesis in current computer scienceis often understood in terms of finite machinecomputation in contrast to human computation.

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Interpretation of the (Church-)Turing Thesis

I The Gandy-Sieg account of an effectively calculablefunction for the Church-Turing Thesis is aninterpretation phrased in computors/idealized humancomputers, whereas the other view is machine based.

I So in fact we can formulate two versions of theChurch-Turing Thesis (as done in Shagrir2002).

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Interpretation of the (Church-)Turing Thesis

I The Gandy-Sieg account of an effectively calculablefunction for the Church-Turing Thesis is aninterpretation phrased in computors/idealized humancomputers, whereas the other view is machine based.

I So in fact we can formulate two versions of theChurch-Turing Thesis (as done in Shagrir2002).

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Human and Machine Church-Turing Thesis

The human version of the Church-Turing Thesis:

Any function (of positive integers) that can becomputed by a human computer is Turingcomputable.

The machine version of the Church-Turing Thesis:

Any function (of positive integers) that can becomputed by a finite machine is Turingcomputable.

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Human and Machine Church-Turing Thesis

The human version of the Church-Turing Thesis:

Any function (of positive integers) that can becomputed by a human computer is Turingcomputable.

The machine version of the Church-Turing Thesis:

Any function (of positive integers) that can becomputed by a finite machine is Turingcomputable.

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Variations on the Church-Turing Thesis

I Quite some variants on the Church-Turing Thesis havebeen proposed.

I Also motivated by the two possible interpretationsgiven before.

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Physical Church-Turing Thesis (Shagrir)

I An abstraction quickly made from human computersand finite machines is the abstraction to physicalsystems in general.

I This abstraction also serves as a distinction betweenproper Church-Turing Thesis as defined before and thePhysical Church-Turing Thesis (PCTT).

I This is also called the Strong Church-Turing Thesisexpressing that Turing Machines capture allcomputation, so models of computation moreexpressive than TMs are impossible.

PCTT as formulated in Shagrir2002:

Any function (of positive integers) that can becomputed by a physical system isTuring-computable.

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Physical Church-Turing Thesis (Piccinini)

I Piccinini makes a further distinction (Piccinini2007) inthe PCTT.

I A modest version which is supposedly equal to theprevious formulation by Shagrir.

I A bold version that possibly broadens the scope ofeffectively calculable functions.

Modest PCTT:... the functions whose values are generated bycomputing mechanisms are Turing-computable.

Bold PCTT:... the functions whose values are generated (bycomputation or any other means) by physicalsystems are Turing-computable.

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Criticism on the modest PCTT (Copeland)

I Copeland has a lot of criticism on the modest version ofthe Church-Turing Thesis and also already on themachine version of the Church-Turing Thesis.

I He suggests there could exist physical machinescapable of performing computations not computable byTuring Machines.

I This field is called hypercomputation and is stillspeculative.

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Criticism on the modest PCTT (Copeland)

I Copeland has a lot of criticism on the modest version ofthe Church-Turing Thesis and also already on themachine version of the Church-Turing Thesis.

I He suggests there could exist physical machinescapable of performing computations not computable byTuring Machines.

I This field is called hypercomputation and is stillspeculative.

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The Extended Church-Turing Thesis (1)

I Formulated early by Parberry in 1986 using a parallelcomputation thesis.

I Yao (Yao2003) calls it an extension of the classicalChurch-Turing Thesis, Extended Church-Turing Thesis.

I Beside the standard thesis, the Turing machine is alsodeemed as efficient as any computing device can be.

Extended Church-Turing Thesis (adaptation of Yao2003):

Any effectively calculable function is Turingcomputable and efficiently computable usingTuring machines.

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I The Extended Church-Turing Thesis can be used as aninteresting starting point for discussion.

I Discuss physical realisation of computing machinesthat are capable of solving some problems much fasterthan the standard computational model.

I Does not immediately go into the realm of quantummechanics.

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Quantum Complexity-Theoretic Church-TuringThesis

I Quantum Complexity-Theoretic Church-Turing Thesiswas formulated by Bernstein and Vazirani in 1997.

I A variant formulation of the Church-Turing Thesisclosely related to the Extended Church-Turing Thesis.

• Would invalidate the extended Church-Turing Thesisbecause the quantom model of computation is moreefficient than a normal Turing machine.

Quantum Complexity-Theoretic Church-Turing Thesis:

A quantum Turing machine can efficiently simulateany realistic model of computation.

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Summary

I The Church-Turing Thesis can be described as a thesisconcerning computors or finite machines.

I Terminology, implications and exact meaning of theChurch-Turing Thesis are still discussed.

I Variations can put constraints or give more freedom onthe physicality and efficiency of thecomputers/computors.

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slides church-turing thesis (moc)

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