TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
1
Church-Turing Thesis – revisited
Bas van Gijzel
June 7, 2010
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
2
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
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
3
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
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
4
So first...
A reintroduction to the Church-Turing Thesis and someconcepts from Sélinde’s talk.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
5
Alonzo Church (1903-1995)
Church wanted to define the notion of an effectivelycalculable function.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
6
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
6
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
7
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).
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
7
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).
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
7
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).
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
7
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).
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
7
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).
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
8
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
9
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
10
Alan Turing (1912-1954)
Turing wanted to define the notion of effective calculabilityor informal computability in terms of an idealizedhuman/computor.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
11
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
11
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
11
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
12
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).
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
13
The Church-Turing Thesis
The two short formulations combined:Any effectively calculable function is Turingcomputable.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
14
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
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
15
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)
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
15
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)
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
15
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)
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
16
Interpretation of Turing’s Thesis(2)
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 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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
16
Interpretation of Turing’s Thesis(2)
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 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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
16
Interpretation of Turing’s Thesis(2)
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 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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
17
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).
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
17
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).
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
18
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
18
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
19
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
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
20
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
21
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
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
22
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
22
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
22
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
23
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
23
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
23
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
23
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
23
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
24
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
24
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
25
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
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
26
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
26
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
26
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
26
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
27
The Extended Church-Turing Thesis (2)
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
28
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
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
29
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
29
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
29
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
29
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
30
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
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
31
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.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
32
Further reading
Some other interesting topics not discussed due to a lack oftime:
I Interactive Turing machines (possibly with advice).
I All knowing entities that can be used in computationscalled oracles.
I Gandy machines, a concept relating to physicalmachines.
I Super-recursive algorithms, a class of algorithms thatcompute more than Turing machines.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
33
References
I Bernstein, E; Vazirani, U. (1997), ’Quantum complexitytheory’, SIAM Journal on Computing 26 (5): pp.1411-1473.
I Church, A. (1936), ’An Unsolvable Problem ofElementary Number Theory’, American Journal ofMathematics 58, pp. 345-363.
I Church, A. (1937), ’Review of Turing( 1936)’, Journal ofSymbolic Logic 2, pp. 42-43.
I Copeland, J.; Proudfoot, D. (1999), ’Alan Turing’sForgotten Ideas in Computer Science ScientificAmerican’, vol. 280 (1999), pp. 99-103.
I Hodges, A. (2006), ’Did Church and Turing have a thesisabout machines?’, Church’s Thesis after 70 yearsOlszewski, Adam (ed.).
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
34
References
I Parberry, I. (1986), ’Parallel speedup of sequentialmachines: a defense of parallel computation thesis,’ACM SIGACT News, Volume 18, Issue 1, pp. 54-67.
I Piccinini, G. (2007), ’Computationalism, theChurch-Turing Thesis, and the Church-Turing Fallacy’,Synthese, 154.1, pp. 97-120.
I Shagrir, O. (2002), ’Effective Computation by Humansand Machines’, Minds Mach. 12, 2, pp. 221-240.
I Sieg, W. (1997), ’Step by Recursive Step: Church’sAnalysis of Effective Calculability’, Bulletin of SymbolicLogic 2, pp. 154-180.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
35
References
I Sieg, W. (2001), ’Calculation by Man and Machine:Conceptual Analysis’, in W. Sieg, R. Sommer and C.Talcott, eds., Reflections on the Foundations ofMathematics (Essays in Honor of Solomon Feferman),Volume 15 of Lectures Notes in Logic, Association ofSymbolic Logic, pp. 387-406.
I Soare, R (1999), ’The history and concept ofcomputability, in Handbook of Computability Theory,E.R. Griffor (ed.); Elsevier, 3-36.
I Yao, A. (2003), ’Classical physics and theChurch–Turing Thesis’, JACM 50, 100.
TheChurch-TuringThesis
Criticism on andinterpretationsof theChurch-TuringThesis
Variations ontheChurch-TuringThesis
Physical/StrongChurch-Turing Thesis
FeasibilityThesis/ExtendedChurch-Turing Thesis
QuantumComplexity-TheoreticChurch-Turing Thesis
Concludingremarks
36
Questions