COMP-330 Theory of Computation

Fall 2017 -- Prof. Claude Crépeau Lec. 16 :

Turing Machines

COMP 330 Fall 2017: Lectures Schedule

14. MIDTERM15-16. Deterministic CFLs16-17. Turing Machines and Church-Turing Thesis17. Models of computation Basic computability theory18. Reducibility, undecidability and Rice’s theorem19. Undecidable problems about CFGs20. Post Correspondence Problem21. Validity of FOL is RE / Gödel’s and Tarski’s thms22. Universality / The recursion theorem 23. Degrees of undecidability24. Introduction to complexity25. Introduction to Quantum Computing/Complexity26. Review of course material

1-2. Introduction 1.5. Some basic mathematics2-3. Deterministic finite automata +Closure properties,4. Nondeterministic finite automata5. Minimization 6. Determinization+Kleene’s theorem7. Regular Expressions+GNFA8. The pumping lemma9. Myhill-Nerode theorem10. Context-Free Grammar11. PushDown automata12 .CFG - PDA equivalence13. CFG Pumping lemma and applications

All languages

Decidable Languages

Context-freeLanguages

RegularLanguages

Languages we can describe

NON-Regular Languagesvia Pumping Lemma

NON-RegularLanguages

via Reductions

Computability Theory

All languages

Decidable Languages

Context-freeLanguages

Languages we can describe

NON-CFLsvia Pumping Lemma

NON-CFLsvia Reductions

RegularLanguages

Computability Theory

All languages

Decidable Languages

Context-freeLanguages

Languages .we can

describe

NON-decidablevia Diagonalization

NON-decidablevia Reductions

RegularLanguages

Computability Theory

Turing MACHINES

Alan Turing

M1

q1

M1

10010101

q2

1

M1M1

10010101

M1

10010101

q3

0

M1

q2

1

M1M1

10110101read and write!moves Right and Left!

M1

10110101

q3

0

M1

Turing Machines

TM Example

StatesInput AlphabetTape AlphabetTransition functionStart stateAccept stateReject state

Definition of TMq1

q2 q3

a,b,c

q1 q2

b→c,Da,b,c,A,B,C,_

StatesInput AlphabetTape AlphabetTransition functionStart stateAccept stateReject state

Definition of TMq1

q2 q3

a,b,c

q1 q2

b→c,Da,b,c,A,B,C,_

q1 q2

b→c,D

inputsymbol

outputsymbol

L or Rheadmove

StatesInput AlphabetTape AlphabetTransition functionStart stateAccept stateReject state

Definition of TMq1

q2 q3

a,b,c

q1 q2

b→c,D

q1

qacc

a,b,c,A,B,C,_

qrej

TM definition

TM Configuration

TM Computation

For all a,b,c∈𝜞, u,v∈𝜞*, qi,qi∈Q

Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,L

Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,R

Special cases: Config. qi bv yields qj cv if δ(qi,b) = qj,c,LConfig. qi bv yields c qj v if δ(qi,b) = qj,c,R

TM definition

TM Computation

Start configuration: q0 w (w = input string)

Accepting configuration: state = qaccept

Rejecting configuration: state = qreject

TM Computation

Turing Machine M accepts input w if there exists configurations C0, C1,..., Cm such that

C0 is a start configuration

Ci yields Ci+1 for 0≤i<m

Cm is an accepting configuration.

The collection of strings that M accepts is the language of M or the language recognized by M, denoted L(M).

TM Computation

A TM decides a language if it recognizes it and halts (reaches an accepting or rejecting states) on all input strings.

TM Computation

A TM decides a language if it recognizes it and halts (reaches an accepting or rejecting states) on all input strings.

1Often named Recursively-Enumerable in the literature.2Often named Recursive in the literature.

TM Examples

TM Examples

TM Computation

TM Examples

∑ \ { x , _ }

∑ \ { # , _ }

TM Examples

TM Examples

TM Examples

More Turing MACHINES

Multitape Turing Machines

Non-Deterministic Turing Machines

Enumerator Turing Machines

Everything else...

Multitape TM

Multitape TM

Multitape TM

Multitape TM

Non-deterministic TM

Non-deterministic TM

Non-deterministic TM

Enumerator TM

Enumerator TM

Enumerator TM

Enumerator TM

Enumerator TM

Lambda-calculus

Recursive Functions

Programming languages:

FORTRAN, PASCAL, C, JAVA,...

LISP, SCHEME,...

Everything Else

Alonzo Church

J. Barkley Rosser

Stephen Kleene

Lambda-calculus

Recursive Functions

Programming languages:

FORTRAN, PASCAL, C, JAVA,...

LISP, SCHEME,...

Everything Else

Alonzo Church

J. Barkley Rosser

Stephen Kleene

Lambda-calculus

Recursive Functions

Programming languages:

FORTRAN, PASCAL, C, JAVA,...

LISP, SCHEME,...

Everything Else

Alonzo Church

J. Barkley Rosser

Stephen Kleene

Church-Turing Thesis

Alonzo Church Alan Turing

Church-Turing Thesis

Speaking on 8 August 1900, at the Paris 2nd International Congress of Mathematicians, at La Sorbonne, German mathematician David Hilbert presented ten problems in mathematics.

The problems were all unsolved at the time, and several of them turned out to be very influential for 20th century mathematics.

Paris, 1900

David Hilbert

Hilbert’s 10th problem

Let P be a polynomial in several variables: P(x,y,z)=24x2y3+17x+5y+25

Is there a set of integers for x,y,z such that P(x,y,z)=0 ?

This problem is undecidable...but is Turing-Recognizable...

Needed a formal model ofcomputing to prove impossibility. Yuri Matiyasevich

Single variable Poly

COMP-330 Theory of Computation

Fall 2017 -- Prof. Claude Crépeau Lec. 16 :

Turing Machines &Church-Turing Thesis