turing machines...

Turing MachinesDecidability

Master Informatique

Université Paris 5 – René Descartes

2016

Master Info. Complexity – Turing 1/24

Some General KnowledgeAlan Mathison Turing

• UK, 1912 – 1954• Mathematician, computer scientist,

cryptanalyst• Most famous works:

I Computation model (« TuringMachines »)

I Work on Enigma during WWIII Imitation Game (« Turing Test »)

• Father of theoretical CS and AI• Turing Award ' Nobel Award for CS


Outline

Turing MachinesIntroduction to Turing MachinesTuring Machines and Decision ProblemsNon-DeterminismChurch-Turing Thesis

DecidabilityRecognizable and Decidable ProblemsHalting Problem and Reductions


Turing Machines

Computing Machine [Turing 1936]

• Abstract model to compute any « calculable » decimalnumberi.e. any number which can be computed with a finiteamount of resources• Proposed years before the apparition of computer, but

quite a faithful abstraction of modern machines• Roughly speaking,

I tape (sequence of squares) browsed by a reader/writer:memory of a computer

I transition function: processor of a computer


Definition of Turing Machines

A Turing machine is a tuple 〈Q,Γ, B,Σ, q0, δ, F 〉 with:• Q = {q0, q1, . . . , qm}, a finite set of states• Γ, the finite set of symbols used by the machine

(vocabulary)• B ∈ Γ, a particular symbol called blank• Σ ⊆ Γ, the input vocabulary• q0 ∈ Q, the initial state of the machine• δ : Q× Σ→ Q× Γ× {L,R}, the transition function• F ⊆ Q, the set of final states


Transition Function

δ : Q× Σ→ Q× Γ× {L,R}

• Input: a pair (current state, symbol on the tape) calledconfiguration• Output: a tuple (next state, symbol to write, move)


Example of Turing Machines (1/2)Multiplying Integers by 2

Let us considerM = 〈Q,Γ, B,Σ, q0, δ, F 〉 with:

• Q = {in_progress, done}• Γ = {0, 1, B}• Σ = {0, 1}

• q0 = in_progress• δ as described in the table• F = done

Current state Current symbol Next State Write Movein_progress 0 in_progress 0 R

in_progress 1 in_progress 1 R

in_progress B done 0 R

done STOP



1 0 1 0 1 0 B B . . .

i_p

• Read 1 in state in_progress: write 1, move right, statein_progress



1 0 1 0 1 0 B B . . .

i_p




1 0 1 0 1 0 B B . . .

i_p

• Read B in state in_progress: write 0, move right, statedone



1 0 1 0 1 0 0 B . . .

d

• State done: stop


Accept vs Reject vs Loop

We consider a special class of Turing machines, with threepossible « results ». For an input x, we say that the TuringmachineM• accepts x ifM reaches the final state YES after a finite

number of steps (i.e. transitions)• rejects x ifM reaches the final state NO after a finite

number of steps• loops on x ifM never reaches a final state


Language of a Turing MachineWe call language of a Turing MachineM the set of all inputswhich are accepted byM

L(M) = {x ∈ Σ∗ | M stops on x andM(x) = YES}

ExampleIfM returns YES on inputs P ∈ N[X] with exactly one root,then L(M) = {P ∈ N[X] | P is a polynomial with exactly oneroot }

Language ' ProblemWe associate the language L(M) of a Turing MachineM to adecision problem P

x ∈ L(M) iff x is a positive instance of P


(Non-)Deterministic Turing MachinesDefinitionGivenM a Turing machine and δ its transition function,M isdeterministic (DTM) if d is a mapping from any configuration(q, x) to a single image (q′, x′,m)Otherwise,M is non-deterministic (NDTM)

Solving a Decision Problem

• A DTM solves a decision problem P if the sequence oftransitions leads to the answer for any instance• Not so easy for NDTM: when the transition function has

several images, each of them must be checkedThe machine solves the problem if

I for each positive instance, at least one sequence oftransitions leads to a final state with the answer YES

I for each negative instance, every possible sequence oftransitions leads to a final state with the answer NO


Illustration: DTM vs NDTM

Deterministic(q0, x0)

(q1, x1)

. . .

(qn, xn)

δ

δ

δ

Non-Deterministic(q0, x0)

(q1, x1) . . . (qk, xk)

. . .. . . . . .

(qm, xm) (qn, xn)(qo, xo)

(qp, xp)

δ δ δ

δ

δ

δ

δ δ

δ

δ


Other Computing ModelsOther models have been proposed• Turing machines with several tapes/several

dimensions/several readers-writers• λ-calculus [Church 1936]• Kolmogorov-Uspensky machines

[Kolmogorov and Uspensky 1958]• Schönhage machines [Schönhage 1980]• . . .

Power of Turing machines: the Church-Turing ThesisEverything which can be computed with any of these modelscan be computed with a Turing machine

Turing completenessA system (computing model, programming language,. . . ) iscalled Turing complete if it can compute every possiblecomputable function


Outline

Turing MachinesIntroduction to Turing MachinesTuring Machines and Decision ProblemsNon-DeterminismChurch-Turing Thesis

DecidabilityRecognizable and Decidable ProblemsHalting Problem and Reductions


Recognizability

Recognizability of a ProblemA problem P is called recognizable if there exists a TuringmachineM such that, for each instance i of P,M(i) answersYES iff i is a positive instance of P

Recognizability of a LanguageA language L is called recognizable if there exists a TuringmachineM such that L = L(M)

Prime Numbers

• Is n ∈ N a prime number?

This problem/language is recognizable: we can write analgorithm which answers YES when n is prime, and doesn’t careof non-prime numbers


DecidabilityDecidability of a ProblemA problem P is called decidable if there exists a Turing machineM such that, for each instance i of P,• M(i) answers YES when i is a positive instance of P• M(i) answers NO when i is a negative instance of P

Decidability of a ProblemA language L is called decidable if there exists a Turing machineM which accepts each x ∈ L and rejects each x /∈ LEquivalently: L is decidable iff both L and L̄ are recognizable

Prime Numbers

• Is n ∈ N a prime number?

This problem/language is decidable: we can write an algorithmwhich answers YES when n is prime, and NO otherwise


Complement of a Decision Problem

DefinitionA problem P1 is called the complement of P2 L(P1) = L(P2).Equivalently:• P1 and P2 are defined on the same set of instances• Positive instances of P1 are negative instances of P2• Negative instances of P1 are positive instances of P2

Notation: P1 = P2

Prime Numbers

• Given n ∈ N, is there m ∈ N,m 6= 1,m 6= n, s.t. m dividesn?

This problem is the complement of the prime number problem.


Halting Problem

Is there a Turing machine H which has two parameters:• A Turing machineM• An input i ofM

and which returns• YES whenM stops on i• NO whenM loops on i

If there is such a machine, then Halting problem is decidable;otherwise, it is undecidable

Result from [Turing 1936]Halting is undecidable


Reducing a Problem to Another

Functional ReductionA functional reduction f is a total computable function from aproblem P1 to a problem P2 such that, for any instance i of P1,i is a positive instance iff f(i) is a positive instance of P2Notation:

P1 ≤f P2

TheoremIf P1 ≤f P2 and P1 is undecidable, then P2 is undecidable


(Trivial) Example of FunctionalReduction

• P1: Given n ∈ N, is n even?• P2: Given n ∈ N, is n odd?• f : N→ N is defined by f(n) = n+ 1

For a positive instance i of P1, f(i) is a positive instance of P2.For a negative instance i of P1, f(i) is a negative instance of P2.So P1 ≤f P2Here we also have P2 ≤f P1; this is trivial since these aredecidable problems.




For a positive instance i of P1, f(i) is a positive instance of P2.For a negative instance i of P1, f(i) is a negative instance of P2.

So P1 ≤f P2Here we also have P2 ≤f P1; this is trivial since these aredecidable problems.




For a positive instance i of P1, f(i) is a positive instance of P2.For a negative instance i of P1, f(i) is a negative instance of P2.So P1 ≤f P2

Here we also have P2 ≤f P1; this is trivial since these aredecidable problems.




For a positive instance i of P1, f(i) is a positive instance of P2.For a negative instance i of P1, f(i) is a negative instance of P2.So P1 ≤f P2Here we also have P2 ≤f P1; this is trivial since these aredecidable problems.


A Less Trivial Example

Language

L1 = {M | M halts on ε}

with ε the empty word. Is L1 decidable?

Proof of UndecidabilityWe need a mapping f : (M, x) 7→ M′ s.t. (M, x) ∈ Halting iffM′ ∈ L1• M′x(y) =M(x.y) with x.y the concatenation of words• M′x(ε) =M(x), so obviously:

I IfM halts on x, thenM′x halts on ε

I IfM loops on x, thenM′x loops on ε



Language



Proof of UndecidabilityWe need a mapping f : (M, x) 7→ M′ s.t. (M, x) ∈ Halting iffM′ ∈ L1

• M′x(y) =M(x.y) with x.y the concatenation of words• M′x(ε) =M(x), so obviously:





Language



Proof of UndecidabilityWe need a mapping f : (M, x) 7→ M′ s.t. (M, x) ∈ Halting iffM′ ∈ L1• M′x(y) =M(x.y) with x.y the concatenation of words• M′x(ε) =M(x), so obviously:




Other Method to Prove Undecidability

« Algorithmic » ProofIf P1 is known to be undecidable, we can prove that P2 isundecidable with an algorithm which computes P1 using only« simple » steps and calls to P2

Remember: L1 ={M | M halts on ε}.Suppose thatML1 isa Turing Machinewhich decides L1.

Algorithm 1 HaltingInput: M, x = x0x1 . . . xn

if x = ε thenreturn ML1(M)

elsex′ = x1 . . . xnM′ =M with q0 replaced by δ(q0, x0)return Halting(M′, x′)

end if








end if


How to Prove Decidability?• Prove P1 ≤f P2 with P2 decidable, then P1 is decidable

• Write an algorithm which solves your problem with only« simple » steps!

Of course, the more complex is the algorithm, the less easy it isto be sure that it is correct :(

Example: Is n ∈ N a prime number?

Algorithm 5 PrimeInput: n ∈ N

for x ∈ {2, . . . , b√nc} do

if ∃y ∈ N s.t. n = x× y thenreturn false

end ifend forreturn true


How to Prove Decidability?• Prove P1 ≤f P2 with P2 decidable, then P1 is decidable• Write an algorithm which solves your problem with only

« simple » steps!Of course, the more complex is the algorithm, the less easy it isto be sure that it is correct :(



for x ∈ {2, . . . , b√nc} do




References

A. M. Turing, On computable numbers, with an applicationto the Entscheidungsproblem. Proceedings of the LondonMathematical Society, 1936.

A. Church, An unsolvable problem of elementary numbertheory. American Journal of Mathematics, 1936.

A. N. Kolmogorov and V. Uspensky, On the definition of analgorithm. Uspekhi Mat. Naut, 1958.

A. Schönhage, Storage modification machines. SIAMJournal on Computing, 1980.


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