Lecture9_Decidability

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TIC 2151 Theory of

Computation

Lecture 9

Decidability

TIC 2151Theory of Computation


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Lecture 09 - Outline

Decidability

What is it?

Undecidable problems: some examples.

The Turing machine halting problem


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A Turning machine can do everything a real computer can do.

ChurchTuring thesis states that a function on the natural numbers is

computable if and only if it is computable by a Turing machine.

TM Execution Possible Outcomes

Accept: Running TMM on input w eventually leads to qaccept.Reject: Running TMM on input w eventually leads to qreject.

No Decision: Running TMM on input w runs forever, never reaches qacceptor

qreject.

Introduction


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Recognizing vs. Deciding

Turing-decidable: A language L is Turing-decidable if there exists a TMM

such that for all strings w:

If w L: eventually Menters qaccept.

If w L: eventually Menters qreject

also called a recursive language


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Recognizing vs. Deciding

Turing-recognizable : A language L is Turing recognizable if there exists a

TMMsuch that for all strings w:

If w L: eventually Menters qaccept.

If w L: either M enters qreject or Mnever terminates.

also called a recursively enumerable language


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Inside LTM:

A language is Turing-recognizable or recursively enumerable if some Turing machine

recognizes it.

A language is Turing-decidable (decidable, or recursive) if some Turing machine decides it.

We call a language decidable if some Turing Machines decides it.

We call a language decidable if there is a Turing Machines saying Yes or No on every input.

Every decidable language is Turing-recognizable.

Decidability

Lr.e.Lde L


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Lr.e.Lde L

Inside LTM:

Here the Turing Machine

has to stop on every inputand say Yes or No!

Here the Turing Machine

has to stop on the inputonly when saying Yes!

Here the Turing Machine will either not stop or print

wrong results as their cannot be a program for the problem!

Decidability

Decider vs. Recognizer?

Deciders always terminate.

Recognizers can run forever without deciding.


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Lr.e.Lde L

Remember! Already inside

here it is difficult tocompute a result as

sometimes the machine runs

forever. But unfortunately

many real-world problems

are lying in this class.

PCPHALTING

Lne

Busy Beaver

~HALTING

Le**

Lne = language not empty,Le = language empty.

PCP = Post Correspondence Problem

Decidability


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Lfin Lreg LDPDA Lcf Lcs Lde Lre L

Complete Summary:

Decidability


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Decidable Problem (example)

For example, theacceptance problem for DFAs of testingwhether a particular DFA

accepts a given string can be expressed as a language, ADFA. This language contains theencodings of all DFAs together with strings that the DFAs accept. Let

The problem of testing whether a DFA B accepts an input w is the same as the

problem of testing whether B,w. is a member of the language ADFA.

ADFA is a decidable language.

B,w is a representation of a DFA B together with a string w. One reasonable representation

of B is simply a list of its five components: Q, S, d, q0, andF. M carries out the simulation

directly. It keeps track of Bs current state and Bs current position in the input w by writing

this information down on its tape. If the simulation ends in an accept state, accept. If itends in a nonaccepting state, reject

PROOF IDEA : We simply need to present a TM M that decides ADFA.

M = On input B,w, where B is a DFA and w is a string:

1. Simulate B on input w.

2. If the simulation ends in an accept state, accept. If it ends in anonaccepting state, reject.


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Undecidability

There is a specific problem that is algorithmically unsolvable. Computers appear to

be so powerful that you may believe that all problems will eventually yield to them.The theorem presented here demonstrates that computers are limited in a

fundamental way.

you are given a computer program and a precise specification of what that

program is supposed to do (e.g., sort a list of numbers). You need to verify thatthe program performs as specified (i.e., that it is correct).

The general problem of software verification is not solvable by computer.

The problem of determining whether a Turing machine accepts a given input string

we call it ATMby analogy with ADFA and ACFG. But, whereas ADFA and ACFG weredecidable, ATM is not.

ATM is Turing-undecidable language.

ATM is Turing-recognizable language


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Undecidability

ATM is undecidable language.. Proof by contradiction Step 1

1) Suppose ATM is decidable andH is a decider forATM.

2) Construct a new TMD withHas a subroutine. It callsHto determine whatM does

when the input toM is its own description M. OnceD knows this, it does theopposite.

3) What happens when we runD with its own description D?

Can you see the contradiction?

Neither TMD nor TMHcan exist.

ATM is undecidable language

.acceptnotdoesif

acceptsif),(

wMreject

wMacceptwMH TMHM, w

Accept (ifMaccepts)

Reject (ifMrejects)

.acceptsif

acceptnotdoesif),(

MMreject

MMacceptMMD

.acceptsif

acceptnotdoesif),(

DDreject

DDacceptDDD


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Undecidability

Halting problem. HALTTM, the problem of determining whether aTuring machine

halts (by accepting or rejecting) on a given input.

Is there an algorithm able to decide whether a given code will terminate?

In some programs it helps to look at some variables, in others look at the loops, or tolook at the recursive calls, or ... or ... (and here come infinitely many ors)

Undecidabe languages examples


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Learning Outcomes

Upon completion of this lecture, you should be able to:

Explain the definition of Turing-decidable (recursive)

and Turing-recognizable (recursive enumerable)

languages. Tell that problems in Lr.e. and L are undecidable, but

for problems in Lr.e. we can give a TM program

accepting and stopping for at least all the Yes-instances.

Name some problems from undecidable language (Lr.e.and L).

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