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Lecture 6 Decidability. Jan Maluszynski , IDA, 200 7 http://www.ida.liu.se/~janma janma @ ida.liu.se. Outline. Lecture 6 : Decidability (Sipser 4.1 – 4.2) Motivation What is a problem? Problem vs. Language Decidable and semi-decidable problems. Examples of decidable problems

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lecture 6 decidability

Lecture 6Decidability

Jan Maluszynski, IDA, 2007


janma @ ida.liu.se

Jan Maluszynski - HT 2007


Lecture 6 : Decidability (Sipser 4.1 – 4.2)

  • Motivation
  • What is a problem? Problem vs. Language
  • Decidable and semi-decidable problems.
  • Examples of decidable problems
  • Undecidability: the diagonalization method
  • The halting problem is undecidable
  • More examples of undecidable problems
  • Existence of unrecognizable languages.
  • Reducibility and examples of its application

Jan Maluszynski - HT 2007

computational problems vs languages
Computational problems vs. Languages

Computational problem:

a function from input data to output data

more generally a relation R on data

Data encoded as strings

Relations encoded as set of strings

For a given encoding

R is represented as a language <R>

Computing R => generating strings of <R>

Jan Maluszynski - HT 2007

example problems and their languages
Example Problems and their languages

Acceptance Problem for DFA: check if a given DFA accepts a given string.


ADFA = {<B,w> | B is aDFA that accepts w}

Emptiness Problem for DFA

EDFA = {<A> | A is a DFA and L(A) is empty}

Equivalence Problem for DFA

EQDFA = {<A,B> | A,B are DFAs and L(A)=L(B)}

Jan Maluszynski - HT 2007

decidable problems
Decidable Problems

An algorithm for a Computational Problem R is a Turing Machine that recognizes/generates the language <R>

A problem Ris decidable iff

there exists a TM that decides <R>.

Encoding is a one-one onto function.

Thus there is a correspondence between different encodings.

Jan Maluszynski - HT 2007

examples of decidable problems
Examples of Decidable Problems
  • All problems in slide 4, e.g.:

DFA Acceptance Problem:

Construct a TM that on input <B,w>

simulates B on w

If the simulation ends in accept state accept,

otherwise reject

  • Parsing problem for CFG

argue why

  • Emptiness problem for CFG

argue why

Jan Maluszynski - HT 2007

an undecidable problem
An Undecidable Problem

Given a TM M and a string w check if M accepts w

ATM = {<M,w> | M is a TM that accepts w}

Simulation of M on w may loop!

But ATM is Turing-recognizable

Jan Maluszynski - HT 2007

a tm is undecidable
ATM is undecidable

Assume the contrary: there exists a TM H that

accepts <M,w> if M accepts w

rejects <M,w> otherwise

Construct TM D that runs on descriptions of TM’s:

on input <M>

D runs H on input <M,<M>> and:

accepts if M does not accept <M>

rejects if M accepts <M>

D(<D>) accepts if D does not accept <D>

rejects if D accepts <D> : contradiction

Jan Maluszynski - HT 2007

turing unrecognizable languages
Turing-unrecognizable languages

The set of all Turing machines is countable

The set of all languages over a given alphabet is not countable!

There are languages not recognizable by a TM! Turing-unrecognizable languages

Jan Maluszynski - HT 2007

a turing unrecognizable language
A Turing-unrecognizable language

L is decidable iff both L and its complement are


Explain why!

ATM = {<M,w> | M is a TM that accepts w}

We proved ATM is undecidable and Turing-recognizable

Hence the complement of ATM is Turing-unrecognizable

Jan Maluszynski - HT 2007