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Understanding Decidability in Computational Problems

Explore the concept of decidability in computational problems with examples, including DFA acceptance and equivalence problems. Learn about undecidable problems and the existence of unrecognizable languages. Discover the implications of Turing-unrecognizable languages.

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Understanding Decidability in Computational Problems

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  1. Lecture 6Decidability Jan Maluszynski, IDA, 2007 http://www.ida.liu.se/~janma janma @ ida.liu.se Jan Maluszynski - HT 2007

  2. 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 • 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

  3. 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

  4. Example Problems and their languages Acceptance Problem for DFA: check if a given DFA accepts a given string. Language: 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. A Turing-unrecognizable language L is decidable iff both L and its complement are Turing-recognizable. 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

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