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Applied Computer Science II Chapter 4: Decidability Prof. Dr. Luc De Raedt Institut für Informatik Albert-Ludwigs Universität Freiburg Germany

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Overview An investigation into the solvable/decidable Decidable languages The halting problem (undecidable)

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Decidable problems ? Acceptance problem : –decide whether an automaton accepts a string Equivalence problem : –Decide whether two automata are equivalent, i.e. accept the same language Emptiness testing problem : –Decide whether the language of an automaton is empty Can be applied to –DFA, NFA, REX, PDA, CFG, TM,…

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The acceptance problem for DFAs

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The acceptance problem for NFAs

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The acceptance problem for Regular Expressions

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The emptiness testing problem for DFAs

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The equivalence problem for DFAs

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Decidable problems for CF languages

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The halting problem There is a specific problem that is algorithmically unsolvable (undecidable), e.g. the halting problem Philosophical implications : computers are fundamentally limited

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Diagonalization Georg Cantor 1873 Measure the size of (infinite) sets

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Summary

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