l16 - Lecture 16: coNP and PSPACE completeness David Dill...

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Lecture 16: coNP and PSPACE completeness David Dill Department of Computer Science 1
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Outline coNP PSPACE 2
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Other complexity classes The basic ideas that we have seen in NP-completeness (reductions, complete problems, etc.) have been applied to many other complexity classes (see http://qwiki.stanford.edu/wiki/Complexity_Zoo to get an impression of just how many there are!) We’ll talk about two of the most important classes that are also most closely related to the NP-complete problems: CoNP and PSPACE (called ps in the textbook). CoNP = “Complement of a problem in NP” PSPACE = “Can be computed using polynomial space” 3
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Closure properties Closure under complementation is a very important topic in these classes. A complexity class C is closed under complementation if, L ∈ C implies L is also in C . Theorem P is closed under complementation. This is easy to see. If L P , then there is a DTM that can compute it (and always halts, of course) in polynomial time. Modify this DTM to swap the accept and reject results. The book points out that this can be done at a cost of one additional step in the computation, so the complemented DTM also runs in polynomial time. 4
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For a concrete example: Let L be the triples a G,v 0 ,v 1 A of a directed graph and two vertices in the graph where there is a path from v 0 to v 1 in G . L = Σ * L It includes the set of problems a G,v 0 ,v 1 A where there is not a path from v 0 to v 1 in G – plus any ill-formed inputs (e.g., the graph encoding is bogus, or there is a missing separator between G and v 0 or v 0 and v 1 ). Checking whether an input is well-formed is generally much easier than doing the actual computation, and we only care about the hardest problems in a complexity class. We will ignore the ill-formed members of L . 5
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l16 - Lecture 16: coNP and PSPACE completeness David Dill...

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