exam1 - ={h φ,s,t i | h G φ,s,t i ∈ stconn(c The...

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CSci 5403, Spring 2010 Exam 1 due: Feb 19, 2010 For each of the following languages, prove the tightest upper and lower bounds you can on its complexity. For our purposes, an upper bound is a proof that the language resides in some complexity class, and a lower bound is a proof that the language is hard for some complexity class. (a) Let G = ( V G ,E G ) and H = ( V H ,E H ) be directed graphs, and | V H | ≤ | V G | . We say that H is subgraph isomorphic to G , written H - G , if there is a 1-1 function f : V H V G such that ( u,v ) E H ⇐⇒ ( f ( u ) ,f ( v )) E G , that is, there is a way to map the vertices of H into G such that every edge between the vertices of H is also in G . We define the language subiso = ± h G,H i | H - G ² . (b) A boolean formula φ ( x 1 ,...,x n ,y 1 ,...,y n ) on 2 n variables implicitly defines a directed graph G φ = ( { 0 , 1 } n , { ( u,v ) | φ ( u,v ) } ), i.e. in which the vertices are strings of length n and there is an edge between vertices u and v if the formula φ ( u,v ) is true. We can then define the language formula-conn
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Unformatted text preview: = {h φ,s,t i | h G φ ,s,t i ∈ stconn } . (c) The “cat-and-mouse game” is a game between two players, the “cat” and the “mouse”, played on a directed graph G . A single vertex of G is designated the “hole,” and the cat and mouse are each placed on vertices of G . The players alternate turns, starting with the cat: at each turn, the current player must choose one of the vertices adjacent to his current position. If the cat and mouse ever occupy the same vertex, the cat wins. If the mouse enters the “hole” he wins. Each game can be encoded by the tuple h G,c,m,h i , meaning the game is played on graph G , the cat starts at vertex c , the mouse starts at vertex m , and the hole is at vertex h . We define the language cm = {h G,c,m,h i | the cat can force a win in h G,c,m,h i} . 1...
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