day30 - Formal Language Undecidability Continued Click to...

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Click to edit Master subtitle style 10/4/11 Formal Language Undecidability Continued PCP and Traces
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10/4/11 Dec 2, © UCF (Charles E. 22 Post Correspondence Problem Many problems related to grammars can be shown to be no more complex than the Post Correspondence Problem (PCP). Each instance of PCP is denoted: Given n>0, “ a finite alphabet, and two n-tuples of words ( x1, … , xn ), ( y1, … , yn ) over & , does there exist a sequence i1, … , ik , k>0, 1 ≤ ij ≤ n, such that xi1 … xik = yi1 … yik ? Example of PCP: n = 3, & = { a , b }, ( a b a , b b , a ), ( b a b , b , b a a ). Solution 2 , 3, 1 , 2 b b a a b a b b = b b a a b a b b
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10/4/11 Dec 2, © UCF (Charles E. 33 PCP Example#2 Start with Semi-Thue System aba Instance of word problem: bbbb ±*? aa Convert to PCP [bbbb* abab aaaa a a ] [ aba aba a a b b * aa] And * * a a b b * * a a b b
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10/4/11 Dec 2, © UCF (Charles E. 44 How PCP Construction Works? Using underscored letters (~ in text) avoids solutions that don’t relate to word problem instance. E.g., aba a ab aa Top row insures start with [W0* Bottom row insures end with * Wf] Bottom row matches Wi, while top matches Wi+1 (one is underscored)
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10/4/11 Dec 2, © UCF (Charles E. 55 Ambiguity of CFG Problem to determine if an arbitrary CFG is ambiguous S A | B A xi A [i] | xi [i] 1 i n ≤ ≤ B yi B [i] | yi [i] 1 i n ≤ ≤ A * xi1 … xik [ik] … [i1] k > 0 B * yi1 … yik [ik] … [i1] k > 0
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10/4/11 Dec 2, © UCF (Charles E. 66 Intersection of CFLs Problem to determine if arbitrary CFG’s define overlapping languages Just take the grammar consisting of all the A-rules from previous, and a second grammar consisting of all the B-rules. Call the languages generated by these grammars, LA and LB. LA & LB ≠ Ø, if and only there is a solution to this PCP instance.
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10/4/11 Dec 2, © UCF (Charles E. 77 CSG Produces Something S o xi S yiR | xi T yiR 1 ≤ i ≤ n T l * Our only terminal is *. We get strings of form * 2j+1, for some j’s if and only if there is a solution to this PCP
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day30 - Formal Language Undecidability Continued Click to...

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