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Hector Miguel Chavez Western Michigan University Jun 10, 2009. Post&#039;s Correspondence Problem Word Problem in semi-Thue Systems. Post&#039;s Correspondence Problem. An instance of the Post&#039;s Correspondence Problem (PCP) consists of two lists of strings over some alphabet Σ; - PowerPoint PPT Presentation

### Transcript of Post's Correspondence Problem Word Problem in semi-Thue Systems

• Post's Correspondence Problem Word Problem in semi-Thue SystemsHector Miguel ChavezWestern Michigan UniversityJun 10, 2009

• Post's Correspondence ProblemAn instance of the Post's Correspondence Problem (PCP) consists of two lists of strings over some alphabet ; A = w1, w2, . . ., wkB = x1, x2, . . ., xk The PCP has a solution if there is a sequence where:wi, wi, . . ., wk = xi, xi, . . ., xk

• Post's Correspondence ProblemExample 1:This problem has a solution: 2, 1, 1, 3w2w1w1w3 = x2x1x1x3 = 101111110

List AList Biwixi11111210111103100

• Post's Correspondence ProblemExample 2:w1 = 10w3 = 101x1 = 101x3 = 01110101.. 101011...

List AList Biwixi1101012011113101011

• Post's Correspondence ProblemThe Modified PCPThe first pair in the solution must be the first pair in the lists.w1, wi, . . ., wk = x1, xi, . . ., xkNo solution

List AList Biwixi11111210111103100

• Post's Correspondence ProblemReducing a MPCP to PCP

List AList Biwixi0*1**1*1*111**1*1*121*0*1*1*1**1*031*0**04\$*\$

List AList Biwixi11111210111103100

• Post's Correspondence ProblemYESNOMPCPDeciderSolution?String SequencesABYESNOMembershipW L(G)InputGw

• Post's Correspondence ProblemMembership ProblemYESNOMPCPDeciderABGenerateA BGwMPCP can be reduced to PCP

• Post's Correspondence ProblemGenerating A & B

ABGFS FS: Start symbolF: Special SymbolaaFor every aVVFor every VE wEString wE: Special SymbolyxFor every productionX Y

• Post's Correspondence ProblemExample:

ABFS F

• Post's Correspondence ProblemExample:

ABFS Faabbcc

• Post's Correspondence ProblemExample:

ABFS FaabbccAABBCCSS

• Post's Correspondence ProblemExample:

ABFS FaabbccAABBCCSSE aaacEaABbSBbbSCBbaacAC

• Post's Correspondence ProblemMembership ProblemYESNOMPCPDeciderABGenerateA BGwMPCP can be reduce to PCP

• Word Problem for Semi-Thue SystemsA semi-Thue system S is a pair {, P} where: is an alphabetP is a set of rewrite rules or productions

In a rewriting x is called the antecedent and y the consequent.x yA semi-Thue system is also known as a rewriting system.

• Word Problem for Semi-Thue SystemsWe say that a word v over is immediately derivable from u if there is a rewrite rule x y such that: u = rxs and v = rys

If v is immediately derivable from u we write:

v u

• Word Problem for Semi-Thue SystemsLet P' be the set of all pairs (u, v) from * x * such that u v. Then P P' and if u v , then

w u w v and u w v w for any word w

If a * b there is a sequence of derivations a = a1, a2, a3 = b.If a * b and c * d imply ac * bd

• Word Problem for Semi-Thue SystemsExample: Let S be a semi-Thue system where: = {a, b, c}P = {ab bc, bc cb}. The words ac3b, a2c2b and bc4 can be derived from a2bc2.a2bc2 a(bc)c2 ac(bc)c ac2(cb) = ac3ba2bc2 a2(cb)c a2c(cb) = a2c2b a2bc2 a(bc)c2 (bc)cc2 = bc4

• Given an arbitrary semi-Thue system S over = {a, b} and two arbitrary words x, y, is y derivable from x in S?Word Problem for Semi-Thue SystemsThe halting problem of the Turing Machines can be reduced to the Word Problem. Ex: If given an input X, the machine halts if Y can be produced.

• ReferencesIntroduction to Automata Theory, Languages and Computation, John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman, 2nd edition, Addison Wesley 2001 (ISBN: 0-201-44124-1)Mathematical Theory of Computation, Zohar Manna. Courier Dover Publications, 2003 (ISBN 0486432386, 9780486432380)Lecture Notes, The Post Correspondence Problem, Konstantin Busch. www.csc.lsu.edu/~busch/courses/theorycomp/fall2008/slides/Post_Correspondence.ppt

• QuestionQ: How can you reduce an MPCP to PCP

List AList Biwixi0*1**1*1*111**1*1*121*0*1*1*1**1*031*0**04\$*\$

List AList Biwixi11111210111103100