Posts Correspondence Problem Word Problem in semiThue Systems

Post's Correspondence Problem Word Problem in semi-Thue Systems Hector Miguel Chavez Western Michigan University Jun 10, 2009

Post's Correspondence Problem An instance of the Post's Correspondence Problem (PCP) consists of two lists of strings over some alphabet Σ; A = w 1, w 2, . . . , wk B = x 1, x 2, . . . , xk The PCP has a solution if there is a sequence where: wi, . . . , wk = xi, . . . , xk

Post's Correspondence Problem Example 1: List A List B i wi xi 1 1 111 2 10111 10 3 10 0 This problem has a solution: 2, 1, 1, 3 w 2 w 1 w 1 w 3 = x 2 x 1 x 1 x 3 = 101111110

Post's Correspondence Problem Example 2: List A List B i wi xi 1 10 101 2 011 11 3 101 011 w 1 = 10 w 3 = 101 x 1 = 101 x 3 = 011 10101. . 101011. . .

Post's Correspondence Problem The Modified “PCP” The first pair in the solution must be the first pair in the lists. w 1, wi, . . . , wk = x 1, xi, . . . , xk List A List B i wi xi 1 1 111 2 10111 10 3 10 0 No solution

Post's Correspondence Problem Reducing a MPCP to PCP List A List B i wi xi 0 *1*1*1 1 1 111 1 1* *1*1*1 2 10111 10 2 1*0*1*1*1* *1*0 3 1*0* *0 4 $ *$ List A 0

Post's Correspondence Problem String Sequences Solution? A YES B MPCP Decider NO Input W ∈ L(G) G YES w Membership NO

Post's Correspondence Problem Membership Problem G w A Generate AB B YES MPCP Decider MPCP can be reduced to PCP NO

Post's Correspondence Problem Generating A & B A B G FS → F S: Start symbol F: Special Symbol a a For every a V V For every V E → w. E String w E: Special Symbol y x For every production X→Y → →

Post's Correspondence Problem Example: A B FS → F

Post's Correspondence Problem Example: A B FS → F a a b b c c

Post's Correspondence Problem Example: A B FS → F a a b b c c A A B B C C S S

Post's Correspondence Problem Example: A B FS → F a a b b c c A A B B C C S S E → aaac. E a. ABb S Bbb S C Bb aac AC → →

Post's Correspondence Problem Membership Problem G w A Generate AB B YES MPCP Decider MPCP can be reduce to PCP NO

Word Problem for Semi-Thue Systems A semi-Thue system S is a pair {Σ, P} where: Σ is an alphabet P is a set of rewrite rules or productions In a rewriting x is called the antecedent and y the consequent. x→y A semi-Thue system is also known as a rewriting system.

Word Problem for Semi-Thue Systems We say that a word v over Σ 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 Systems Let P' be the set of all pairs (u, v) from Σ* x Σ* 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 = a 1, a 2, a 3 = b. If a ⇒* b and c ⇒* d imply ac ⇒* bd

Word Problem for Semi-Thue Systems Example: Let S be a semi-Thue system where: Σ = {a, b, c} P = {ab → bc, bc → cb}. The words ac 3 b, a 2 c 2 b and bc 4 can be derived from a 2 bc 2. a 2 bc 2 ⇒ a(bc)c 2 ⇒ ac(bc)c ⇒ ac 2(cb) = ac 3 b a 2 bc 2 ⇒ a 2(cb)c ⇒ a 2 c(cb) = a 2 c 2 b a 2 bc 2 ⇒ a(bc)c 2 ⇒ (bc)cc 2 = bc 4

Word Problem for Semi-Thue Systems Given an arbitrary semi-Thue system S over Σ = {a, b} and two arbitrary words x, y, is y derivable from x in S? The 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.

References Introduction to Automata Theory, Languages and Computation, John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman, 2 nd 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/fall 2008/sli des/Post_Correspondence. ppt

Question Q: How can you reduce an MPCP to PCP List A List B i wi xi 0 *1*1*1 1 1 111 1 1* *1*1*1 2 10111 10 2 1*0*1*1*1* *1*0 3 1*0* *0 4 $ *$ List A 0