Post's Correspondence Problem Word Problem in semi-Thue Systems

Hector Miguel ChavezWestern 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 = w1, w2, . . ., wk

B = x1, x2, . . ., xk

The PCP has a solution if there is a sequence where:

wi, wi, . . ., wk = xi, xi, . . ., xk

Post's Correspondence Problem Example 1:

List A List B

i wi xi

1 1 1112 10111 103 10 0

This problem has a solution: 2, 1, 1, 3

w2w1w1w3 = x2x1x1x3 = 101111110

Post's Correspondence Problem Example 2:

List A List B

i wi xi

1 10 101

2 011 11

3 101 011

w1 = 10w3 = 101

x1 = 101x3 = 011

10101.. 101011...

Post's Correspondence Problem The Modified “PCP”

The first pair in the solution must be the first pair in the lists.

w1, wi, . . ., wk = x1, xi, . . ., xk

List A List B

i wi xi

1 1 1112 10111 103 10 0

No solution

Post's Correspondence Problem Reducing a MPCP to PCP

List A List B

i wi xi

0 *1* *1*1*11 1* *1*1*12 1*0*1*1*1* *1*03 1*0* *04 $ *$

List A List B

i wi xi

1 1 1112 10111 103 10 0

Post's Correspondence Problem

YES

NOMPCPDecider

Solution?String Sequences

AB

YES

NOMembership

W ∈ L(G)Input

Gw

Post's Correspondence Problem

Membership Problem

YES

NOMPCPDecider

ABGenerate

A B

Gw

MPCP can be reduced to PCP

Post's Correspondence Problem Generating A & B

A B G

FS → F S: Start symbolF: Special Symbol

a a For every aV V For every V

E → wE String wE: Special Symbol

y x For every productionX → Y

→ →

Post's Correspondence Problem Example:

aacACCBb

BbbaABbS

|

aaacw

A BFS → F

Post's Correspondence Problem Example:

A BFS → F

a a

b b

c c

aacACCBb

BbbaABbS

|

aaacw

Post's Correspondence Problem Example:

A BFS → F

a a

b b

c c

A A

B B

C C

S SaacACCBb

BbbaABbS

|

aaacw

Post's Correspondence Problem Example:

A BFS → F

a ab bc cA AB BC CS SE → aaacE

aABb SBbb SC Bb

aac AC→ →

aacACCBb

BbbaABbS

|

aaacw

Post's Correspondence Problem

Membership Problem

YES

NOMPCPDecider

ABGenerate

A B

Gw

MPCP can be reduce to PCP

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 = a1, a2, a3 = 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 ac3b, a2c2b and bc4 can be derived from a2bc2.

a2bc2 ⇒ a(bc)c2 ⇒ ac(bc)c ⇒ac2(cb) = ac3b a2bc2 ⇒ a2(cb)c ⇒ a2c(cb) a2c2b a2bc2 ⇒ a(bc)c2 ⇒ (bc)cc2 bc4

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.

Word Problem for Semi-Thue Systems

References Introduction 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

Question

Q: How can you reduce an MPCP to PCP

List A List B

i wi xi

0 *1* *1*1*11 1* *1*1*12 1*0*1*1*1* *1*03 1*0* *04 $ *$

List A List B

i wi xi

1 1 1112 10111 103 10 0