Post's Correspondence Problem Word Problem in semi-Thue Systems
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Post's Correspondence Problem Word Problem in semi-Thue Systems Hector Miguel Chavez Western Michigan University Jun 10, 2009
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Hector Miguel Chavez Western Michigan University Jun 10, 2009. Post's Correspondence Problem Word Problem in semi-Thue Systems. Post's Correspondence Problem. An instance of the Post'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 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, w
2, . . ., w
k
B = x1, x
2, . . ., x
k
The PCP has a solution if there is a sequence where:
wi, w
i, . . ., w
k = x
i, x
i, . . ., x
k
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
w2w
1w
1w
3 = x
2x
1x
1x
3 = 101111110
Post's Correspondence Problem
Example 2:
List A List B
i wi
xi
1 10 101
2 011 11
3 101 011
w1 = 10
w3 = 101
x1 = 101
x3 = 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, w
i, . . ., w
k = x
1, x
i, . . ., x
k
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 1* *1*1*1
2 1*0*1*1*1* *1*0
3 1*0* *0
4 $ *$
List A List B
i wi
xi
1 1 111
2 10111 10
3 10 0
Post's Correspondence Problem
YES
NOMPCPDecider
Solution?String Sequences
A
B
YES
NOMembership
W ∈ L(G)Input
G
w
Post's Correspondence Problem
Membership Problem
YES
NOMPCPDecider
A
BGenerateA B
G
w
MPCP can be reduced to PCP
Post's Correspondence Problem
Generating A & B
A B G
FS → FS: Start symbol
F: Special Symbol
a a For every a
V V For every V
E → wEString w
E: Special Symbol
y xFor every production
X → Y
→ →
Post's Correspondence Problem
Example:
aacAC
CBb
BbbaABbS
|
aaacw
A B
FS → F
Post's Correspondence Problem
Example:A B
FS → F
a a
b b
c c
aacAC
CBb
BbbaABbS
|
aaacw
Post's Correspondence Problem
Example:A B
FS → F
a a
b b
c c
A A
B B
C C
S SaacAC
CBb
BbbaABbS
|
aaacw
Post's Correspondence Problem
Example:A B
FS → F
a a
b b
c c
A A
B B
C C
S S
E → aaacE
aABb S
Bbb S
C Bb
aac AC
→ →
aacAC
CBb
BbbaABbS
|
aaacw
Post's Correspondence Problem
Membership Problem
YES
NOMPCPDecider
A
BGenerateA B
G
w
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 = 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 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*1
1 1* *1*1*1
2 1*0*1*1*1* *1*0
3 1*0* *0
4 $ *$
List A List B
i wi
xi
1 1 111
2 10111 10
3 10 0
