The Post Correspondence Problem

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The Post Correspondence Problem. Some undecidable problems for context-free languages:. Is ?. are context-free grammars. Is context-free grammar ambiguous?. We need a tool to prove that the previous problems for context-free languages

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### The Post Correspondence Problem

Costas Busch - RPI

Some undecidable problems for

context-free languages:

• Is ?

are context-free grammars

• Is context-free grammar ambiguous?

Costas Busch - RPI

We need a tool to prove that the previous

problems for context-free languages

are undecidable:

The Post Correspondence Problem

Costas Busch - RPI

The Post Correspondence Problem

Input:

Two sets of strings

Costas Busch - RPI

There is a Post Correspondence Solution

if there is a sequence such that:

PC-solution:

Indices may be repeated or omitted

Costas Busch - RPI

Example:

PC-solution:

Costas Busch - RPI

Example:

There is no solution

Because total length of strings from

is smaller than total length of strings from

Costas Busch - RPI

The Modified Post Correspondence Problem

Inputs:

MPC-solution:

Costas Busch - RPI

Example:

MPC-solution:

Costas Busch - RPI

We will show:

1. The MPC problem is undecidable

(by reducing the membership to MPC)

2. The PC problem is undecidable

(by reducing MPC to PC)

Costas Busch - RPI

Theorem: The MPC problem is undecidable

Proof: We will reduce the membership

problem to the MPC problem

Costas Busch - RPI

Membership problem

Input: Turing machine

string

Question:

Undecidable

Costas Busch - RPI

Membership problem

Input: unrestricted grammar

string

Question:

Undecidable

Costas Busch - RPI

Suppose we have a decider for

the MPC problem

String Sequences

MPC solution?

YES

MPC problem

decider

NO

Costas Busch - RPI

We will build a decider for

the membership problem

YES

Membership

problem

decider

NO

Costas Busch - RPI

The reduction of the membership problem

to the MPC problem:

Membership problem decider

yes

yes

MPC problem

decider

no

no

Costas Busch - RPI

We need to convert the input instance of

one problem to the other

Membership problem decider

yes

yes

MPC problem

decider

Reduction?

no

no

Costas Busch - RPI

Reduction:

Convert grammar and string

to sets of strings and

Such that:

There is an MPC

solution for

generates

Costas Busch - RPI

Grammar

: start variable

: special symbol

For every symbol

For every variable

Costas Busch - RPI

Grammar

string

: special symbol

For every production

Costas Busch - RPI

Example:

Grammar :

String

Costas Busch - RPI

Grammar :

Costas Busch - RPI

Derivation:

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Derivation:

Costas Busch - RPI

Derivation:

Costas Busch - RPI

Derivation:

Costas Busch - RPI

Derivation:

Costas Busch - RPI

has an MPC-solution

if and only if

Costas Busch - RPI

Membership problem decider

yes

yes

Construct

MPC problem

decider

no

no

Costas Busch - RPI

Since the membership problem is undecidable,

The MPC problem is undecidable

END OF PROOF

Costas Busch - RPI

Theorem: The PC problem is undecidable

Proof: We will reduce the MPC problem

to the PC problem

Costas Busch - RPI

Suppose we have a decider for

the PC problem

String Sequences

PC solution?

YES

PC problem

decider

NO

Costas Busch - RPI

We will build a decider for

the MPC problem

String Sequences

MPC solution?

YES

MPC problem

decider

NO

Costas Busch - RPI

The reduction of the MPC problem

to the PC problem:

MPC problem decider

yes

yes

PC problem

decider

no

no

Costas Busch - RPI

We need to convert the input instance of

one problem to the other

MPC problem decider

yes

yes

PC problem

decider

Reduction?

no

no

Costas Busch - RPI

: input to the MPC problem

Translated to

: input to the PC problem

Costas Busch - RPI

replace

For each

For each

Costas Busch - RPI

PC-solution

These strings

Costas Busch - RPI

PC-solution

MPC-solution

Costas Busch - RPI

has a PC solution

if and only if

has an MPC solution

Costas Busch - RPI

MPC problem decider

yes

yes

Construct

PC problem

decider

no

no

Costas Busch - RPI

Since the MPC problem is undecidable,

The PC problem is undecidable

END OF PROOF

Costas Busch - RPI

Some undecidable problems for

context-free languages:

• Is ?

are context-free grammars

• Is context-free grammar
• ambiguous?

We reduce the PC problem to these problems

Costas Busch - RPI

Theorem:

Let be context-free

grammars. It is undecidable

to determine if

(intersection problem)

Reduce the PC problem to this

problem

Proof:

Costas Busch - RPI

Suppose we have a decider for the

intersection problem

Context-free

grammars

Empty-

interection

problem

decider

YES

NO

Costas Busch - RPI

We will build a decider for

the PC problem

String Sequences

PC solution?

YES

PC problem

decider

NO

Costas Busch - RPI

The reduction of the PC problem

to the empty-intersection problem:

PC problem decider

no

yes

Intersection

problem

decider

no

yes

Costas Busch - RPI

We need to convert the input instance of

one problem to the other

PC problem decider

no

yes

Intersection

problem

decider

Reduction?

no

yes

Costas Busch - RPI

Introduce new unique symbols:

Context-free grammar :

Context-free grammar :

Costas Busch - RPI

has a PC solution

if and only if

Costas Busch - RPI

Because are unique

There is a PC solution:

Costas Busch - RPI

PC problem decider

no

yes

Construct

Context-Free

Grammars

Intersection

problem

decider

no

yes

Costas Busch - RPI

Since PC is undecidable,

the Intersection problem is undecidable

END OF PROOF

Costas Busch - RPI

For a context-free grammar ,

Theorem:

it is undecidable to determine

if G is ambiguous

Reduce the PC problem

to this problem

Proof:

Costas Busch - RPI

PC problem decider

no

yes

Construct

Context-Free

Grammar

Ambiguous

problem

decider

no

yes

Costas Busch - RPI

start variable of

start variable of

start variable of

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has a PC solution

if and only if

if and only if

is ambiguous

Costas Busch - RPI