Costas Busch - LSU1 The Post Correspondence Problem

Costas Busch - LSU2 Some undecidable problems for context-free languages: Is context-free grammar ambiguous? Is ? are context-free grammars

Costas Busch - LSU3 We need a tool to prove that the previous problems for context-free languages are undecidable: The Post Correspondence Problem

Costas Busch - LSU4 The Post Correspondence Problem Input: Two sets of strings

Costas Busch - LSU5 There is a Post Correspondence Solution if there is a sequence such that: PC-solution: Indices may be repeated or omitted

Costas Busch - LSU6 Example: PC-solution:

Costas Busch - LSU7 Example: There is no solution Because total length of strings from is smaller than total length of strings from

Costas Busch - LSU8 The Modified Post Correspondence Problem Inputs: MPC-solution:

Costas Busch - LSU9 Example: MPC-solution:

Costas Busch - LSU10 1. The MPC problem is undecidable 2. The PC problem is undecidable (by reducing MPC to PC) (by reducing the membership to MPC) We will show:

Costas Busch - LSU11 Theorem: The MPC problem is undecidable Proof: We will reduce the membership problem to the MPC problem

Costas Busch - LSU12 Membership problem Input: Turing machine string Question: Undecidable

Costas Busch - LSU13 Membership problem Input: unrestricted grammar string Question: Undecidable

Costas Busch - LSU14 Suppose we have a decider for the MPC problem MPC solution? YES NO String Sequences MPC problem decider

Costas Busch - LSU15 We will build a decider for the membership problem YES NO Membership problem decider

Costas Busch - LSU16 MPC problem decider Membership problem decider The reduction of the membership problem to the MPC problem: yes no

Costas Busch - LSU17 MPC problem decider Membership problem decider We need to convert the input instance of one problem to the other yes no Reduction?

Costas Busch - LSU18 Convert grammar and string Reduction: to sets of strings and Such that: generates There is an MPC solution for

Costas Busch - LSU19 : special symbol For every symbol Grammar : start variable For every variable

Costas Busch - LSU20 Grammar For every production : special symbol string

Costas Busch - LSU21 Example: Grammar : String

Costas Busch - LSU22

Costas Busch - LSU23

Costas Busch - LSU24 Grammar :

Costas Busch - LSU25 Derivation:

Costas Busch - LSU26 Derivation:

Costas Busch - LSU27 Derivation:

Costas Busch - LSU28 Derivation:

Costas Busch - LSU29 Derivation:

Costas Busch - LSU30 has an MPC-solution if and only if

Costas Busch - LSU31 MPC problem decider Membership problem decider Construct yes no

Costas Busch - LSU32 Since the membership problem is undecidable, The MPC problem is undecidable END OF PROOF

Costas Busch - LSU33 Theorem: The PC problem is undecidable Proof: We will reduce the MPC problem to the PC problem

Costas Busch - LSU34 Suppose we have a decider for the PC problem PC solution? YES NO String Sequences PC problem decider

Costas Busch - LSU35 We will build a decider for the MPC problem MPC solution? YES NO String Sequences MPC problem decider

Costas Busch - LSU36 PC problem decider MPC problem decider The reduction of the MPC problem to the PC problem: yes no

Costas Busch - LSU37 PC problem decider MPC problem decider yes no Reduction? We need to convert the input instance of one problem to the other

Costas Busch - LSU38 : input to the MPC problem : input to the PC problem Translated to

Costas Busch - LSU39 For each replace

Costas Busch - LSU40 PC-solution Has to start with These strings

Costas Busch - LSU41 PC-solution MPC-solution

Costas Busch - LSU42 has a PC solution has an MPC solution if and only if

Costas Busch - LSU43 PC problem decider MPC problem decider yes no Construct

Costas Busch - LSU44 Since the MPC problem is undecidable, The PC problem is undecidable END OF PROOF

Costas Busch - LSU45 Some undecidable problems for context-free languages: Is context-free grammar ambiguous? Is ? are context-free grammars We reduce the PC problem to these problems

Costas Busch - LSU46 Theorem: Proof: Let be context-free grammars. It is undecidable to determine if Reduce the PC problem to this problem (intersection problem)

Costas Busch - LSU47 Suppose we have a decider for the intersection problem Empty- interection problem decider YES NO Context-free grammars

Costas Busch - LSU48 We will build a decider for the PC problem PC solution? YES NO String Sequences PC problem decider

Costas Busch - LSU49 PC problem decider The reduction of the PC problem to the empty-intersection problem: noyes no Intersection problem decider

Costas Busch - LSU50 PC problem decider noyes no Reduction? We need to convert the input instance of one problem to the other Intersection problem decider

Costas Busch - LSU51 Context-free grammar : Introduce new unique symbols: Context-free grammar :

Costas Busch - LSU52 if and only if has a PC solution

Costas Busch - LSU53 Because are unique There is a PC solution:

Costas Busch - LSU54 PC problem decider noyes no Intersection problem decider Construct Context-Free Grammars

Costas Busch - LSU55 Since PC is undecidable, the Intersection problem is undecidable END OF PROOF

Costas Busch - LSU56 For a context-free grammar, Theorem: it is undecidable to determine if G is ambiguous Proof: Reduce the PC problem to this problem

Costas Busch - LSU57 PC problem decider noyes no Ambiguous problem decider Construct Context-Free Grammar

Costas Busch - LSU58 start variable of

Costas Busch - LSU59 if and only if is ambiguous if and only if has a PC solution