A Note on Decidable Separability by Piecewise Testable Languages - - PowerPoint PPT Presentation

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A Note on Decidable Separability by Piecewise Testable Languages - - PowerPoint PPT Presentation

Sep 26, 2022 251 likes •930 views

A Note on Decidable Separability by Piecewise Testable Languages Wojciech Czerwi ski Wim Martens Lorijn van Rooijen Marc Zeitoun Separability Separability K L Separability K L S Separability K L S S separates K and L

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A Note on Decidable Separability by Piecewise Testable Languages

Wim Martens Wojciech Czerwiński Lorijn van Rooijen Marc Zeitoun

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Separability

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Separability

K L

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Separability

S K L

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Separability

S S separates K and L K L

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Separability

S S separates K and L K L K and L are separable by family F if some S from F separates them

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General problem

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General problem

Given: two languages K and L from family F1

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General problem

Given: two languages K and L from family F1 Question: are K and L separable by some language from family F2

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General problem

Given: two languages K and L from family F1 Question: are K and L separable by some language from family F2 Separability of F1 by F2

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General problem

Given: two languages K and L from family F1 Question: are K and L separable by some language from family F2 Separability of F1 by F2 If F1 effectively closed under complement

  • generalization of membership
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Main problem

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Main problem

Given: context-free grammars for languages K and L

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Main problem

Given: context-free grammars for languages K and L Question: are K and L separable by piecewise testable languages (PTL)?

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Main problem

Given: context-free grammars for languages K and L Question: are K and L separable by piecewise testable languages (PTL)? piece language

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Main problem

Given: context-free grammars for languages K and L Question: are K and L separable by piecewise testable languages (PTL)? Σ* a1 Σ* a2 Σ*... Σ* an Σ* piece language

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Main problem

Given: context-free grammars for languages K and L Question: are K and L separable by piecewise testable languages (PTL)? Σ* a1 Σ* a2 Σ*... Σ* an Σ* piece language piecewise testable language

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Main problem

Given: context-free grammars for languages K and L Question: are K and L separable by piecewise testable languages (PTL)? Σ* a1 Σ* a2 Σ*... Σ* an Σ* piece language

  • bool. comb. of pieces

piecewise testable language

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What is known?

Separability of CFL by

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What is known?

  • CFL - undecidable (intersection problem)

Separability of CFL by

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What is known?

  • CFL - undecidable (intersection problem)
  • regular languages - undecidable

Separability of CFL by

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What is known?

  • CFL - undecidable (intersection problem)
  • regular languages - undecidable
  • any family containing wΣ* and closed under

boolean combination - undecidable Separability of CFL by

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Our main result

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Our main result

Theorem: Separability of context free languages by piecewise testable languages is decidable

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Our main message

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Our main message

  • something nontrivial possible for separability
  • f CFL
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Our main message

  • something nontrivial possible for separability
  • f CFL
  • no algebra needed
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Our main message

  • something nontrivial possible for separability
  • f CFL
  • no algebra needed
  • piecewise testable languages are special
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Our main message

  • something nontrivial possible for separability
  • f CFL
  • no algebra needed
  • piecewise testable languages are special
  • separability problem is special (deciding

whether CFL is a PTL is undecidable)

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Proof (sketch)

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Proof (sketch)

Two semi-procedures

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Proof (sketch)

Two semi-procedures One tries to show separability

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Proof (sketch)

Two semi-procedures One tries to show separability One tries to show non-separability

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Proof (sketch)

Two semi-procedures One tries to show separability One tries to show non-separability Enumerates all piecewise testable languages and test them

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Proof (sketch)

Two semi-procedures One tries to show separability One tries to show non-separability Enumerates all piecewise testable languages and test them Enumerates all patterns and test them

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Second main result

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Second main result

Theorem Languages K and L are non-separable by PTL if and only if there exists a pattern p, that fits both to K and L

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Second main result

Theorem Languages K and L are non-separable by PTL if and only if there exists a pattern p, that fits both to K and L It is decidable whether pattern p fits to CFL L

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Patterns

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Patterns

Pattern p over Σ consists of:

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Patterns

Pattern p over Σ consists of: words w0, w1, ..., wn in Σ*

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Patterns

Pattern p over Σ consists of: words w0, w1, ..., wn in Σ* subalphabets B1, ..., Bn of Σ

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Patterns

Pattern p over Σ consists of: words w0, w1, ..., wn in Σ* subalphabets B1, ..., Bn of Σ B⊗ = words from B* that contain all the letters from B

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Patterns

Pattern p over Σ consists of: words w0, w1, ..., wn in Σ* subalphabets B1, ..., Bn of Σ Pattern p fits to a language L if for all k ≥ 0 intersection of L and w0 (B1⊗)k w1 ... wn-1 (Bn⊗)k wn is nonempty B⊗ = words from B* that contain all the letters from B

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Generalization

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Generalization

The same construction works for separating:

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Generalization

The same construction works for separating:

  • languages of Petri Nets
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Generalization

The same construction works for separating:

  • languages of Petri Nets
  • languages of Higher Order Pushdown

Automata of order 2

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Generalization

The same construction works for separating:

  • languages of Petri Nets
  • languages of Higher Order Pushdown

Automata of order 2

  • every well-behaving family of languages
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Well-behaving languages

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Well-behaving languages

Family of languages over Σ is a full-trio if it is effectively closed under:

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Well-behaving languages

Family of languages over Σ is a full-trio if it is effectively closed under:

  • removing letters from subalphabet B ⊆ Σ
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Well-behaving languages

Family of languages over Σ is a full-trio if it is effectively closed under:

  • removing letters from subalphabet B ⊆ Σ
  • adding letters from subalphabet B ⊆ Σ
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Well-behaving languages

Family of languages over Σ is a full-trio if it is effectively closed under:

  • removing letters from subalphabet B ⊆ Σ
  • adding letters from subalphabet B ⊆ Σ
  • intersection with regular languages
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Diagonal problem

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Diagonal problem

Given: word language L over alphabet Σ

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Diagonal problem

Given: word language L over alphabet Σ Question: does there exists for every n a word in L containing each letter from Σ at least n times?

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Generalized theorem

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Generalized theorem

Theorem: For every full-trio F with decidable diagonal problem separability of F by PTL is decidable

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Further research

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Further research

  • complexity of separability of CFL by PTL
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Further research

  • complexity of separability of CFL by PTL
  • is separability of CFL by some other

nontrivial family decidable?

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Further research

  • complexity of separability of CFL by PTL
  • is separability of CFL by some other

nontrivial family decidable?

  • group languages?
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Further research

  • complexity of separability of CFL by PTL
  • is separability of CFL by some other

nontrivial family decidable?

  • group languages?
  • solvable group languages?
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Further research

  • complexity of separability of CFL by PTL
  • is separability of CFL by some other

nontrivial family decidable?

  • group languages?
  • solvable group languages?
  • connections with other problems
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Thank you!