Separability of Context-Free Languages by Piecewise Testable - - PowerPoint PPT Presentation

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Separability of Context-Free Languages by Piecewise Testable - - PowerPoint PPT Presentation

Jan 11, 2023 15 likes •506 views

Separability of Context-Free Languages by Piecewise Testable Languages Wojciech Czerwi ski Wim Martens Separability Separability K L Separability K L S Separability K L S S separates K and L Separability K L S S separates K and

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Separability

  • f Context-Free Languages

by Piecewise Testable Languages

Wim Martens Wojciech Czerwiński

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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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Problem

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Problem

Given: context-free grammars for languages K and L

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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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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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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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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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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 (reverse)-definite

languages - undecidable Separability of CFL by

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Definite languages

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Definite languages

reverse definite language = finite union of wΣ*

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Definite languages

reverse definite language = finite union of wΣ* any logic L

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Definite languages

reverse definite language = finite union of wΣ* any logic L

  • able to express n-th letter equals a
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Definite languages

reverse definite language = finite union of wΣ* any logic L

  • able to express n-th letter equals a
  • closed under boolean combinations
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Definite languages

reverse definite language = finite union of wΣ* any logic L

  • able to express n-th letter equals a
  • closed under boolean combinations

describes all reverse definite languages

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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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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 Lossy Counter Machines (?)
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Generalization

The same construction works for separating:

  • languages of Petri Nets
  • languages of Lossy Counter Machines (?)
  • every class of well-behaving languages
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Thank you!

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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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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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Patterns and separability

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Patterns and separability

Theorem (van Rooijen, Zeitoun `13): Languages K and L are non-separable by PTL if and only if there exists a pattern p, that fits to both to K and L