Derivatives of WS1S Formulas Dmitriy Traytel Isabelle = - PowerPoint PPT Presentation

Derivatives of WS1S Formulas Dmitriy Traytel Isabelle ∀ = α λ β →

Logic-Automaton Connection WS1S ϕ = T | F | x ∈ X | x < y | ϕ ∨ ϕ | ¬ ϕ | ∃ x . ϕ | ∃ X . ϕ finite

Logic-Automaton Connection WS1S ∀ I . I � ϕ ⇐ ⇒ I � ψ ? ϕ = T | F | x ∈ X | x < y | ϕ ∨ ϕ | ¬ ϕ | ∃ x . ϕ | ∃ X . ϕ finite

Logic-Automaton Connection Finite Automata MONA WS1S ∀ I . I � ϕ ⇐ ⇒ I � ψ ? ϕ = T | F | x ∈ X | x < y | ϕ ∨ ϕ | ¬ ϕ | ∃ x . ϕ | ∃ X . ϕ finite

Logic-Automaton Connection Finite Automata MONA WS1S ∀ I . I � ϕ ⇐ ⇒ I � ψ ? Regular Expressions L ( α ) = L ( β )?

Logic-Automaton Connection Finite Automata MONA WS1S ∀ I . I � ϕ ⇐ ⇒ I � ψ ? ∀ I . I � ϕ ⇐ ⇒ enc I ∈ L ( mkRE ϕ ) T. & Nipkow, ICFP 2013 Π -Extended Regular Expressions L ( α ) = L ( β )?

a ∗ ? ≡ ε + a · a ∗ for Σ = { a , b } a ∗ ε + a · a ∗

a ∗ ? ≡ ε + a · a ∗ for Σ = { a , b } a ∗ d a ε + a · a ∗ ε · a ∗ ∅ + ε · a ∗ Brzozowski derivative d : letter → regex → regex L ( d a r ) = { w | aw ∈ L ( r ) }

a ∗ ? ≡ ε + a · a ∗ for Σ = { a , b } a ∗ d a d b ε + a · a ∗ ε · a ∗ ∅ · a ∗ ∅ + ε · a ∗ ∅ + ∅ · a ∗

a ∗ ? ≡ ε + a · a ∗ for Σ = { a , b } a ∗ d a d b ε + a · a ∗ ε · a ∗ ∅ · a ∗ ∅ + ε · a ∗ ∅ + ∅ · a ∗ d a ∅ · a ∗ + ε · a ∗ ∅ + ∅ · a ∗ + ε · a ∗

a ∗ ? ≡ ε + a · a ∗ for Σ = { a , b } a ∗ d a d b ε + a · a ∗ ε · a ∗ ∅ · a ∗ ∅ + ε · a ∗ ∅ + ∅ · a ∗ d a ∅ · a ∗ + ε · a ∗ ∅ + ∅ · a ∗ + ε · a ∗ d a ∅ · a ∗ + ∅ · a ∗ + ε · a ∗ ∅ + ∅ · a ∗ + ∅ · a ∗ + ε · a ∗

a ∗ ? ≡ ε + a · a ∗ for Σ = { a , b } a ∗ d a d b ε + a · a ∗ ε · a ∗ ∅ · a ∗ ∅ + ε · a ∗ ∅ + ∅ · a ∗ d a ∅ · a ∗ + ε · a ∗ ∅ + ∅ · a ∗ + ε · a ∗ d a ACI ∅ · a ∗ + ∅ · a ∗ + ε · a ∗ ∅ + ∅ · a ∗ + ∅ · a ∗ + ε · a ∗

a ∗ ? ≡ ε + a · a ∗ for Σ = { a , b } a ∗ d a d b ε + a · a ∗ ε · a ∗ ∅ · a ∗ d a ∅ + ε · a ∗ ∅ + ∅ · a ∗ d b d b d a ACI ACI ∅ · a ∗ + ε · a ∗ ∅ · a ∗ + ∅ · a ∗ ∅ + ∅ · a ∗ + ε · a ∗ ∅ + ∅ · a ∗ + ∅ · a ∗ d b d a ACI ∅ · a ∗ + ∅ · a ∗ + ε · a ∗ ∅ · a ∗ + ∅ · a ∗ + ∅ · a ∗ ∅ + ∅ · a ∗ + ∅ · a ∗ + ε · a ∗ ∅ + ∅ · a ∗ + ∅ · a ∗ + ∅ · a ∗

Key ingredients: Derivative + Acceptance Test

Key ingredients: Derivative + Acceptance Test Let’s define them on WS1S formulas directly!

? ( ∃ X . x ∈ X ) ≡ ( ¬ x < x ) for Σ = { ( T ) , ( F ) } ∃ X . x ∈ X d ( F ) d ( T ) ¬ x < x ACI ∃ X . ( T ∨ F ) ∃ X . ( x ∈ X ∨ x ∈ X ) ¬ x < x ¬ F d ( T ) , d ( F ) ACI ∃ X . ( T ∨ F ) ∨ ( T ∨ F ) ¬ F

Derivative = d v T T = d v F F  x ∈ X if ¬ v [ x ]   d v ( x ∈ X ) = if v [ x ] ∧ v [ X ] T  F otherwise  d v ( x < y ) = ··· d v ( ϕ ∨ ψ ) = d v ϕ ∨ d v ψ d v ( ¬ ϕ ) = ¬ d v ϕ d v ( ∃ X . ϕ ) = ∃ X . ( d ( v X �→ T ) ϕ ∨ d ( v X �→ F ) ϕ )

Acceptance Test = ε T T = ε F F ε ( x ∈ X ) = F ε ( x < y ) = F ε ( ϕ ∨ ψ ) = ε ϕ ∨ ε ψ ε ( ¬ ϕ ) = ¬ ε ϕ ε ( ∃ X . ϕ ) = action happens here

Acceptance Test = ε T T = ε F F ε ( x ∈ X ) = F ε ( x < y ) = F ε ( ϕ ∨ ψ ) = ε ϕ ∨ ε ψ ε ( ¬ ϕ ) = ¬ ε ϕ ε ( ∃ X . ϕ ) = action happens here futurization derivatives from the right

Altogether A decision procedure for WS1S that

Altogether A decision procedure for WS1S that operates on formulas directly and

Altogether A decision procedure for WS1S that operates on formulas directly and Isabelle HOL ∀ is verified in = α and λ β →

Altogether A decision procedure for WS1S that operates on formulas directly and Isabelle HOL ∀ is verified in = α and λ β → outperforms MONA on very well selected examples.

Altogether A decision procedure for WS1S that operates on formulas directly and Isabelle HOL ∀ is verified in = α and λ β → outperforms MONA on very well selected examples. Thanks. Questions?

Derivatives of WS1S Formulas Dmitriy Traytel Isabelle ∀ = α λ β →