SLIDE 1
Derivatives of WS1S Formulas
Dmitriy Traytel
λ → ∀
=
Isabelle
β α
Logic-Automaton Connection
WS1S
ϕ = T | F | x ∈ X | x < y | ϕ∨ϕ | ¬ϕ | ∃x. ϕ | ∃X. ϕ
finite
Derivatives of WS1S Formulas Dmitriy Traytel Isabelle = - - PowerPoint PPT Presentation
Derivatives of WS1S Formulas Dmitriy Traytel Isabelle = - - PowerPoint PPT Presentation
Derivatives of WS1S Formulas Dmitriy Traytel Isabelle = Logic-Automaton Connection WS1S = T | F | x X | x < y | | | x . | X . finite Logic-Automaton Connection WS1S I . I
SLIDE 2
SLIDE 3
Logic-Automaton Connection
WS1S
ϕ = T | F | x ∈ X | x < y | ϕ∨ϕ | ¬ϕ | ∃x. ϕ | ∃X. ϕ
finite ∀I. I ϕ ⇐ ⇒ I ψ?
SLIDE 4
Logic-Automaton Connection
WS1S
ϕ = T | F | x ∈ X | x < y | ϕ∨ϕ | ¬ϕ | ∃x. ϕ | ∃X. ϕ
finite ∀I. I ϕ ⇐ ⇒ I ψ? Finite Automata
MONA
SLIDE 5
Logic-Automaton Connection
WS1S ∀I. I ϕ ⇐ ⇒ I ψ? Regular Expressions L(α) = L(β)? Finite Automata
MONA
SLIDE 6
Logic-Automaton Connection
WS1S ∀I. I ϕ ⇐ ⇒ I ψ?
Π-Extended Regular Expressions
L(α) = L(β)? ∀I. I ϕ ⇐ ⇒ enc I ∈ L(mkRE ϕ) Finite Automata
MONA
- T. & Nipkow, ICFP 2013
SLIDE 7
a∗ ?
≡ ε+ a· a∗ for Σ = {a,b}
a∗
ε+ a· a∗
SLIDE 8
a∗ ?
≡ ε+ a· a∗ for Σ = {a,b}
a∗
ε+ a· a∗
Brzozowski derivative d: letter → regex → regex
L(da r) = {w | aw ∈ L(r)} ε· a∗ ∅+ε· a∗
da
SLIDE 9
a∗ ?
≡ ε+ a· a∗ for Σ = {a,b}
a∗
ε+ a· a∗ ε· a∗ ∅+ε· a∗ ∅· a∗ ∅+∅· a∗
da db
SLIDE 10
a∗ ?
≡ ε+ a· a∗ for Σ = {a,b}
a∗
ε+ a· a∗ ε· a∗ ∅+ε· a∗ ∅· a∗ +ε· a∗ ∅+∅· a∗ +ε· a∗ ∅· a∗ ∅+∅· a∗
da da db
SLIDE 11
a∗ ?
≡ ε+ a· a∗ for Σ = {a,b}
a∗
ε+ a· a∗ ε· a∗ ∅+ε· a∗ ∅· a∗ +ε· a∗ ∅+∅· a∗ +ε· a∗ ∅· a∗ +∅· a∗ +ε· a∗ ∅+∅· a∗ +∅· a∗ +ε· a∗ ∅· a∗ ∅+∅· a∗
da da da db
SLIDE 12
a∗ ?
≡ ε+ a· a∗ for Σ = {a,b}
a∗
ε+ a· a∗ ε· a∗ ∅+ε· a∗ ∅· a∗ +ε· a∗ ∅+∅· a∗ +ε· a∗ ∅· a∗ +∅· a∗ +ε· a∗ ∅+∅· a∗ +∅· a∗ +ε· a∗ ∅· a∗ ∅+∅· a∗
da da da ACI db
SLIDE 13
a∗ ?
≡ ε+ a· a∗ for Σ = {a,b}
a∗
ε+ a· a∗ ε· a∗ ∅+ε· a∗ ∅· a∗ +ε· a∗ ∅+∅· a∗ +ε· a∗ ∅· a∗ +∅· a∗ +ε· a∗ ∅+∅· a∗ +∅· a∗ +ε· a∗ ∅· a∗ ∅+∅· a∗ ∅· a∗ +∅· a∗ ∅+∅· a∗ +∅· a∗ ∅· a∗ +∅· a∗ +∅· a∗ ∅+∅· a∗ +∅· a∗ +∅· a∗
da da da ACI db db db ACI ACI da db
SLIDE 14
Key ingredients: Derivative + Acceptance Test
SLIDE 15
Key ingredients: Derivative + Acceptance Test Let’s define them on WS1S formulas directly!
SLIDE 16
(∃X.x ∈ X)
?
≡ (¬x < x) for Σ = {(T),(F)}
∃X.x ∈ X ¬x < x ∃X.(T∨ F) ¬F ∃X.(T∨ F)∨(T∨ F) ¬F ∃X.(x ∈ X ∨ x ∈ X) ¬ x < x
d(T) d(T), d(F) ACI d(F) ACI
SLIDE 17
Derivative
d v T
=
T d v F
=
F d v (x ∈ X)
=
x ∈ X if ¬v[x] T if v[x]∧ v[X] F
- therwise
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 (vX→T) ϕ∨ d (vX→F) ϕ)
SLIDE 18
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
SLIDE 19
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
SLIDE 20
Altogether
A decision procedure for WS1S that
SLIDE 21
Altogether
A decision procedure for WS1S that
- perates on formulas directly and
SLIDE 22
Altogether
A decision procedure for WS1S that
- perates on formulas directly and
is verified in
λ → ∀
=
Isabelle
β α
HOL
and
SLIDE 23
Altogether
A decision procedure for WS1S that
- perates on formulas directly and
is verified in
λ → ∀
=
Isabelle
β α
HOL
and
- utperforms MONA on very well selected examples.
SLIDE 24
Altogether
A decision procedure for WS1S that
- perates on formulas directly and
is verified in
λ → ∀
=
Isabelle
β α
HOL
and
- utperforms MONA on very well selected examples.
- Thanks. Questions?
SLIDE 25
Derivatives of WS1S Formulas
Dmitriy Traytel
λ → ∀
=
Isabelle
β α