Formal Languages aa bb baab ba ab Formally Coinductively - PowerPoint PPT Presentation

Formal Languages aa bb baab ··· ε ba ab Formally Coinductively and a b a b a b a b a b a b a b . . . . . . . . . . . . Dmitriy Traytel

Contribution Isabelle H O L ∀ = library of formal languages in α λ β →

Contribution define regular operations ∅ , ε , Atom, + , · , ∗ prove axioms of Kleene Algebra Isabelle H O L ∀ = library of formal languages in α λ β →

Contribution define regular operations ∅ , ε , Atom, + , · , ∗ prove axioms of Kleene Algebra Isabelle H O L ∀ = Coinductive library of formal languages in α λ β →

Contribution define regular operations ∅ , ε , Atom, + , · , ∗ prove axioms of Kleene Algebra Isabelle H O L ∀ = Coinductive library of formal languages in α λ β → Tutorial for corecursion and coinduction

Contribution define regular operations ∅ , ε , Atom, + , · , ∗ prove axioms of Kleene Algebra Isabelle H O L ∀ = Coinductive library of formal languages in α λ β → Formal Structure Tutorial for corecursion and coinduction Computation Deduction

Contribution define regular operations ∅ , ε , Atom, + , · , ∗ prove axioms of Kleene Algebra Isabelle H O L ∀ = Coinductive library of formal languages in α This talk: Tutorial in 20 min λ β → Formal Structure Tutorial for corecursion and coinduction Computation Deduction

Related Work: A Selection of CoTutorials t n n n e a o p o t i s y i s t i t c s r a u u s t a c d A e n d f r i o o o o C C o C r P Jacobs, Rutten EATCS’97 stream Rutten CONCUR’98 language Giménez, Castéran ’98 stream, lazy list Rutten MSCS ’05 stream Hinze JFP ’11 stream Chlipala ’13 stream, while Rot, Rutten, Bonsangue language LATA’13 Kozen, Silva MSCS ’14 stream Setzer Festschrift Jäger’16 stream e l H O e l L b Traytel FSCD’16 language a ∀ s I = α λ β →

Codatatype Corecursion Coinduction

Codatatype Corecursion Coinduction

a b a b a b a b a b a b a b . . . . . . . . . . . .

lang = L bool codatatype lang lang a b a b a b a b a b a b a b . . . . . . . . . . . .

codatatype α lang = L bool ( α ⇒ α lang ) a b a b a b a b a b a b a b . . . . . . . . . . . .

codatatype α lang = L bool ( α ⇒ α lang ) a b a b a b a b a b a b a b . . . . . . . . . . . .

codatatype α lang = L bool ( α ⇒ α lang ) a b a b a b a b a b a b a b . . . . . . . . . . . .

codatatype α lang = L bool ( α ⇒ α lang ) o :: α lang ⇒ bool δ :: α lang ⇒ α ⇒ α lang a b a b a b a b a b a b a b . . . . . . . . . . . .

primrec ∈ ∈ :: α list ⇒ α lang ⇒ bool [] ∈ ∈ L = o L aw ∈ ∈ L = w ∈ ∈ δ L a a b a b a b a b a b a b a b . . . . . . . . . . . .

primrec ∈ ∈ :: α list ⇒ α lang ⇒ bool [] ∈ ∈ L = o L aw ∈ ∈ L = w ∈ ∈ δ L a aa ∈ ∈ a b a b a b a b a b a b a b . . . . . . . . . . . .

primrec ∈ ∈ :: α list ⇒ α lang ⇒ bool [] ∈ ∈ L = o L aw ∈ ∈ L = w ∈ ∈ δ L a a b a ∈ ∈ a b a b a b a b a b a b . . . . . . . . . . . .

primrec ∈ ∈ :: α list ⇒ α lang ⇒ bool [] ∈ ∈ L = o L aw ∈ ∈ L = w ∈ ∈ δ L a a b a b a b [] ∈ ∈ a b a b a b a b . . . . . . . . . . . .

primrec ∈ ∈ :: α list ⇒ α lang ⇒ bool [] ∈ ∈ L = o L aw ∈ ∈ L = w ∈ ∈ δ L a a b a b a b a b a b a b a b . . . . . . . . . . . .

Codatatype Corecursion Coinduction

primcorec ∅ :: α lang o ∅ = δ ∅ = λ _ . ∅

primcorec ∅ :: α lang o ∅ = δ ∅ = λ _ . ∅ a b a b a b . . . . . .

primcorec ∅ :: α lang primcorec ε :: α lang o ∅ = o ε = δ ∅ = λ _ . ∅ δ ε = λ _ . ∅ a b a b a b a b a b a b . . . . . . . . . . . .

primcorec ∅ :: α lang primcorec ε :: α lang o ∅ = o ε = δ ∅ = λ _ . ∅ δ ε = λ _ . ∅ a b a b a b a b a b a b . . . . . . . . . . . . primcorec Atom :: α ⇒ α lang o ( Atom a ) = δ ( Atom a ) = λ b . if a = b then ε else ∅

primcorec ∅ :: α lang primcorec ε :: α lang o ∅ = o ε = δ ∅ = λ _ . ∅ δ ε = λ _ . ∅ a b a b a b a b a b a b . . . . . . . . . . . . primcorec Atom :: α ⇒ α lang o ( Atom a ) = δ ( Atom a ) = λ b . if a = b then ε else ∅ a b a b a b a b a b a b . . . . . . . . . . . .

primrec primcorec Syntactic criterion for termination productivity α list ⇒ ··· ··· ⇒ α lang of functions of type

primrec primcorec Syntactic criterion for termination productivity α list ⇒ ··· ··· ⇒ α lang of functions of type Philosophy consume 1 pattern match argument

primrec primcorec Syntactic criterion for termination productivity α list ⇒ ··· ··· ⇒ α lang of functions of type Philosophy consume 1 produce 1 pattern match argument copattern match output

primrec primcorec Syntactic criterion for termination productivity α list ⇒ ··· ··· ⇒ α lang of functions of type Philosophy consume 1 produce 1 pattern match argument copattern match output (Co)recursive call arguments very restricted context arbitrary

primrec primcorec Syntactic criterion for termination productivity α list ⇒ ··· ··· ⇒ α lang of functions of type Philosophy consume 1 produce 1 pattern match argument copattern match output (Co)recursive call arguments very restricted arbitrary context arbitrary very restricted

primrec primcorec Syntactic criterion for termination productivity α list ⇒ ··· ··· ⇒ α lang of functions of type Philosophy consume 1 produce 1 pattern match argument copattern match output (Co)recursive call arguments very restricted arbitrary context arbitrary very restricted a b a b a b a b a b a b . . . . . . . . . . . . Atom a Atom b

primrec primcorec Syntactic criterion for termination productivity α list ⇒ ··· ··· ⇒ α lang of functions of type Philosophy consume 1 produce 1 pattern match argument copattern match output (Co)recursive call arguments very restricted arbitrary context arbitrary very restricted a b a b a b = + a b a b a b a b a b a b . . . . . . . . . . . . . . . . . . Atom a Atom b

primrec primcorec Syntactic criterion for termination productivity α list ⇒ ··· ··· ⇒ α lang of functions of type Philosophy consume 1 produce 1 pattern match argument copattern match output (Co)recursive call arguments very restricted arbitrary context arbitrary very restricted primcorec + :: α lang ⇒ α lang ⇒ α lang o ( L + K ) = o L ∨ o K δ ( L + K ) = λ a . δ L a + δ K a a b a b a b = + a b a b a b a b a b a b . . . . . . . . . . . . . . . . . . Atom a Atom b

o ∅ = o ε = δ ∅ = λ _ . ∅ δ ε = λ _ . ∅ o ( Atom a ) = δ ( Atom a ) = λ b . if a = b then ε else ∅ o ( L + K ) = o L ∨ o K δ ( L + K ) = λ a . δ L a + δ K a

 L , K :: α lang = o ∅    = o ε L + ∅ = L     o ( Atom a ) =  nullability o ( L + K ) = o L ∨ o K         = λ _ . ∅  δ ∅   = λ _ . ∅ δ ε     δ ( Atom a ) = λ b . if a = b then ε else ∅ Brzozowski  δ ( L + K ) = λ a . δ L a + δ K a derivative       

L , K :: α regex = o ∅ = o ε L + ∅ � = L o ( Atom a ) = o ( L + K ) = o L ∨ o K = λ _ . ∅ d ∅ = λ _ . ∅ d ε d ( Atom a ) = λ b . if a = b then ε else ∅ d ( L + K ) = λ a . d L a + d K a

L , K :: α regex = o ∅ = o ε L + ∅ � = L o ( Atom a ) = o ( L + K ) = o L ∨ o K o ( L · K ) = o L ∧ o K o ( L ∗ ) = = λ _ . ∅ d ∅ = λ _ . ∅ d ε d ( Atom a ) = λ b . if a = b then ε else ∅ d ( L + K ) = λ a . d L a + d K a d ( L · K ) = ... d ( L ∗ ) = ...

 L , K :: α regex = o ∅    = o ε L + ∅ � = L     o ( Atom a ) =  nullability o ( L + K ) = o L ∨ o K   o ( L · K ) = o L ∧ o K     o ( L ∗ )  =  = λ _ . ∅  d ∅   = λ _ . ∅ d ε     d ( Atom a ) = λ b . if a = b then ε else ∅ Brzozowski  d ( L + K ) = λ a . d L a + d K a derivative   d ( L · K ) = ...     d ( L ∗ ) = ... 

L , K :: α regex L + ∅ � = L o ( L · K ) = o L ∧ o K d ( L · K ) = ...

primcorec · :: α lang ⇒ α lang ⇒ α lang o ( L · K ) = o L ∧ o K δ ( L · K ) = λ a . if o L then δ L a · K + δ K a else δ L a · K

Nonprimitively corecursive specification ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ · :: α lang ⇒ α lang ⇒ α lang primcorec o ( L · K ) = o L ∧ o K δ ( L · K ) = λ a . if o L then δ L a · K + δ K a else δ L a · K