Total (Co)Programming with Guarded Recursion
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1 Total (Co)Programming with Guarded Recursion Andrea Vezzosi Department of Computer Science and Engineering Chalmers University of Technology, Gothenburg, Sweden Types for Proofs and Programs Annual Meeting 2015 Tallinn, Estonia 18 May 2015 Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden [1ex] 1 / 26 )
2 Guarded Recursion Guarded coinductive types Coinductive types Guarded fixed point operator as only source of recursion Recursive types as fixed points on the universe What about Induction? Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden [1ex] 2 / 26 )
3 Main Combinators A, later A, modality as an applicative functor: next : A A : (A B) A B Guarded fixpoint combinator: fix : ( A A) A fix f = f (next (fix f )) Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden [1ex] 3 / 26 )
4 Corecursion Example gstr A = A gstr A ghead : gstr A A ghead = fst gtail : gstr A gstr A gtail = snd map : (A B) gstr A gstr B map f = fix (λ map. λ xs. ghead xs, map gtail xs) Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden [1ex] 4 / 26 )
5 Recursion Example? glist A = + A glist A all : (A Bool) glist A Bool all p = fix (λ (all : (glist A Bool)). λ xs. case xs of [ ] True (x :: xs) p x? We need a way to call all with xs as argument and obtain Bool. Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden [1ex] 5 / 26 )
6 Recursion Example, take 2, with diamonds glist A = + A glist A extract : Bool Bool : (A B) A B all : (A Bool) glist A Bool all p = fix (λ (all : (glist A Bool)). λ xs. case xs of [ ] True (x :: xs) p x extract (all xs) Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden [1ex] 6 / 26 )
7 Problem: we lose next For A we cannot have next, e.g.: next : ( ) ( ) means if there is time left now, there will be time left later too Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden [1ex] 7 / 26 )
8 Semantics The standard model for Guarded Recursion is the topos of trees i.e. functors ω op Set A : N Set A (n m) : A m A n ( A) 0 = ( A) (suc n) = A n next 0 =! next suc n = A (n suc n) next uses the functoriality of A Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden [1ex] 8 / 26 )
9 Alternative Semantics: Relators A : N Set A (n m) : A m A n Set A (n n) = = A n Any functor A : ω op Set is also a relator: A (n m) a n a m = a n = A n A (n m) a m Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden [1ex] 9 / 26 )
10 ala Sized Types, : (Time Set) (Time Set) A i = j < i. A j A i = j < i. A j : i. ( j < i. A j B j ) ( j < i. A j ) j < i. B j f (j, a) = (j, f j a) Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 10 [1ex] / 26 )
11 ala Sized Types (contd.) fix : ( i. ( j < i. A j ) A i) i. A i unfold : ( i. S i + (A j < i. S j )) i. S i List A unfold f = fix λ i unfold s. case f i s of inl [ ] inr (a, (j, s )) a :: unfold j s Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 11 [1ex] / 26 )
12 Recursive Types through fixed points ˆ : U U gstr A = fix λ X. A ˆ X gstr A = fix λ i (X : j < i. U ). A j < i. X j Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 12 [1ex] / 26 )
13 Coinductive Types with gstr κ A = A κ gstr κ A Str A = κ. gstr κ A force : ( κ. κ A) = ( κ. A) tail : Str A Str A tail xs = force (λ κ. gtail (xs κ)) Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 13 [1ex] / 26 )
14 Coinductive Types with j < i gstr A i = A j < i. gstr A j Str A = i. gstr A i force : ( i. j < i. A j ) i. A i force f i = f (suc i) i guard : ( i. A i) i. j < i. A j guard f i j = f j guard (force f ) i j = f (suc j ) j Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 14 [1ex] / 26 )
15 Inductive Types with j < i gnat i = + j < i. A j Nat = i. gnat i force : ( i. j < i. A j ) i. A i force (i, j, a) = (j, a) guard : ( i. A i) i. j < i. A j guard (j, a) = (suc j, j, a) guard (force (i, j, a)) = suc j, j, a Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 15 [1ex] / 26 )
16 i as a weak existential gnat i = + j < i. A j Nat = i. gnat i Want all zeros to be equal: (i, inl tt) = (j, inl tt) We cannot project times out: fst : ( i. A i) Time fst (i, a) = i i = fst (i, inl tt) = fst (j, inl tt) = j Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 16 [1ex] / 26 )
17 i as a weak existential P : ( i. A i) U f : ( i. (a : A i) P (i, a)) uncurry f : (x : i. A i) P x where U is a type theoretic universe such that Time / U Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 17 [1ex] / 26 )
18 Summary Ordered type Time : Type which supports well-founded induction A universe U : Type such that Time / U Parametric time quantifiers i. A i and i. A i Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 18 [1ex] / 26 )
19 Reflexive Graph Model of Martin Löf Type Theory Γ O : Set Γ R : Γ O Γ O Set Γ refl : (γ O : Γ O ) Γ R γ O γ O Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 19 [1ex] / 26 )
20 Time Time O = N Time R i j = Any two time values are related. Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 20 [1ex] / 26 )
21 Types depending on Time i : Time A : Type A O : N Set A R : (n m : N) A n A m Set A refl : (n : N) (a : A n) A R n n a a A R n n =? = AO Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 21 [1ex] / 26 )
22 Universe of Small Discrete Reflexive Graphs U O = {(A O, A R ) A O small set, A R = eqao } U R A B = {Rel Rel small proof irrelevant relation between A O and B O } U refl (A O, A R ) = A R Γ A : U Γ El A : Type (El A) R (Γ refl γ) = = ElAO Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 22 [1ex] / 26 )
23 Time dependency for discrete reflexive graphs Given A such that i / fv A i : Time t : El A t O : N (El A) O t R : (n m : N) t O n = ElAO t O m i. j < i. El (A j ) = i. El (A i) Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 23 [1ex] / 26 )
24 Discretization Given any small reflxive graph A we can form its free discrete reflexive graph A : U (El ( A)) O = A O / symmetric transitive closure of A R (El ( A)) R = =( A)O Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 24 [1ex] / 26 )
25 Discretization, Universal property (El ( A) El B) = (A El B) P : El ( A) U f : (a : A) P ( A)) El (P x) elim f : (x : El ( a) i. A i = (Σ (i : Time). A i) Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 25 [1ex] / 26 )
26 Future work How to internalize the parametricity properties of i and i? Very interested in the talks about parametricity in the following days! Cohesive Homotopy Type Theory has something like discretization How to preserve strong normalization? fix f i = f i (fix f ) = f i (λ j. fix f j ) = f i (λ j. f j (fix f )) =... Andrea Vezzosi ( Department of Computer Science Total (Co)Programming and Engineering Chalmers with Guarded University Recursion of Technology, Gothenburg, TYPES 2015 Sweden 26 [1ex] / 26 )
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