Basold, H. (Henning)

Hansen, H.H. (Helle)

Rutten, J.J.M.M. (Jan)

English

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A Note on TypedBehavioural Differential EquationsHenning Basold1,2, Helle H. Hansen1,2, Jan Rutten2,11 Radboud University Nijmegen2 CWI Amsterdam“One man’s observation is another man’s closed book or flight of fancy.”— Willard Van Orman QuineThe aim of this short note is two-fold. Our main motivation is to widen theclass of maps being definable by Behavioural Differential Equations (BDE) [8].One way of giving semantics to BDE’s is by using Structural Operational Se-mantics [7], but this prevents mixing induction and coinduction. Moreover, theensuing syntactic formats have limited expressiveness, and exclude useful func-tions from being defined. These issues can be dealt with if we add type systems.The second aim of this note is to advertise an observational approach to in-finite structures in Type Theory. For example in Coq, coinductive types are in-troduced by “constructors, albeit with the power of being infinitely iterated” [3].There is work on extending type systems with (weakly) final coalgebras (e.g. [6]),but only recently efforts have been made to bring this into theorem provers [2,1].The use of copatterns in loc. cit. is a way of extending λ-calculus by BDE’s. Wewant to study which maps one can define in this calculus and under which cir-cumstances it leads to final coalgebras.First of all, we search for the ingredients necessary to obtain final coalgebras.To this end, we organise the types and terms defined in [1] into a category T.Using a fixed set TyVar of type variables, the syntax of our basic types is:A,B ::= 1 | X ∈ TyVar | A×B | A+B | A→ B | µX.A | νX.AFor example, natural numbers and streams are defined by N := µX.(1 +X) andStrA := νX.(A × X), respectively. More interesting may be the streams thathave infinitely many entries of type A: Inf A B := νX.µY.(A×X +B × Y ).For space reasons we do not define the terms here, but rather give a proto-typical example of a definition using copatterns in Fig. 1. One should see StrAas final coalgebra for the functor F = A × Id with the observation .out beingthe transition map. Thus for σ : StrA we have σ.out : F (StrA). The head andtail of σ are obtained as σ.out.pi1 : A and σ.out.pi2 : StrA. The copatterns onthe left-hand of 7→ in Fig. 1 denote the head and tail of even(σ), respectively.We construct a category T with closed types as objects and closed terms oftype A→ B as arrows. The composition and identity arrows are defined in theobvious way, but the axioms of a category are a little trickier. We would like to useas equality the convertibility relation ≡ induced by the reduction relation definedin [2]. Unfortunately, this reduction relation only contracts function application,and taking ≡ as equality on terms yields neither a category, nor final coalgebras.even : StrA→ StrA = {even σ .out.pi1 7→ σ.out.pi1;even σ .out.pi2 7→ even (σ.out.pi2.out.pi2)}Fig. 1. Even positions of a streamWe fix these problems in onego, by introducing a notion of ob-servational equivalence, which isinspired by the likewise namednotion from λ-calculus. An obser-vation for a type A is a closedterm O : A → B for any type B, whose structure depends on A. For ex-ample, in the case of streams, O : StrA → B is either constant or observeshead and tail. Two terms t1, t2 : A are observationally equivalent, t1 ≡obs t2,if Ot1 ≡ Ot2 for all observations O : A → B. If we rule out non-normalisingterms, for example by ensuring termination/productivity using the sized typesin [1], we arrive at the desired result: sized types as objects and closed termsmodulo ≡obs as arrows form a category T. It has finite (co)products, and forevery type A with a free variable X, there is a functor FA : T → T given bysubstitution: FA(B) = A[B/X]. Whether all these functors have initial algebrasand final coalgebras is work in progress (it is true in the case of streams though).Next we study which maps can actually be defined. In the spirit of Brouwer’sschool of intuitionism (e.g. [4,5]), we consider here continuity as basic property.First we define a functor T : T→ Set by sending a type A to the set T (A) of itsinhabiting terms modulo observational equivalence. A term f : A → B inducesa map T (f) : T (A) → T (B) by application. The sets T (A) carry a naturaltopology generated by the subbase consisting of all UO(t) = {t′ : A | Ot ≡ Ot′}where t : A is a term and O : A → B an observation. One can show that allT (f) are continuous with respect to the above topology. Finally, for example forstreams, we find that the topology is the expected one: the space T (StrA) ishomeomorphic to a subspace of T (A)ω (product topology), provided that T (A)is discrete (e.g. if A = N).References1. A. Abel and B. Pientka. Wellfounded recursion with copatterns: a unified approachto termination and productivity. In ICFP, pages 185–196, 2013.2. A. Abel, B. Pientka, D. Thibodeau, and A. Setzer. Copatterns: Programming infinitestructures by observations. In Proceedings POPL ’13, page 27–38. ACM, 2013.3. V. Capretta. Coalgebras in functional programming and type theory. TCS,412(38):5043–5069, 2011.4. M. P. Fourman. Notions of choice sequence. In D. v. Dalen and A. Troelstra, editors,L.E.J. Brouwer Centenary Symposium, page 91–105. North-Holland, 1982.5. N. Ghani, P. Hancock, and D. Pattinson. Continuous functions on final coalgebras.ENTCS, 249:3–18, 2009.6. P. Hancock and A. Setzer. Interactive Programs and Weakly Final Coalgebras inDependent Type Theory, pages 115–134. Number 48 in Oxford Logic Guides. OxfordUniversity Press, 2005.7. B. Klin. Bialgebras for structural operational semantics: An introduction. TCS,412(38):5043–5069, 2011.8. J. J. M. M. Rutten. Behavioural differential equations: a coinductive calculus ofstreams, automata, and power series. TCS, 308(1-3):1–53, 2003.

A note on typed behavioural differential equations

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