A [Monad-Based] Semantics for Hybrid Iteration

Transcription

1 A [Monad-Based] Semantics for Hybrid Iteration Sergey Goncharov a Julian Jakob a Renato Neves b CONCUR 2018, September 4-7, Beijing a Friedrich-Alexander-Universität Erlangen-Nürnberg b INESC TEC (HASLab) & University of Minho 1/15

2 Plan of the Talk Precursors: Hybrid automata Platzer s dynamic logic Höfner and Möller s algebra of hybrid systems Hybrid process calculi: ACP srt hs, hybrid χ, HYPE, HyPA, hybrid CSP,... More recently: hybrid monad : H 0 (deterministic, abstract, composable) Here: two new monads H` and H for hybrid iteration : Neves, Barbosa, Hofmann, and Martins 2016, Continuity as a computational effect 2/15

3 Quick Intro: Bouncing Ball and Zeno Behaviour Bouncing ball is a simple Newtonian system specified by differential equation : h g (g «9.8) whose solution is hptq h 0 ` v 0 t gt2 2 1 with initial values: v 0 0, h 0 0 (peak height) h 0 0, v 0 0 (zero height) The velocity changes discretely at the bottom v ÞÑ cv, but it changes continuously in the meanwhile. This is called hybrid behaviour The state of rest is only reachable in the limit. This is called Zeno behaviour 3/15

4 Progressive Iteration vs. Singular Iteration Bouncing ball movement is progressive in the sense that each iteration is non-instantaneous On the other hand while true tx : 1 xu is not progressive and must yield the divergence K This disrupts the time flow structure, for we expect while true tx : 1 xu» while true tx : 1 xu; waitp1q Idea: progressiveness is a form of abstract guardedness : : Goncharov and Schröder 2018, Guarded Traced Categories 4/15

5 Monads in Semantics Monads formalize generalized functions : f : X Ñ TY, like nondeterministic (with T PX ) or partial (with TX X ` 1) T is a type constructor, together with operations η : X Ñ TX (unit) and pf : X Ñ TY q ÞÑ pf : TX Ñ TY q (lifting), inducing the Klesili category of T under id η : X Ñ TX f g pf : Y Ñ TZq pg : X Ñ TY q In Haskell s point-full notation: do x Ð p; f pxq f ppq : Moggi 1991, Notions of Computation and Monads 5/15

6 Abstract Guardedness on Monads Abstract guardedeness for a monad T is a relation between Kleisi morphisms f : X Ñ TY and summands σ : Y 1 Y satisfying (trv) (sum) (cmp) f : X Ñ TY pt in 1 q f : X Ñ in2 T py ` Zq f : X Ñ σ TZ g : Y Ñ σ TZ rf, gs : X ` Y Ñ σ TZ f : X Ñ in2 T py ` Zq g : Y Ñ σ TV h : Z Ñ TV rg, hs f : X Ñ σ TV where f : X Ñ σ TY means that f and σ are in the relation 6/15

7 Abstract Guardedness on Monads A monad is guarded Elgot if it supports partial iteration sending each f : X Ñ 2 T py ` X q to f : : X Ñ TY satisfying the fixpoint law and other laws of iteration f : rη, f : s f This yields 2-dimensional classification: total partial iterative (unique -- : ) degenerate, e.g. T 1 completely iterative, e.g. T νγ. P ω p-- `A ˆ γq Elgot (non-unique -- : ) e.g. T P e.g. T PpA ˆ -- `A ω q 7/15

8 Comparing Guarded and Unguarded Iteration We identify totally guarded iteration with unguarded iteration T P: this supports unguarded iteration via least fixpoints T P `, i.e. non-empty powerset: the iteration operator is inherited from P along injection P ` ãñ P The resulting operator is partial, for η inr : 1 Ñ P `p1 ` 1q does not have a fixpoint This operator is guarded with f : X Ñ 2 P `py ` Zq Dy. inl y P f pxq 8/15

9 Basic Monad for Hybrid Computations

10 The Monad H 0 1. A closed trajectory t is a pair pt d P R`, t e : r0, t d s Ñ X q 2. An infinite trajectory t is a pair p8, t e : r0, 8q Ñ X q 3. More uniformly, t P R` ˆ X R` + flattening condition: x ě t d implies t e pxq t e pt d q Let H 0 X be the set of trajectories identified by (3). 9/15

11 The Monad H 0 1. A closed trajectory t is a pair pt d P R`, t e : r0, t d s Ñ X q 2. An infinite trajectory t is a pair p8, t e : r0, 8q Ñ X q 3. More uniformly, t P R` ˆ X R` + flattening condition: x ě t d implies t e pxq t e pt d q Let H 0 X be the set of trajectories identified by (3). H 0 is a monad! : Unit: ηpxq p0, xq, where x is the constant function λ. x; Kleisli lifting: given f : X Ñ H 0 Y and pd, eq P H 0 X, pf pd, eqq d d ` f d pe d q d P R` 8 pf pd, eqq t e fe 0 pe t q t ă d fe t d pe d q where x t reads as xptq and x y z as if y then x else z : Neves, Barbosa, Hofmann, and Martins 2016, Continuity as a computational effect 9/15

12 Why H 0 Does Not Suffice? Closure under iteration ñ closure under Zeno behaviour 10/15

13 Why H 0 Does Not Suffice? Closure under iteration ñ closure under Zeno behaviour However, 1. Zeno argued that it is impossible to reach the limit;) So, it is impossible to close the trajectory even if there is a unique closed trajectory candidate 10/15

14 Why H 0 Does Not Suffice? Closure under iteration ñ closure under Zeno behaviour However, 1. Zeno argued that it is impossible to reach the limit;) So, it is impossible to close the trajectory even if there is a unique closed trajectory 1 candidate 2. It is certainly possible to have hybrid behaviour without a closed trajectory candidate whatsoever Solution: open trajectories /15

15 Iterating Hybrid Computations

16 Partial Trajectories Let M be the maybe monad: MX X ` 1 Proposition: Every monad T pt, η, p--q q induces a monad TM whose underlying functor is X ÞÑ T px ` 1q Theorem: The hom-sets HompX, TMq can be suitably ordered, so that 1. Every H 0 MX is an ω-complete partial order under Ď, and p0, Kq is the bottom element 2. Kleisli composition is monotone and continuous on both sides 3. Kleisli composition is right-strict: f p0, Kq p0, Kq However, Kleisli composition is not left-strict: p0, Kq p1, idq p1, Kq p0, Kq! Theorem: By (1) (3), H 0 M is a (total) Elgot monad 11/15

17 Open Trajectories H 0 M contains lots of junk trajectories, while we only need open and closed ones: pd P R`, e : r0, dq Ñ X q and pd P R`, e : r0, ds Ñ X q Let H` be the following subfunctor of H 0 M: pd, eq P H`X iff e K and e t Ó for all t P r0, dq This induces a monad H` with pf pd, eqq d d, pf pd, eqq t e f 0 e pe t q t ă d K if e d Ò pf pd, eqq d d ` f d pe d q, pf pd, eqq t e f 0 e pe t q t ă d f t d e pe d q if e d Ó Total iteration of H 0 M restricts to guarded iteration of H` with guardedness being progressiveness: pd, eq : X Ñ 2 H`pY ` Zq if e 0 : X Ñ Y ` Z factors through inl 12/15

18 Full Hybrid Monad H In H` we excluded the black hole trajectory p0, Kq, for it brings up space-time artefacts like p1, Kq Let H be the following subfunctor of H 0 M: pd, eq P HX iff e K and e t Ó for all t P r0, dq Now, the induced injection υ : H ãñ H 0 M is not a monad morphism: p0, Kq p1, idq must be p0, Kq in H, but not in H 0 M However, υ has a retraction ρ : H 0 M Ñ H, which is a monad morphism. Moreover pυ, ρq is an iteration-congruent retraction : Theorem: H is an iteration-congruent retract of H 0 M, hence an Elgot monad : Goncharov, Schröder, Rauch, and Piróg 2017, Unifying Guarded and Unguarded Iteration 13/15

19 Big Picture H 0 M ι υ H` ρι ρ H H` is a submonad of H via ρι progressive (partial) iteration on H` is a restriction of total iteration on H (analogously to P ` vs. P) 14/15

20 Further Work What is a topological semantic domain for hybrid computations? Designing a hybrid programming language, relating to the metalanguage for guarded iteration : Program logics for hybrid iteration, apposing Platzer s differential dynamic logic Process semantics via generalized coalgebraic resumption monad transform ; νγ. Hp-- `A ˆ γq Guarded Lawvere theories, guarded PROPs and applications to H 0, H`, H : Goncharov, Rauch, and Schröder 2018, A Metalanguage for Guarded Iteration ; Goncharov, Rauch, and Schröder 2015, Unguarded Recursion on Coinductive Resumptions 15/15

21 References I References Sergey Goncharov and Lutz Schröder. Guarded traced categories. In Christel Baier and Ugo Dal Lago, editors, Proc. 21th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS 2018). Springer, Sergey Goncharov, Christoph Rauch, and Lutz Schröder. Unguarded recursion on coinductive resumptions. In Mathematical Foundations of Programming Semantics, MFPS 2015, /15

22 References II Sergey Goncharov, Lutz Schröder, Christoph Rauch, and Maciej Piróg. Unifying guarded and unguarded iteration. In Javier Esparza and Andrzej Murawski, editors, Foundations of Software Science and Computation Structures, FoSSaCS 2017, volume 10203, pages Springer, Sergey Goncharov, Christoph Rauch, and Lutz Schröder. A metalanguage for guarded iteration. In Bernd Fischer and Tarmo Uustalu, editors, 15th International Colloquium on Theoretical Aspects of Computing (ICTAC 2018), Eugenio Moggi. Notions of computation and monads. Inf. Comput., 93: 55 92, Renato Neves, Luis S. Barbosa, Dirk Hofmann, and Manuel A. Martins. Continuity as a computational effect. Journal of Logical and Algebraic Methods in Programming, 85(5, Part 2): , Articles dedicated to Prof. J. N. Oliveira on the occasion of his 60th birthday. 14/15

23 Simple Iteration and Instability Hybrid iteration occurring in the literature is a Kleene-style iteration: it is supposed to send f : X Ñ H`X to f # : X Ñ H`X (e.g. this is the case with bouncing ball, because there is no final result ) We recover such basic iteration pd, eq # : X Ñ H`X as a progressive one: : pd, λx. λt. inl e 0 pxq t 0 inr e t pxqq : X Ñ 2 H`pX ` X q Unlike H 0, we cannot move H` (even more so H) from Set to Top because of instability: given continuous pd, eq : R` Ñ H`R` with dpxq x, e t pxq x, pd, eq # d p0q 0, pd, eq# d pɛq 8 for any ɛ ą 0 15/15