Synchronous Machines: a Traced Category

Transcription

1 Synchronous Machines: a Traced Cateory Marc anol, Guatto drien To cite this version: Marc anol, Guatto drien. Synchronous Machines: a Traced Cateory. [Research Report] <hal v3> HL Id: hal Submitted on 27 Nov 2012 HL is a multi-disciplinary open access archive or the deposit and dissemination o scientiic research documents, whether they are published or not. The documents may come rom teachin and research institutions in France or abroad, or rom public or private research centers. L archive ouverte pluridisciplinaire HL, est destinée au dépôt et à la diusion de documents scientiiques de niveau recherche, publiés ou non, émanant des établissements d enseinement et de recherche rançais ou étraners, des laboratoires publics ou privés.

2 Synchronous Machines: a Traced Cateory Marc anol & drien Guatto Institut de Mathématiques de Luminy, ix-marseille Université Département d Inormatique de l École normale supérieure and INRI Rocquencourt bstract. Synchronous prorammin lanuaes have been extensively used in the area o critical embedded systems. Synchronous machines, a speciic class o labelled transition systems, are oten used to ive denotational semantics o these lanuaes. In this work, we study the cateorical structure o the aorementioned machines. We irst show that the cateory S o synchronous machines can be iven a traced symmetric monoidal structure with diaonals. Then, we apply a standard variant o the Int construction to S and relate the composition in the resultin cateory with the synchronous product, the operation used to model parallel composition o synchronous prorams. We also show how properties o synchronous machines like determinism and reactivity relate to the way they compose with diaonal morphisms o S. Keywords: Denotational semantics; Traced cateories; Synchronous lanuaes. 1 Introduction Synchronous prorammin lanuaes [18] are domain-speciic lanuaes dedicated to the desin and implementation o critical real-time systems. ased on a stron yet abstract notion o time, they have met success and industrial use. Compared to most practically used prorammin lanuaes, their semantics tend to be better understood and more ormal. In both computer science and loics, the denotational semantics approach has helped in clariyin the structure o prorammin lanuaes and loical systems, by means o mathematical tools such as cateory theory. For instance linear loics [3] comes rom the study o a ine-rained denotational model o system F [19]. The traced monoidal cateories [20] have been studied in the area o denotational semantics to model reasonable eedback situations. The G construction [17] allows to build out o a traced monoidal cateory a new cateory in which the composition can be thinked o intuitively as a kind o parallel eedback loop between the composed morphisms. This author was supported by the ence Nationale de la Recherche project NR- 10-LN-0213 This author was unded by the European FP7 project PHRON id

3 utomata theory orms the backbone o synchronous prorammin. More speciically, parallel composition mirrors the synchronous product o automata: two interactin processes share the same notion o loical time and must synchronize their reaction steps. This operation also provides a natural notion o mutual eedback between concurrent processes, which comes in handy or prorammin real-time systems. For example, one may use it to implement the interaction between the model o a physical environment and its sotware controller. It is then quite natural to wonder i the automata-based models o synchronous prorammin lanuaes satisy the axioms o traced monoidal cateories. Even i these models were introduced without cateories in mind, their parallel composition operator should be related somehow to the parallel composition introduced by the G construction. In this work, we irst show that the synchronous machines, the kind o automata used in these models, orm a traced monoidal cateory S with diaonals. Then, we show that the parallel composition in G(S) correspond indeed to the synchronous product. Moreover we study the diaonal structure o S, which is a cateorical way to speak o duplicability. We show in particular that the class o copyable morphisms o S corresponds exactly to the one-state deterministic synchronous machines. 2 Synchronous machines 2.1 Context Pnueli [15] proposed to consider two broad classes o computin systems. Transormational systems, whose role is to compute an output rom an input and then stop. Compilers, automated theorem provers, or ray-tracers are transormational prorams. Reactive systems, repetitively interactin with an external environment. ll the sotware and hardware systems rom the ield o real-time systems all under the umbrella o reactive systems; let us cite automotive and avionics control sotware, but also on-line video encodin and imae ilterin, etc. Most industrial computin systems are reactive ones. It is thereore not surprisin that the desin and implementation o prorammin lanuaes and speciication loics tailored to this settin has attracted much interest. One particular strand o reactive ormalisms have met widespread industrial use: the amily o synchronous prorammin lanuaes, such as Esterel [1], Lustre [14] and Sinal [10]. The main characteristics o these lanuaes is the notion o a lobal loical time shared between all processes, with instantaneous communications. n ain but useul survey o the whole area is [18]. Compared to common prorammin lanuaes and systems or real-time prorammin, concurrency in synchronous lanuaes is built-in rather than tacked on: it is a linuistic notion. It is also deterministic, in contrast with most works in asynchronous process alebra.

4 The semantics underlyin synchronous lanuaes is rooted in automatatheoretic notions. In such lanuaes, parallel composition corresponds to the synchronous product o automata. This choice is conceptually simple and its well with the culture o their main users, control and sinal scientists. It is also very relevant or ormal veriication, a vital activity in critical system desin. Indeed, the synchronous product typically results in much smaller systems that the asynchronous one and contributes to keep the state-space explosion problem under control. Moreover, saety properties can themselves be expressed directly throuh so-called synchronous observers [16]. It is then possible to veriy, test or simulate the parallel composition o a proram and its observers as syntactic parallel composition and intersection o execution traces coincide. Observers are synchronous prorams and can thus be themselves veriied and tested usin the same methods. In this work, we handle the eneral settin o synchronous products o composable inite automata, ollowin the deinitions o [16]. Our oal is then to extract the cateorical structure o these objects, and recreate the seeminly primitive notion o synchronous interaction rom alebraic means. 2.2 Synchronous machines We study an abstracted model o synchronous communication rather than a concrete synchronous lanuae. s in [16], we use a variant o inite automata with the synchronous product. However, here we do not suppose that sinals come rom a predeined lobal set. This is essential in order to comply with the cateorical point o view; we discuss this point urther in section 5. Deinition 1. Synchronous machines synchronous machine is a 5-tuple (,, F, i, ) where: F is a set o states; i F is the initial state o ;, are sets o input and output sinals (respectively); F P () P () F is the transition relation, expressin how the machine in a iven state reacts to a set o input sinals by emittin a set o output sinals and movin to another state. When can be deduced rom the context, we write q i o q or (q, i, o, q ). We also deine δ : F P () P (P () F) the reaction unction o machine by δ (q, i) = {(o, q ) o, q.(q, i, o, q ) }. Properties o synchronous machines Next we describe properties o synchronous machines. Our cateorical presentation will characterize them throuh purely alebraic means. Note that the two irst properties are very desirable or practical purposes.

5 Deinition 2. Deterministic machine synchronous machine is deterministic i it has at most one possible reaction or each input in each state: q, i. δ (q, i) 1 Deinition 3. Reactive machine synchronous machine is reactive i it has at least one possible reaction or each input in each state: q, i. δ (q, i) 1 Deinition 4. Versatile machine synchronous machine is versatile i it can produce all the possible outputs in each state: q, o. i, q. q i o q Operations on synchronous machines Deinition 5. Projection Let be a synchronous machine and a subset o its output sinals. The projected machine is (,, F, i, ) where = { (q, i, o, q ) or all (q, i, o, q ) } Deinition 6. Synchronous product Let (resp. ) be a synchronous machine with input sinals in (resp. C) and output sinals in (resp. D). We deine ( ) = ( (\D) (C\), D, F G, i i, ( ) ) where ( ) is deined by the inerence scheme: q 1 a (d ) b q 1 q 2 q 1 q 2 c (b C) a c b d q 1q 2 d q 2 where a \D and c C\, b and d D The synchronous product o synchronous machines and thus runs and in parallel, and iuratively plus the outputs o (resp. ) into the inputs o (resp. ). One reaction step o the compound machine perorms exactly one reaction step o and one reaction step o. 3 Traced cateories Traced monoidal cateories or rather traced cateories, as the notion o trace relies on a monoidal structure were introduced in [20] to ive a cateorical account o a situation arisin in various areas o mathematics such as knot theory, alebraic topoloy or theoretical computer science.

6 3.1 Symmetric monoidal cateories Deinition 7. Monoidal cateory monoidal cateory is a cateory C toether with a biunctor, a distinuished unit object 1 and natural amilies o isomorphisms ρ : 1 λ : 1 α,,c : ( C) ( ) C satisyin some coherence axioms [11]. I these isomorphisms turn out to be identities i.e. ( C) = ( ) C and 1 = = 1 the cateory is said to be strict monoidal. The strict case allows to write simpler, more readable equations. It turns out that every monoidal cateory is equivalent with a strict one [11], so we will most o the time avoid writin brackets around without any loss o rior. Deinition 8. Symmetries amily o symmetries in a (strict) monoidal cateory C is a natural amily o isomorphisms σ, : such that σ, ; σ, = Id σ,c = (Id σ,c ) ; (σ,c Id ) We say that C is a (strict) symmetric monoidal cateory (SMC, or short) i C is a (strict) monoidal cateory toether with a amily o symmetries. 3.2 Graphical lanuae and the notion o trace lon with the notion o traced cateory, [20] introduced a raphical lanuae or these cateories. s we shall see, the expression o some simple properties can be quite hard to read in the traditional equational lanuae, while they look obvious in the raphical lanuae. Fiure 2 ives the translation o these properties to diarammatical lanuae. Tr U, ( ): = U U Fi. 1. Feedback: the trace

7 Id : = m... : 1 2 m 1 2 m = 2 n ; : C = C : C D = C D, : = Fi. 2. The raphical lanuae o symmetric monoidal cateories Once introduced this lanuae o circuits and wires, it is easy to state the basic idea behind the notion o trace: one wants to make (cateorical) sense o the wirin presented on iure 3.2, which makes sense rom a circuit perspective: just add a eedback wire and observe what happens 1. The point is thereore to axiomatize the notion o eedback in a SMC. Deinition 9. Trace trace in a (strict) SMC C is a amily o unctions (we will oten omit the, subscript when it is obvious rom the context) satisyin the ollowin axioms: Tr U, : C( U, U) C(, ) Superposin: or all : U U and : C D, Tr U,() = Tr U C, D( ) U C U D = U C U D Tihtenin: or all : U U, : and h :, Tr U, ( (IdU ) ; ; (Id U h) ) = ; Tr U,() ; h 1 This will eventually cause a ire, but this is o the scope o this work...

8 h = h Slidin: or all : U U and : U U, Tr U (, ; ( Id ) ) = Tr U (, ( Id ) ; ) = Vanishin: or all : (as the cateory is strict we have = 1 and = 1 ) = Tr 1,() 1 1 = ssociativity: or all : U U, ( ) Tr U, Tr V U, U () = Tr U V, () = Yankin: or all, Tr,(σ, ) = Id =

9 We will reer to SMC equipped with a trace as traced cateories. 3.3 The G construction The Int construction was deined in [20] to relate traced cateories with compactclosed cateories. In [17], S. bramsky introduced a (isomorphic) variant o it, called the G construction to ive a cateorical version o J.-Y. Girard s Geometry o Interaction (see section 6). Deinition 10. Compact-closed cateory compact-closed cateory is a SMC in which or every object there is dual object and a pair o arrows η : 1 (called the unit) and ε : 1 (called the co-unit) satisyin (aain we assume the cateory is strict or readability) = 1 Id η ε Id 1 = is equal to Id = 1 η Id Id ε 1 = is equal to Id The idea o the Int construction is to build a compact-closed cateory out o a traced cateory C, the composition in Int(C) bein deined usin the trace o C. The same is true o G C C C C + Fi. 3. Composition in G, intuitive and ormal diarams

10 Deinition 11. G I C is a traced cateory, the cateory G(C) is deined as ollows: Objects are pairs +, o objects o C Morphisms : +, +, in G(C) are iven by morphisms : + + in C Composition o : +, +, and : +, C +, C is deined as (see i.3 or a more readable version in the raphical lanuae) ( ; G := T r + + C, C + S1 ; ( ) ; S 2 ; (Id Id C + σ (+, )) ) where S 1 := (Id + σ C,( + )) and S 2 := (Id σ (+ ),C +) Identities Id +, in G(C) are iven by the symmetries σ +, o C The act that G(C) is indeed a cateory ollows rom the axioms o trace. s the compact-closed structure o G(C) is not our main interest in this paper, we skip its deinition. The interested reader may ind it or instance in [8]. 3.4 Diaonal morphisms The tensor product o a SMC is not in eneral a cartesian product, which means basically that there is no diaonal morphism c : satisyin the riht set o equation to talk about duplication o data. However, one may want to identiy amon the morphisms o a iven SMC the ones that can be manipulated as i the tensor product was cartesian. Deinition 12. Diaonals (strict) SMC C is said to have diaonals there exists two (not necessarily natural) amilies o morphisms c : and w : 1 such that (, c, w ) is a symmetric comonoid or all, that is to say: c ; (Id c ) = c ; (c Id ) and c ; (w Id ) = c ; (Id w ) = Id w 1 = Id 1 and w w = w c c ; (Id σ, Id ) = c In such a cateory, the weak initial morphism is deined or all objects as i := Tr 1,(c a ). c w i = Fi. 4. The morphisms c, w, i in the raphical lanuae One can then deine class o morphisms that behave well with respect to the diaonal structure.

11 Deinition 13. Diaonal morphisms morphism : in a SMC with diaonals is said to be copyable i ; c = c ; ( ) = discardable i w = ; w = strict i i ; = i = Remark : The morphisms o a SMC with diaonals that are both copyable and discardable orm a subcateory, the larest one on which the tensor product is a cartesian product. 4 The cateory o synchronous machines We can see now how the synchronous machines can be oranized as a traced cateory. First deinitions The basic idea is that a synchronous reactive system is a morphism rom its input set o sinals to its output set o sinals. In a way, the cateory we describe is quite close to the cateory Rel o sets and relations. The dierence bein the state space, which needs a speciic treatment in terms o composition and equality. Deinition 14. Objects, morphisms The cateory S o synchronous machines is deined as objects are sets morphisms rom to are synchronous machines (deinition 1) with input and output Given two morphisms in S, one has to be able to say when they are equal. In act, with plain equality we would not end up with a cateory. The issue is that the state space has to be treated somehow up to isomorphism. This amounts to the standard issue o equality o labelled transition system. The notion we settle or in this paper is isomorphism o state raphs. Deinition 15. Equality Two morphisms, : in S are equal whenever they are raphisomorphic, i.e. there exists a bijection ϕ : F G such that = ϕ ϕ 1

12 We can now deine the composition o two morphisms, in S. Look back at the raphical representation o composition in a monoidal cateory: the intuition is that when composin, the output o is passed on as an input or. s or the state space, it is natural to say that a state o the composed system is iven by a state o and a state o, hence the use o cartesian product. Deinition 16. Composition and identities The composition o two morphisms : and : C in S is iven by ; := (, C, F G, i i, ; ) where ; := { a c b, a b and b c } For any object, the identity Id is deined as ( Id :=,, { },, { a or all a }) a It is routine to check that all this deines a cateory: irst check that equality is compatible with composition, then provin associativity and identity axioms is just a matter o choosin appropriate (and trivial in that case) bijections between state spaces. Symmetric monoidal structure The tensor product is the operation that puts two (non-interactin) systems in parallel. ain the raphical lanuae suests the use o the cartesian product or handlin the state space. Deinition 17. Tensor product On objects o S, the tensor product is the disjoint union, :=. I : and : C D are morphisms in S, their tensor product is iven by := ( C, D, F G, i i, ) where := { ac } a b and c bd d The unit 1 o the tensor product is the empty set. Deinition 18. Symmetries Given two object, o S, the associated symmetry σ, : is deined as σ, := (,, { },, { ab ba or all a, b })

13 The trace The trace is quite similar to the one o Rel. Note that it does neither preserve reactivity nor determinism o synchronous machines. Deinition 19. Trace Given a morphism : U U o S, its trace Tr U,( ) : is deined as Tr U,( ) := ( {,, F, i, a au u U, b } ) bu Copyable, discardable and strict morphisms Proposition 1. Diaonal morphisms in S The cateory S has diaonals, iven by ( c :=,, { },, { a or all a } ) aa w := (, 1, { },, { a or all a } ) Moreover, the weak initial morphism or an object is i = Tr 1,(c a ) = ( 1,, { },, { a or all a } ) Remark : weak initial morphisms are quite dramatic examples o the nonpreservation o determinism by the trace o S. Theorem 1. Deterministic, total and strict morphisms in S the copyable morphisms are the deterministic, one-state synchronous machines the discardable morphisms are the reactive, one-state synchronous machines the total morphisms are the surjective, one-state synchronous machines 5 G(S) and the synchronous product We now relate the composition in G(S) with the synchronous product o synchronous machines. The theorem below says that, provided the input/output sets o the two systems are correctly positioned 2, their composition in G(S) is equal to their synchronous product ollowed by hidin the sinal used durin the interaction. 2 That is, they do not intersect anywhere but on +, where the systems are expected to interact.

14 Theorem 2. Let : ( +, ) ( +, ) and : ( +, ) (C +, C ) be two morphisms is G(S). Suppose also that + C + = C = (see discussion below). Then we have ; G = ( ) C + Proo. ; G and ( ),C + share the same state space F G and initial state i i. Thereore we show they are equal (deinition 15) by the identity bijection between their state spaces: We have that ac in ; G a c i and only i there exist b +, b acb such that + b in a c b + b S 1 ; ( ) ; S 2 ; (Id Id C + σ (+, )), (with S 1 := (Id + σ ( + ),C ) and S2 := (Id σ ( + ),C +) as in deinition 11) i and only i there exist b +, b such that ab b + c a b + b c in, i and only i there exist b +, b such that in, ab + in and b c a b b + c i and only i there exist b +, b ac such that ab + b c in ( ), i and only i ac a c in ( ) C +. The condition + C + = C = may look simply technical. ut there is actually more to it. s explained in section 2, synchronous prorammin lanuaes such as Esterel oten assume that there is a lobal space o sinals S. Reactive systems do not have a speciied input/output space and they rather all read and emit sinals in S. I two systems are executed side by side and share input/output sinal, they will interere. This does not quite correspond to the cateorical point o view, which works up to isomorphism: to et a monoidal cateory, one needs to avoid intererence between and when buildin. This is achieved usin direct sums on input/output spaces o reactive systems when deinin, while with a lobal sinal set S one would use union, with possible intererence so that the would be partially deined. The same dilemma occurs in loics, this is the theme o locativity, introduced by J.-Y. Girard in [6]. Locative approaches to loics start by settin a lobal space where everythin happens, and also use union rather than direct sum, endin up with partially deined connectives. From this point o view, we could say that synchronous machines are naturally locative.

15 6 Related and uture work Circuit synthesis The eometry o synthesis research proram by Ghica et al. [2] proposes novel ways o compilin an idealized hih-level lanuae to diital circuits. The lanuae a dialect o lol, that is an hiher-order unctional lanuaes with imperative eatures, extended with inite-state asynchronous concurrency. Various type systems have been proposed to accommodate this initeness condition. Part o the oriinality o the approach stems rom the oriins o the translation process, heavily inspired rom cateorical models. Compared with most works in hih-level circuit synthesis, this denotational lavor ives a compositional translation, i.e. one where unctions can be compiled separately. Followin the seminal work o lur et al. [9], Ghica et al. also studied the round abstraction [12], a transormation rom asynchronous to synchronous circuits, in order to beneit rom the rich ecosystem available or the latter. One possible application o the present work would be to be able to directly enerate synchronous circuits rom a hih-level prorammin lanuae. The Towards eometry o interaction or synchronous machines The G construction was irst introduced to ive a cateorical account o the Geometry o Interaction proram. This proram that was initiated and pursued by J.-Y. Girard in a series o paper [4,5,7]. It is built around the idea o lookin at the interactive aspects o loical phenomena. The composition o two proos the (Cut) rule is seen interactively in GoI: the computation can low back and orth between the two composed prorams, as in ame semantics [13] or instance. Moreover, in the GoI approach, or a proram to be o a certain type is no loner a matter o ollowin some iven set o rules, but rather the result o mutual testin. In loics, this technique is usually reerred to as construction by orthoonality. y showin that synchronous machines orm a traced cateory and that the synchronous product corresponds to the composition induced by the G construction, our work opens the way to applyin GoI ideas and techniques to reactive systems. In particular, it would allow to use the ood structural properties obtained by orthoonality methods to try to deal with the problem o determinism/reactiveness. Reerences 1. Gérard erry ; Laurent Cosserat, The esterel synchronous prorammin lanuae and its mathematical semantics, Seminar on Concurrency, Carneie-Mellon University (London, UK, UK), Spriner-Verla, 1985, pp Dan R. Ghica, Function interace models or hardware compilation, MEMOCODE, 2011, pp

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