Towards a Mathematical Operational Semantics

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1 Towrds Mthemticl Opertionl Semntics Dniele Turi Gordon Plotkin Deprtment of Computer Science Lbortory for Foundtions of Computer Science University of Edinburgh, The King s Buildings Edinburgh EH9 3JZ, Scotlnd Abstrct We present ctegoricl theory of well-behved opertionl semntics which ims t complementing the estblished theory of domins nd denottionl semntics to form coherent whole. It is shown tht, if the opertionl rules of progrmming lnguge cn be modelled s nturl trnsformtion of suitble generl form, depending on functoril notions of syntx nd behviour, then one gets the following for free: n opertionl model stisfying the rules nd cnonicl, internlly fully bstrct denottionl model which stisfies the opertionl rules. The theory is bsed on distributive lws nd bilgebrs; it specilises to the known clsses of well-behved rules for structurl opertionl semntics, such s GSOS. Introduction Opertionl semntics, fundmentl tool in lnguge design nd verifiction, provides forml description of the behviour of progrms. It is often defined in terms of tomic, elementry trnsitions, describing locl behviour. Mthemticlly, these trnsitions cn be modelled s the elements of reltion, the intended opertionl model of the lnguge. A convenient wy of specifying such trnsition reltion is by induction on the structure of the progrms, strting from suitble opertionl rules for the bsic constructs of the lnguge [21]. Trditionlly, opertionl semntics is contrsted with the mthemticl interprettion of progrms clled denottionl semntics, where progrms re Reserch supported by EuroFOCS. Reserch supported by n EPSRC Senior Fellowship. mpped into suitble semntic domin endowed with n opertion for ech construct of the lnguge. Both opertionl nd denottionl semntics re necessry for complete description of progrmming lnguge: the former for specifying the execution of the progrms nd the ltter for resoning bout them in terms of bstrct, mthemticl entities. It is therefore fundmentl tht denottionl semntics be dequte, ie tht it determines the opertionl behviour of progrms [24]. For lnguges without vrible binding, but possibly multi-sorted, denottionl model cn be seen s Σ-lgebr, where Σ is the signture of the lnguge corresponding to the bsic constructs. The progrms themselves form the initil such Σ-lgebr nd the corresponding unique homomorphism from the progrms to the denottionl model is clled initil lgebr semntics [12]. The semntic domin, ie the crrier of the denottionl model, cn often be regrded s the finl solution of domin eqution X = B(X), for suitble behviour functor B. In other words, the semntic domin is the finl B-colgebr. The trnsition reltions my lso be seen s B-colgebrs nd, therefore, so cn the intended opertionl model of lnguge. The corresponding unique colgebr homomorphism, given by finlity, from the intended opertionl model to the semntic domin is clled finl colgebr semntics [2, 23]; under suitble ssumptions on B, it is fully bstrct with respect to behviourl equivlence. When initil lgebr nd finl colgebr semntics coincide, one hs n dequte denottionl semntics [23]. Adequcy proofs cn be quite demnding, hence generl criteri ensuring dequcy re of interest. For process lgebrs, s used for specifying nondeterministic nd concurrent progrms [17, 5], there exist syntctic restrictions on the formt of the oper-

2 tionl rules which ensure tht bisimultion [17] is congruence. Among the rules in these formts, GSOS rules [8] re the best known nd (negtive) tree rules [11] re the most generl. In [22], the processes s terms method, bsed on such congruence result, is presented which llows for the systemtic derivtion of dequte denottionl models from tyft rules [13], clss of rules equivlent to tree rules. We present here ctegoricl reformultion nd generlistion of the bove dequcy met-results. First, we show tht certin sets R of GSOS rules cn be modelled s nturl trnsformtions [[R] depending on the functoril notions of signture Σ nd behviour B. Next, it is shown tht the mpping R [[R]] is n essentilly 1-1 correspondence. The nturlity of [[R]] ccounts for the syntctic restrictions on the occurrences of met-vribles in GSOS rules nd provides ctegoricl explntion of their good behviour. The first dvntge of the bove pproch is tht the GSOS rules cn be modelled not only in Set, but lso in every ctegory with enough structure such s the ctegory of cpos nd continuous functions used in denottionl semntics. This is step towrds bridging the gp between opertionl nd domin theory. A second dvntge is tht the mthemticl modelling of the rules is useful semntic tool in the investigtion of syntctic formts. For instnce, in Set the dul of the type of nturl trnsformtion corresponding to GSOS lso corresponds to n interesting formt, nmely the sfe tree rules: these form nturl subclss of (negtive) tree rules which lwys possess stisfying trnsition reltion. Interestingly, the filure to fit the clss of (simple negtive) tree rules in the present pproch brought to light slight inccurcy in the literture nd, eventully, led to the discovery of the sfe tree rules. A third dvntge is tht by vrying Σ nd B wide vriety of notions of progrm constructs nd behviour cn be ccommodted. (See lso [30].) Further, one cn study bstrct notions of opertionl rules ρ, such s bstrct GSOS nd bstrct tree rules, pplicble to lnguges other thn process lgebrs nd whose properties cn be studied in generl. In this theory we ssume tht Σ freely genertes mond T which is thought of s corresponding to the syntx of the lnguge. The first result is tht such bstrct opertionl rules ρ induce n opertionl mond T ρ lifting the mond T to the B-colgebrs, ie to the opertionl models, in the sense tht its ction on the crriers is the sme s the mond T. If ρ is of bstrct tree rules form rther thn bstrct GSOS, then, by dulity, one first coinductively derives denottionl comond D ρ. The ssumption here is tht the functor B cofreely genertes comond D which should correspond to the globl behviours of the lnguge. The comond D ρ is lifting of this comond D to the Σ-lgebrs, ie to the denottionl models. However, one cn still spek of the opertionl mond defined by some bstrct tree rules becuse generl theorem shows tht liftings of D to the Σ-lgebrs nd liftings of T to the B-colgebrs re in 1-1 correspondence. In fct, these liftings re lso in 1-1 correspondence with the distributive lws λ of the mond T over the comond D., which generlise both bstrct GSOS nd bstrct tree rules. One is led now to consider the bilgebrs of such distributive lws. When λ corresponds to some bstrct opertionl rules ρ, the λ-bilgebrs cn be seen s combintions of opertionl nd denottionl models which stisfy the rules. Henceforth they re clled ρ-models; they specilise to the GSOS models of [25] nd to models of tree rules (with n pproprite definition). The primry fct bout ρ-models is tht, from results in [15], it esily follows tht the forgetful functors to ech of the ctegories of denottionl nd opertionl models hve djoints. One djunction implies tht there exists n initil ρ-model the intended opertionl model T ρ (0) for the initil lgebr of progrms. By the definition of morphism of ρ-models, this lso implies tht every ρ-model is dequte with respect to the intended opertionl model in the sense tht the behviour of the progrms cn be determined from ny ρ-model up to generlised, colgebric notion [4, 16] of bisimultion. The other djunction implies tht there exists finl ρ-model the cnonicl denottionl model D ρ (1) over the finl colgebr of bstrct, globl behviours. It is necessrily dequte; further, it is internlly fully bstrct with respect to colgebric bisimultion. The derivtion of this finl model specilises to the bove mentioned processes-s-terms method. The unique homomorphism from the initil to the finl ρ-model is both the initil lgebr nd finl colgebr semntics for the bstrct rules ρ. It is clled here universl semntics; it is the most bstrct compositionl interprettion of progrms preserving behviourl distinctions. Moreover, if the behviour functor B stisfies certin mild condition, every ρ-model hs gretest (generlised) bisimultion which, moreover, is (generlised) congruence. This specilises to the fct tht bisimultion is congruence for GSOS nd for tree rules. The generlised, colgebric notion of bisimultion considered here is to be understood s the behviourl equivlence corresponding to the functor B under con-

3 sidertion. It might tke forms quite different from ordinry (strong) bisimultion. For instnce, for the behviour functor in [14] it specilises to the much corser (complete) trce equivlence. As corollry, one hs n bstrct formt of rules ensuring tht trce equivlence is congruence [30]. To some extent, one cn lso del with wek bisimultion in this setting. As shown, eg in [13], wek bisimultion for given set of rules cn be reduced to strong bisimultion by dding three specil rules for the τ- ction. (See lso [3].) These rules re in the tyft/tyxt formt, but they cn be compiled into sfe tree rules, hence the present theory cn be pplied. This wy of deling with wek bisimultion is quite indirect, but tht just reflects the bsence of n estblished denottionl model for it. A more direct tretment of wek bisimultion might rise following [9]. 1 The Motivting Exmple: GSOS Consider the lnguge with signture Σ consisting of constnt symbol nil, set of unry ction prefixing opertors indexed by finite set A of ctions rnged over by, nd binry prllel composition opertor. This signture freely genertes, for every set X of vribles x, the set T X of terms t given by the bstrct grmmr t ::= x nil.t t t This set T X is the crrier of the free Σ-lgebr over X, where, in generl, Σ-lgebr is given by (crrier) set Y nd function h mpping ech opertor σ of rity n in the signture to function of type Y n Y. More concisely, the function h cn be written s h : σ Σ Y rity(σ) Y (1) using the disjoint union functor (coproduct in Set) to glue the interprettion of the vrious opertors together. Next, let the opertionl rules R inductively defining the (lbelled) trnsitions performble by the progrms of the bove lnguge be.x x x x x y x y y y x y x y For instnce, the simple progrm (.nil) (.nil) cn either perform the ction becoming nil (.nil) or perform becoming (.nil) nil. The (locl) behviour of this progrm cn be modelled s the function from A to finite subsets of terms mpping to the set {(.nil) nil, (.nil) nil} nd ll other ctions to the empty set. In generl, the type B of the behviour of the bove lnguge is BX = (P fi X) A (2) the (covrint) functor mpping set X to the set of functions from A to finite subsets of X. Let x nd y rnge over X, β rnge over (P fi X) A, nd let us write {x 1,..., x n } for the function from A to P fi X mpping to {x 1,..., x n } nd ll other elements of A to the empty set. Then, for ech opertor σ of the signture, the corresponding rules cn be modelled s function s follows. [σ]] : (X (P fi X) A ) rity(σ) (P fi T X) A [[nil]] = [.]](x, β) = {x} (x, β)[[ ]](y, β ) = {x y x β()} {x y y β ()} Using the universl property of coproducts, these functions cn then be glued into single function, sy [[R]] X : 1 ( A (X BX)) (X BX)2 BT X Note one hs function [[R] X for ech set X of vribles. In fct, one should think of the vribles in the rules s being met-vribles. Most importntly, the bove definition of [[R] X is nturl in X: for every renming of the vribles (possibly involving equting some of the vribles), first renming nd then pplying the rules is the sme s first pplying the rules nd then renming. As shown in 5 nd 7 the nturlity of [[R]] explins the good behviour of R. More generlly, let A i nd B i rnge over subsets of A nd let R be set of rules of the form {x i yij Ai }1 i n, 1 j m i σ(x 1,..., x n ) b {x i } 1 i n c t b B i (3) which is imge finite in the sense tht there re finitely mny rules for ech opertor σ in Σ nd ction c in A. For every set X, one cn ssocite to R function [[R]] X : σ Σ(X (P fi X) A ) rity(σ) (P fi T X) A (4) s follows. For ll t in T X, c in A, x i in X, nd β i in (P fi X) A, put t [[R]] X (σ((x 1, β 1 ),..., (x n, β n )))(c) if nd only if the following condition holds.

4 Condition 1.1 There exists (possibly renmed) rule (3) in R such tht {yi1,..., y im } is subset of β i i (), for in A i, nd β i (b) is empty, for b in B i. Note the function [[R]] X does not need to be nturl in the set X. Definition 1.1 (GSOS [8]) A GSOS rule is rule of type (3) such tht the x i nd yij re ll distinct nd, moreover, these re the only vribles which cn occur in the term t. Two sets of rules re clled equivlent if they prove the sme rules in the sense of [11, Def. 2.5]. Theorem 1.1 (GSOS is nturl) There is correspondence between nturl trnsformtions of type (X (P fi X) A ) rity(σ) (P fi T X) A (5) σ Σ nd imge finite sets of GSOS rules for signture Σ (over fixed denumerbly infinite set of vribles V ). Moreover, this correspondence is 1-1 up to equivlence of sets of rules. Proof. We just describe the correspondence. One direction is given by the bove mpping R [[R]]. As for nturlity, let us introduce some useful bbrevition first: for every function f : X X, write f for the function (P fi f) A, Γ for the set [R]] X (σ((x 1, β 1 ),..., (x n, β n )))(c), nd Γ for the set [[R]] X (σ((fx 1, f β 1 ),..., (fx n, f β n )))(c). Then the clim is tht nd, conversely, 1. t Γ, t Γ, t = (T f)(t) 2. t Γ, t Γ, t = (T f)(t) Consider the first cluse. If t is in Γ then Condition 1.1 holds. Clerly, β i (b) = if nd only if (f β i )(b) =, nd {yi1,..., y im } β i i () implies {fyi1,..., fy im } (f β i i )(), therefore (T f)(t) is in Γ, becuse the x i nd yim re the only vribles occurring in the rule. For the second cluse, one lso uses i the fct tht the x i nd the yim in GSOS rule re ll i distinct, hence the vlue of Γ does not depend on ny of the possible identifictions mde by the renming function f. In the converse direction, given nturl trnsformtion ρ X of type (5), one cn define set of rules s follows. Let V be x 1, x 2,... nd let x {y 1,..., y k } be n bbrevition for x y 1,..., x y k. Choose β i () = {y ij j = 1,..., m i } so tht the x i nd y ij re ll distinct. Then write rule {x i β i ()} 1 i n A i A {x i σ(x 1,..., x n ) c t b } 1 i n b A\A i (6) whenever t ρ X (σ((x 1, β 1 ),..., (x n, β n )))(c) nd A i = { A β i () }. Nturlity ensures this is GSOS rule. It cn be further shown tht nturlity nd the finiteness of A ensure tht the resulting set of rules is imge finite. We do not understnd this sitution for infinite A, lthough the bove definition of [[R]] still works. 2 GSOS is Ctegoricl In this section, let C be distributive ctegory with infinite coproducts nd commuttive free semi-lttice mond P f. The clim is tht GSOS rules cn be modelled in every such ctegory. Note, first, tht to every signture Σ one cn ssocite n endofunctor on Set with the sme nme: ΣX = σ Σ X rity(σ) Clerly, this definition lso mkes sense in C. Next, rewrite BX = (P fi X) A s BX = (1 + P f X) A, which, gin, mkes sense in C: the power Y A is the product A Y, P f is the free semi-lttice mond which in Set is the relevnt prt of the endofunctor P fi obtined by removing the empty set, nd + is just nother nottion for the binry coproduct. It remins to generlise T. For this, let Σ be n rbitrry endofunctor on C nd let Σ-Alg be the corresponding ctegory of Σ-lgebrs: objects re pirs X, h, where the crrier X is n object nd the structure h : ΣX X is morphism of C; the homomorphisms f : X, h X, h re the morphisms f : X X between the crriers such tht f h = h (Σf). If the forgetful functor U Σ : Σ-Alg C X, h X mpping Σ-lgebrs to their crriers, hs left djoint F Σ, then the corresponding mond T = U Σ F Σ (7) is the mond freely generted by Σ. For finitry endofunctors Σ s the one bove, it suffices tht C hs ω-colimits for the djunction F Σ U Σ to hold nd T to be defined. Thus one cn tke T to be the mond freely generted by the endofunctor

5 ΣX = σ Σ Xrity(σ). In Set, its vlue T X t set X is the set of terms corresponding to the opertors σ of the signture nd with vribles x in X; the unit η X : X T X is the insertion-of-vribles function which mps vrible x in X to the sme vrible but seen s term; nd the multipliction µ X : T 2 X T X is the opertion which llows one to plug terms into contexts. Theorem 2.1 For ny imge finite set R of GSOS rules, nturl trnsformtion [R]] : (X (1 + P f X) A ) rity(σ) (1 + P f T X) A σ Σ cn be defined in the internl lnguge of distributive ctegories with infinite coproducts nd commuttive free semi-lttice mond P f. In the cse of Set it specilises to the trnsformtion (4). Proof. The trnsformtion [[R]] in (4) cn be defined ctegoriclly using: projections nd injections, piring nd copiring, nd the ssocitivity, symmetry, unit, nd distributive lws for products nd coproducts; the unique mp to the finl object 1; the unit nd the free structure of the mond T ; the join of free semi-lttices; the unit nd the strength of the commuttive mond P f. (The use of the strength depends on the ssumption tht, becuse R is imge finite nd A is finite, for ech opertor the set of rules is finite.) The chrcteristion of GSOS given by the bove theorem llows one, for instnce, to relise GSOS rules in the ctegory of cpos nd continuous functions s used in domin theory, rther thn in Set. 3 Abstrct GSOS In generl, given crtesin ctegory C nd rbitrry functoril notions of progrm constructs Σ nd behviour B on C, with Σ freely generting the syntx T, one cn define corresponding bstrct notion of opertionl rules s the nturl trnsformtions ρ of type Σ(Id B) BT (8) We shll need the following chrcteristion. Proposition 3.1 There is 1-1 correspondence between nturl trnsformtions of type Σ(Id B) BT nd those of type Σ(T BT ) BT. Proof. One direction of the correspondence is given by the mpping ρ ϱ = Bµ ρ T : Σ(T BT ) BT, for ρ : Σ(Id B) BT. In the converse direction, simply precompose ϱ : Σ(T BT ) BT with Σ(η Bη). Severl exmples illustrting the use of the generlity of (8) re given in [30]. Here is brief summry thereof. Firstly, the opertionl rules of deterministic progrms with exceptions nd side-effects cn be modelled instntiting (8) with the behviour endofunctor BX = (S (1 + X)) S, where S Y is the copower S Y. As shown below, the behviourl equivlence corresponding to BX = (P fi X) A is bisimultion; corser equivlence, nmely trce equivlence, cn be obtined by considering the endofunctor BX = 1 + A X on the ctegory SL(C) of semi-lttices in ctegory C. (This is simplified version of the behviour in [14].) The progrm construct endofunctor to be considered then is Σ : SL(C) SL(C), monoidl generlistion of the endofunctor Σ on crtesin ctegories. For instnce, for the lnguge of 1, Σ X = 1 + A X + (X X), where is the tensor product of semi-lttices. Note tht (8) cn be instntited to rules not only for single-sorted lnguges but lso for multi-sorted ones; it suffices to work with signtures (nd behviours) over power ctegories. Finlly, we briefly consider recursion. GSOS stnds for SOS for non-deterministic progrms with gurded recursion, becuse the full definition lso llows for definitions of progrms by gurded recursion. In [30], functoril notion of gurd is given which llows one to generlise the definitions by gurded recursion to bstrct rules of type (8). Moreover, one cn lso tret ungurded recursion by relising the bstrct rules, for instnce, in the ctegory of cpos nd prtil continuous functions nd exploiting lgebric compctness. This involves precomposing the endofunctor Σ with the lifting endofunctor, so tht one freely genertes not only finite but lso prtil nd infinite terms, the ltter being used to unfold recursive definitions of progrms. 4 Colgebrs The intended opertionl model of set of concrete GSOS rules is the lest reltion R T A T which stisfies the rules, where x x stnds for x,, x R. In generl, reltion of type X A X is clled lbelled trnsition system [21] with set of sttes X nd set of lbels A. For imge finite sets of GSOS rules it suffices to consider imge finite trnsition systems, where, for ech stte nd ech ction, the imge of the trnsition reltion is finite set. These re in 1-1 correspondence with functions k : X (P fi X) A s follows. x x x k(x)() (9) If, s considered here, the set A is finite, then imge finite trnsition systems cut down to finitely brnch-

6 ing trnsition systems, where for ech stte, the set of outgoing trnsitions is finite. A function k : X (P fi X) A is colgebr of the endofunctor BX = (P fi X) A on Set. Formlly, given n endofunctor B : C C, B-colgebr is pir X, k, where the crrier X is n object nd the structure k : X BX is morphism of C. One often identifies colgebr X, k with its structure k. The B-colgebrs form ctegory B-Colg, with homomorphisms f : X, k X, k the morphisms k X BX f Bf X k BX f : X X between the crriers such tht k f = (Bf) k. Note the forgetful functor U B : B-Colg C X, k X mpping colgebrs to their crriers. For BX = (P fi X) A, the colgebr homomorphisms re, up to the correspondence (9), the sme s the P- open morphisms of [16], where P is suitble ctegory of finite sequences of ctions. (Thus, for this choice of B, B-Colg is proper subctegory of the stndrd ctegory of trnsition systems [31].) As consequence, two trnsition systems re (strongly) bisimilr [17] if nd only if there is spn of colgebr homomorphisms between them. This leds to the following colgebric notion of bisimultion, mild generlistion of the one in [4]. Let B be n endofunctor on ctegory C with kernel pirs nd let the internl equlity of colgebr X, k be the kernel pir (in the underlying ctegory C) of the identity on its crrier X. One cn esily prove tht: Proposition 4.1 (Strong Extensionlity) Internl equlity is the finl B-bisimultion of the finl B-colgebr. In generl, finl colgebrs need not exist, but if C hs finl object 1, nd the forgetful functor U B hs right djoint G B : C B-Colg, then G B 1 is the finl B-colgebr. For the endofunctor BX = (P fi X) A on Set, such right djoint G B exists [6]. It follows [29, 13] tht the finl colgebr G B 1 is the set of rooted, imge finite trees, with brnches lbelled by A, quotiented by (ordinry) bisimultion. This is the set of bstrct globl behviours, ie the (bstrct) nondeterministic processes. Semnticlly, the bove strong extensionlity result specilises then to the fct tht such finl colgebr is internlly fully-bstrct [1] with respect to bisimultion, ie its lrgest bisimultion is the equlity, hence bisimilr elements re indistinguishble. 5 Opertionl Monds Definition 5.1 Let T nd B be endofunctors on the sme ctegory C. An endofunctor T on the ctegory of B-colgebrs lifts the endofunctor T to the B- colgebrs if U B T = T UB, ie the digrm Definition 4.1 (Colgebric Bisimultion) A B- bisimultion between two colgebrs X 1, k 1 nd X 2, k 2 of n endofunctor B is triple X, f 1, f 2 such tht such tht there exists colgebr structure k : X BX mking X, k, f 1, f 2 spn X, k B-Colg U B C T T B-Colg C U B f 1 X 1, k 2 f 2 X 2, k 2 of colgebr homomorphisms f 1,f 2. One cn form the ctegory of B-bisimultions between two colgebrs X 1, k 1 nd X 2, k 2 of n endofunctor B on ctegory C: the morphisms g : X, f 1, f 2 X, f 1, f 2 re those g : X X in C (thus not necessrily colgebr homomorphisms) such tht f i = f i g, for i = 1, 2. commutes. (Cf [15].) When both T nd T re monds, T lifts the mond T to the B-colgebrs if the forgetful functor U B : B-Colg C (together with the identity nturl trnsformtion) is mond morphism [27] from T to T. Remrk 5.1 A mond T lifts mond T = T, η, µ to the B-colgebrs if nd only if U B T = T UB nd, for

7 every B-colgebr k : X BX, the digrm T ϱ (k) : T X BT X to be the unique mp k X η X T (k) T X µ X T 2 X T 2 (k) X k η X BX T X T ϱ(k) ψ X ΣT X Σ id, T ϱ(k) BX Bη X BT X Bµ X BT 2 X Bη X BT X ϱ X Σ(T X BT X) commutes. given by the bove theorem. Consider now T to be the mond freely generted by n endofunctor Σ. The djunction F Σ U Σ gives well-known structurl recursion theorem which specilises to the ordinry recursion (or itertion) theorem for nturl numbers, covering the simplest form of primitive recursive functions, but not others such ddition, multipliction, exponentition, etc, which need prmeters nd ccumultors. (By structurl recursion we men definition by structurl induction.) Here we shll need the following folklore structurl recursion theorem [20] with ccumultors, ie with terms s prmeters of the recursive definition. Theorem 5.1 (Structurl Recursion) Let T be mond freely generted by n endofunctor Σ on crtesin ctegory C nd let ψ X : ΣT X T X be the structure of the free Σ-lgebr over n object X of C. For ll morphisms f : X Y nd h : Σ(T X Y ) Y in C there exists unique morphism f : T X Y in C such tht commutes. X f η X T X Y f h ψ X ΣT X Σ id, f Σ(T X Y ) Proof. Turn h into the Σ-lgebr structure ψ X Σπ 1, h : Σ(T X Y ) T X Y over the product T X Y nd then pply the ordinry structurl recursion theorem to it nd η X, f : X T X Y. Recll Proposition 3.1. For every mp ϱ X : Σ(T X BT X) BT X (10) nd every colgebr k : X BX, define the colgebr Proposition 5.1 If the morphism ϱ X is nturl in X, then the bove construction k T ϱ (k) extends to mond T ϱ lifting T to the B-colgebrs. Proof. First one needs to prove tht, for every colgebr homomorphism f : X, k X, k, T f is colgebr homomorphism, ie T ϱ (k ) T f = BT f T ϱ (k) so tht one cn define T ϱ f to be T f. For this, simply note tht both composites T R (k ) T f nd BT f T R (k) fit s the unique morphism f X X k η X BX Bη X T X! BT X ϱ X ψ X ΣT X Σ id,! Σ(T X BT X ) Σ T f, id Σ(T X BT X ) given by Theorem 5.1, hence they must be equl. (The nturlity of ϱ is essentil here!) Next, one hs to verify tht the endofunctor T ϱ lifts the opertions of the mond T. From Remrk 5.1, it suffices to show tht, for every colgebr structure k : X BX, T ϱ (k) η X = Bη X k nd T ϱ (k) µ X = Bµ X Tϱ 2 (k), ie the unit nd the multipliction of T re colgebr homomorphisms. For the unit, this is immedite by definition of the functor T ϱ, while for the multipliction one lso needs to use the nturlity of ρ nd the fct tht µ is defined by (ordinry) structurl recursion on the free lgebr structure. Definition 5.2 The opertionl mond induced by some bstrct opertionl rules ρ : Σ(Id B) BT, is the mond T ϱ corresponding to the composite nturl trnsformtion ϱ = Bµ ρ T : Σ(T BT ) BT. We write T ρ for this mond.

8 Let us try nd understnd the opertionl mond T ρ when ρ = [[R]], for R set of concrete GSOS rules. Firstly, pplying ρ to T X mounts to instntiting the met-vribles of the rules with the terms in T X. Formlly, in this wy the term t in GSOS rule (3) might contin terms s vribles: one needs to pply to it the multipliction of the term mond T in order to unbrcket it nd obtin n elementry term. This is chieved here by composing ρ T X with Bµ X. Next, recll the correspondence (9) between colgebrs k : X BX = (P fi X) A nd imge finite trnsition systems. By regrding X s set of constnts rther thn s set of sttes, the correspondence (9) cn lso be seen s being between colgebrs nd sets of δ-rules [8], ie xiom rules. Up to these two correspondences, one cn then check tht k T ρ (k) is the usul construction of trnsition system for finite set of GSOS rules R nd possibly infinite (but imge finite) set k of δ-rules. In prticulr, if X is the empty set, hence k is the trivil colgebr 0 : B nd T X = T is the set of closed terms, this construction gives the intended opertionl model for the rules. These remrks hold for rbitrry rules of type (3) nd, correspondingly, to possibly non-nturl functions [[R]] X. The nturlity of GSOS ensures tht T ρ is n opertionl mond, which is essentil for pplying the theory in 7. 6 Dulising GSOS: Tree Rules The dulity between lgebrs nd colgebrs cn be exploited to find formt of rules dul to bstrct GSOS s follows. Let Σ nd B be two endofunctors on cocrtesin ctegory C nd let D = D, ε, δ be the cofree comond generted by B, tht is, the forgetful functor U B hs right djoint G B : C B-Colg nd D = U B G B (11) By the dul of Theorem 5.1 nd Definition 5.1, every nturl trnsformtion ϱ : ΣD B(D + ΣD) coinductively defines lifting D ρ of the comond D to the Σ-lgebrs: Σ-Alg U Σ C D ϱ D Σ-Alg C U Σ In prticulr, such lifting cn be obtined from nturl trnsformtions ρ : ΣD B(Id + Σ) (12) by dulising Proposition 3.1 nd putting ϱ = ρ D Σδ : ΣD B(D + ΣD). This is the denottionl comond D ρ coinduced by ρ. Let Σ freely generte mond T. In the next section, Theorem 7.1 shows tht liftings of the comond D to the Σ-lgebrs re in 1-1 correspondence with liftings of the mond T to the B-colgebrs. Therefore, if Σ corresponds to some progrm constructs nd B to some behviour, every nturl trnsformtion ρ s in (12) defines lso n opertionl mond, sy T ρ (with slight buse of nottion). As mentioned in 4, for the endofunctor BX = (P fi X) A on Set the djunction U B G B exists. The vlue of the corresponding cofree comond D = U B G B t set X is the set of globl behviours with sttes x in X. Formlly, it is quotient of the set of rooted, imge finite trees, with brnches lbelled by A, nd nodes lbelled by x X; the quotient is tken with respect to form of bisimultion tking into ccount the nme of the nodes [29, 13]. The counit ε : D Id is the opertion which extrcts the root from tree nd the comultipliction δ : D D 2 is the opertion which replces the nme of every node in tree by the subtree strting t tht node. Next, consider rules of type {z i i yi } i I {w j c σ(x 1,..., x n ) t b j }j J (13) where the x k, y i, z i, nd w j re ll vribles, nd I nd J re countble, possibly infinite index sets. It is convenient to consider the dependency grph [13] of such rule, nmely the directed grph hving the vribles of the rule s nodes, z i i yi, for i in I s positive b j edges, nd w j s negtive, trgetless edges. A rule of type (13) is well-founded if ll bckwrds chins of edges in its dependency grph re finite [13]. Definition 6.1 (Tree rules [10, 11]) A (simple negtive) tree rule is well-founded rule of type (13) such tht the x k nd the y i re ll distinct vribles nd re the only vribles occurring in the rule (ie the z i nd w j re ll occurrences of the x k nd y i ). A tree rule is sfe if the term t either is vrible x or is of the form σ (x 1,..., x m) for some opertor σ of the signture nd some (not necessrily distinct) vribles x 1,..., x m. Tree rules re more generl thn GSOS: they llow for lookhed, in tht one cn look not only t the

9 locl behviour ( single trnsition x y) of the sttes like in GSOS, but lso t the globl one, s in x b y y. (See [13] for some exmples.) The sfety restriction does not ffect the expressive power of the rules, provided one is llowed to dd sufficiently mny uxiliry opertors to the signture. A tree rule (13) hs the property tht its dependency grph is equl to the grph rechble from the nodes x 1,..., x n. Moreover, the subgrph rechble from node x k is tree the dependency tree with root x k. Let us cll set of tree rules llowed if it is n imge finite set (in the sense of 1) of tree rules whose dependency trees re imge finite. Then, n llowed set R of tree rules defines, for every X, function [[R] X : ΣDX (P fi T X) A (14) s follows. For ll t in T X, c in A, nd d k in DX, put t [[R]] X (σ(d 1,..., d n ))(c) if nd only if there exists (possibly renmed) rule (13) in R such tht the root of d k is x k, for 1 k n, nd the dependency trees of the rule cn be embedded in the d k (where the convention is tht tree with b j negtive edge w j cn be embedded into dk only if the vrible in d k corresponding to w j does not hve n outgoing edge lbelled by b j ). Theorem 6.1 (Tree rules re nturl) Let D be the comond cofreely generted by the endofunctor BX = (P fi X) A on Set. For every llowed set ρ of tree rules the function [[R]] X in (14) is nturl in X. Proof. Similr to the proof of nturlity in Theorem 1.1. Note the well-foundedness of tree rules is needed. For instnce, the non-well-founded rule with premise x x nd conclusion.x nil is not nturl becuse: first pplying [[.]] to (x y) nd then renming y s x yields {x}, while the sme opertions but in the reverse order yield {x, nil}, which fct violtes nturlity. In prticulr, if the rules in R re sfe, the nturl trnsformtion [[R]] is of type ΣD (P fi (Id + Σ)) A Therefore, for every llowed set R of sfe tree rules there exists trnsition system which stisfies the rules, nmely T ρ (0), where ρ = [[R]] nd T ρ is the corresponding opertionl mond. Contrrily to wht is stted in [11], this fils for (simple negtive) tree rules, s the filure to fit these ltter rules in the present theory brought to light. In fct, the sfe tree rules themselves hve been suggested to us by Rob vn Glbbeek s nturl subclss of (negtive) tree rules possessing stisfying trnsition system. 7 Combining Opertionl nd Denottionl Models When T is the mond freely generted by n endofunctor Σ on ctegory C, then one cn esily see tht the ctegory Σ-Alg of lgebrs of the endofunctor Σ is isomorphic to the ctegory T -Alg of lgebrs of the mond T = T, η, µ, with objects those morphisms h : T X X in C such tht h η X = id nd h T h = h µ X. Dully, the ctegory B-Colg of B- colgebrs is isomorphic to the ctegory D-Colg of colgebrs of the comond D cofreely generted by B [29, 7]. The results in this section should be red up to these two isomorphisms of ctegories Σ-Alg = T -Alg 7.1 Distributive Lws B-Colg = D-Colg Given mond T = T, η, µ nd comond D = D, ε, δ on ctegory C, distributive lw [7] of the mond T over the comond D is nturl trnsformtion λ : T D DT stisfying the lws λ η D = Dη nd their dul T ε = ε T λ λ µ D = Dµ λ T T λ Dλ λ D T δ = δ T λ The following theorem my well be folklore. Theorem 7.1 For mond T nd comond D on the sme ctegory, the following notions re mutully equivlent. Distributive lws λ of T over D. Liftings T of T to the D-colgebrs. Liftings D of D to the T -lgebrs. Proof. Given distributive lw λ, one cn define the corresponding liftings s follows. T λ (k) = λ X T k D λ (h) = Dh λ X Conversely, consider lifting T of the comond D to the T -lgebrs, hence U D T = T UD. By Lemm 1 in [15], this determines distributive

10 lw λ of the mond T over the endofunctor D s follows: first tke the nturl trnsformtion T ε : U D T GD = T U D G D = T D T, then trnspose it cross the djunction U D G D obtining λ : T G D G D T, nd finlly define λ to be U D λ : U D T GD = T D DT. It is esy to prove tht λ ctully is distributive lw over the whole comond D. Dully, given lifting D, tke η D : D DT = DU T F T = U T DF T, trnspose it cross F T U T obtining λ : F T D DF T, nd define λ to be U T λ : U T F T D = T D DT = U T DF T. The constructions re esily seen to be mutully inverse. When T is syntx nd D is (globl) behviour, the type of the distributive lw λ might thought of s the most generl type of well-behved rules. Note one cn lso consider monds T corresponding to lgebric theories, with equtions between the derived opertors. (See [29, 10] for n elementry exmple.) 7.2 Bilgebrs s Models Given distributive lw λ : T D DT, one cn consider the ctegory λ-bilg of λ-bilgebrs. Its objects re pirs T X h X k DX of T -lgebrs nd D- colgebrs with common crrier X which stisfy the following pentgonl lw : k h = Dh λ X T k (Cf [28].) This lw mkes h colgebr homomorphism nd k n lgebr homomorphism. The morphisms f : X, h, k X, h, k of λ-bilg re those morphisms f : X X between the crriers which re both T -lgebr nd D-colgebr homomorphisms. Remrk 7.1 The λ-bilgebrs re the sme s the lgebrs of the mond T λ of Theorem 7.1, nd, dully, the sme s the colgebrs of the comond D λ : T λ -Alg = λ-bilg = D λ -Colg Remrk 7.2 When λ is the distributive lw induced by finite set of concrete GSOS rules, the λ-bilgebrs re the GSOS-models of [25]. Given Σ-lgebr h : ΣX X, let h : T X X be its inductive extension to T -lgebr. When λ is induced by some bstrct opertionl rules ρ, no mtter whether of type (8) or the dul (12), λ-bilgebrs re equivlent to pirs ΣX h X k BX such tht k h = B(h ) ρ X Σ id, k (15) The lgebr structure cn be thought of s denottionl model, the colgebr structure s n opertionl model, nd the pentgonl lw (15) sys tht the combintion of the two models stisfies the rules ρ. Henceforth such bilgebrs re clled ρ-models. 7.3 Adequcy Met-Results Consider the forgetful functor U λ : λ-bilg D-Colg X, h, k X, k which forgets the lgebr structure of λ-bilgebr. Theorem 7.2 U λ hs left djoint, nmely: (X k DX) F λ (T 2 X µ X T X T λ(k) DT X) Proof. Dulise Theorem 4 of [15] nd pply it to D-Colg U D C U λ U T D λ -Colg T -Alg U Dλ where λ-bilg = D λ -Colg by Remrk 7.1. Corollry 7.1 The ctegory of λ-bilgebrs hs n initil object, nmely F λ 0, where 0 is the trivil initil D-colgebr. In prticulr, there exists n initil ρ-model, which cn be regrded s the intended opertionl model over the initil lgebr of progrms T 0. This implies tht every ρ-model M is dequte with respect to the intended opertionl model of ρ. Indeed, the unique ρ-model homomorphism to M given by initility is denottionl interprettion which preserves the behviourl distinctions of the intended opertionl model. This mkes M dequte. Now, consider the dul of U λ, nmely the functor U λ : λ-bilg T -Alg X, h, k X, h which forgets the D-colgebr structure of λ- bilgebr. Correspondingly, the following is dul to Theorem 7.2. Theorem 7.3 U λ hs right djoint, nmely: (T X h X) G λ (T DX D λ(h) DX δ X D 2 X) Corollry 7.2 The ctegory of λ-bilgebrs hs finl object, nmely G λ 1, where 1 is the trivil finl T -lgebr.

11 In prticulr, there exists finl ρ-model which is the cnonicl denottionl model for ρ; it hs the finl D-colgebr s crrier which, s mentioned in 4, is internlly fully bstrct with respect to B-bisimultion. The construction 1 G ϱ 1 = D1, D ϱ (1), δ 1 generlises the processes s terms construction of [22], which is systemtic method for deriving dequte denottionl models from tyft rules [13] ( clss of rules equivlent to the tree rules without negtive premises [10]). For more detils, see [30]. Corollry 7.3 The unique (both by initility nd finlity) homomorphism from the initil to the finl ρ- model is both the initil lgebr semntics nd the finl colgebr semntics for ρ. The bove, sy, universl semntics for ρ is thus compositionl interprettion of the progrms which preserves their behviourl distinctions. In Set, the ltter mens tht two progrms with the sme universl semntics re B-bisimilr. One cn esily see tht, under the dditionl hypothesis tht B preserves wek pullbcks, the converse lso holds: two progrms hve the sme universl semntics if nd only if they re B-bisimilr. In other words: Corollry 7.4 If B preserves wek pullbcks, the universl semntics ssocited to some bstrct rules ρ is fully bstrct with respect to B-bisimultion. Next, recll Definition 4.1: by replcing spns of colgebr homomorphisms with spns of T -lgebr homomorphisms, one hs corresponding notion of T - congruence which specilises to the ordinry notion of congruence. Similrly, by considering spns of λ- bilgebr homomorphisms one hs notion of, sy, λ- bicongruence nd corresponding ctegory. We cn sk then whether there exists finl bicongruence for λ-bilgebr. Now, if pullbcks of cospns of crriers of B-colgebrs re B-bisimultions, then, by the universl property of pullbcks, finl B-bisimultion between two colgebrs exists: it is the pullbck of the respective unique colgebr homomorphisms to the finl colgebr. This is T -congruence s well, becuse the forgetful functor U T : T -Alg C cretes limits. Therefore, by definition of finl bilgebr: Corollry 7.5 If B preserves wek pullbcks, then every λ-bilgebr hs finl bicongruence. In prticulr, the behviourl endofunctor B in (2) preserves wek pullbcks, hence the bove corollry specilises to the well-known fct tht (strong) bisimultion is congruence for GSOS nd tree rules. Future Work The mjor chllenge hed is the opertionl semntics of the lnguges with vrible binders, such s the π-clculus nd the λ-clculus. (At the moment, by working in suitble functor ctegory, we re ble to give functoril description of syntx with vrible binders, but it is not yet cler whether this fits our purposes.) We would lso like to obtin dequcy results when working in ctegories of prtil mps. There is n obvious question bout Moggi s computtionl monds [19] nd our behviour functors which remins to be investigted. In different direction, we would like to understnd the reltionship between the trnsitionl pproch considered here nd others, such s the the reductionl one rising in the λ-clculus nd term-rewriting in generl. Further developments of the present theory could led to pplictions in modulr compiler development technology. Perhps there will be useful theory of the combintion of opertionl semntics of different lnguges (cf [18]). Agin, perhps one cn relte the opertionl semntics of lnguge with tht of its trnsltion into nother trget lnguge (cf [26]). Acknowledgements. Thnks to Mrcelo Fiore nd Alex Simpson for discussions. Prt of this study is bsed on the first uthor s thesis; he wishes to thnk Jco de Bkker nd Brt Jcobs for their guidnce. References [1] S. Abrmsky. A domin eqution for bisimultion. Informtion nd Computtion, 92: , [2] P. Aczel. Non-well-founded sets. Number 14 in Lecture Notes. CSLI, [3] P. Aczel. Finl universes of processes. In Mthemticl Foundtions of Progrmming Semntics, Proc. 9th Int. Conf., volume 802 of LNCS, pges Springer-Verlg, [4] P. Aczel nd N. Mendler. A finl colgebr theorem. In D.H. Pitt et l., editors, Proc. ctegory theory nd computer science, volume 389 of LNCS, pges Springer-Verlg, [5] J.C.M. Beten nd W.P. Weijlnd. Process Algebr. Cmbridge University Press, [6] M. Brr. Terminl colgebrs in well-founded set theory. Theoreticl Computer Science, 144(2): , 1993.

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