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1 PDF hosted t the Rdboud Repository of the Rdboud University Nijmegen The following full text is publisher's version. For dditionl informtion bout this publiction click this link. Plese be dvised tht this informtion ws generted on nd my be subject to chnge.

2 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN Rdboud University Nijmegen nd Centrum Wiskunde & Informtic e-mil ddress: ENS Lyon, Université de Lyon, LIP (UMR 5668 CNRS ENS Lyon UCBL INRIA) e-mil ddress: LIACS - Leiden University e-mil ddress: mrcello@lics.nl Centrum Wiskunde & Informtic nd Rdboud University Nijmegen e-mil ddress: jnr@cwi.nl Abstrct. The powerset construction is stndrd method for converting nondeterministic utomton into deterministic one recognizing the sme lnguge. In this pper, we lift the powerset construction from utomt to the more generl frmework of colgebrs with structured stte spces. Colgebr is n bstrct frmework for the uniform study of different kinds of dynmicl systems. An endofunctor F determines both the type of systems (F -colgebrs) nd notion of behviourl equivlence ( F ) mongst them. Mny types of trnsition systems nd their equivlences cn be cptured by functor F. For exmple, for deterministic utomt the derived equivlence is lnguge equivlence, while for non-deterministic utomt it is ordinry bisimilrity. We give severl exmples of pplictions of our generlized determiniztion construction, including prtil Mely mchines, (structured) Moore utomt, Rbin probbilistic utomt, nd, somewht surprisingly, even pushdown utomt. To further witness the generlity of the pproch we show how to chrcterize colgebriclly severl equivlences which hve been object of interest in the concurrency community, such s filure or redy semntics ACM Subject Clssifiction: F.3.2. Key words nd phrses: Colgebrs, Powerset Construction, Liner Semntics. The work of Alexndr Silv is prtilly funded by the ERDF through the Progrmme COMPETE nd by the Portuguese Foundtion for Science nd Technology, project ref. PTDC/EIA-CCO/122240/2010 nd SFRH/BPD/71956/2010. The work of Filippo Bonchi is supported by the CNRS PEPS project CoGIP nd the project ANR 12IS02001 PACE. The reserch of Mrcello Bonsngue nd Jn Rutten hs been crried out under the Dutch NWO project CoRE: Coinductive Clculi for Regulr Expressions., dossier number LOGICAL METHODS IN COMPUTER SCIENCE DOI: /LMCS-??? c Alexndr Silv, Filippo Bonchi, Mrcello Bonsngue, nd Jn Rutten Cretive Commons 1

3 2 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN Introduction Colgebr is by now well estblished generl frmework for the study of the behviour of lrge clsses of dynmicl systems, including vrious kinds of utomt (deterministic, probbilistic etc.) nd infinite dt types (strems, trees nd the like). For functor F : Set Set, n F -colgebr is pir (X, f), consisting of set X of sttes nd function f : X F (X) defining the observtions nd trnsitions of the sttes. Colgebrs generlly come equipped with stndrd notion of equivlence clled F -behviourl equivlence tht is fully determined by their (functor) type F. Moreover, for most functors F there exists finl colgebr into which ny F -colgebr is mpped by unique homomorphism tht identifies ll F -equivlent sttes. Much of the colgebric pproch cn be nicely illustrted with deterministic utomt (DA), which re colgebrs of the functor D(X) = 2 X A. In DA, two sttes re D- equivlent precisely when they ccept the sme lnguge. The set 2 A of ll forml lnguges constitutes finl D-colgebr, into which every DA is mpped by homomorphism tht sends ny stte to the lnguge it ccepts. It is well-known tht non-deterministic utomt (NDA) often provide more efficient (smller) representtions of forml lnguges thn DA s. Lnguge cceptnce of NDA s is typiclly defined by turning them into DA s vi the powerset construction. Colgebriclly this works s follows. NDA s re colgebrs of the functor N(X) = 2 P ω (X) A, where P ω is the finite powerset. An N-colgebr (X, f : X 2 P ω (X) A ) is determinized by trnsforming it into D-colgebr (P ω (X), f : P ω (X) 2 P ω (X) A ) (for detils see Section 2). Then, the lnguge ccepted by stte s in the NDA (X, f) is defined s the lnguge ccepted by the stte s} in the DA (P ω (X), f ). For second vrition on DA s, we look t prtil utomt (PA): colgebrs of the functor P (X) = 2 (1 + X) A, where for certin input letters trnsitions my be undefined. Agin, one is often interested in the DA-behviour (i.e., lnguge cceptnce) of PA s. This cn be obtined by turning them into DA s using totliztion. Colgebriclly, this mounts to the trnsformtion of P -colgebr (X, f : X 2 (1+X) A ) into D-colgebr (1 + X, f : 1 + X 2 (1 + X) A ). Although the two exmples bove my seem very different, they re both instnces of one nd the sme phenomenon, which it is the gol of the present pper to describe t generl level. Both with NDA s nd PA s, two things hppen t the sme time: (i) more (or, more generlly, different types of) trnsitions re llowed, s consequence of chnging the functor type by replcing X by P ω (X) nd (1 + X), respectively; nd (ii) the behviour of NDA s nd PA s is still given in terms of the behviour of the originl DA s (lnguge cceptnce). For lrge fmily of F -colgebrs, both (i) nd (ii) cn be cptured simultneously with the help of the ctegoricl notion of mond, which generlizes the notion of lgebric theory. The structuring of the stte spce X cn be expressed s chnge of functor type from F (X) to F (T (X)). In our exmples bove, both the functors T 1 (X) = P ω (X) nd T 2 (X) = 1 + X re monds, nd NDA s nd PA s re obtined from DA s by chnging the originl functor type D(X) into N(X) = D(T 1 (X)) nd P (X) = D(T 2 (X)). Regrding (ii), one ssigns F -semntics to n F T -colgebr (X, f) by trnsforming it into n F -colgebr (T (X), f ), gin using the mond T. In our exmples bove, the determiniztion of NDA s nd the totliztion of PA s consists of the trnsformtion of N- nd P -colgebrs (X, f) into D-colgebrs (T 1 (X), f ) nd (T 2 (X), f ), respectively.

4 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS 3 We shll investigte generl conditions on the functor types under which the bove constructions cn be pplied: for one thing, one hs to ensure tht the F T -colgebr mp f : X F (T (X)) induces suitble F -colgebr mp f : T (X) F (T (X)). Our results will led to uniform tretment of ll kinds of existing nd new vritions of utomt, tht is, F T -colgebrs, by n lgebric structuring of their stte spce through mond T. Furthermore, we shll prove number of generl properties tht hold in ll situtions similr to the ones bove. For instnce, there is the notion of N-behviourl equivlence with which NDA s, being N-colgebrs, come equipped. It coincides with the well-known notion of Prk-Milner bisimilrity from process lgebr. A generl observtion is tht if two sttes in n NDA re N-equivlent then they re lso D- (tht is, lnguge-) equivlent. For PA s, similr sttement holds. One further contribution of this pper is proof of these sttements, once nd for ll for ll F T -colgebrs under considertion. Colgebrs of type F T were studied in [29, 4, 22]. In [4, 22] the min concern ws definitions by coinduction, wheres in [29] proof principle ws lso presented. All in ll, the present pper cn be seen s the understnding of the forementioned ppers from new perspective, presenting uniform view on vrious utomt constructions nd equivlences. The structure of the pper is s follows. After preliminries (Section 1) nd the detils of the motivting exmples bove (Section 2), Section 3 presents the generl construction s well s mny more exmples, including the colgebric chrcteristion of pushdown utomt (Section 3.2). In Section 4, lrge fmily of utomt (techniclly: functors) is chrcterised to which the constructions bove cn be pplied. Section 5 contins the ppliction of the frmework in order to recover severl interesting equivlences stemming from the world of concurrency, such s filure nd redy semntics. Section 6 discusses relted work nd presents pointers to future work. This pper is n extended version of [43]. Compred to the conference version, we include the proofs nd more exmples. More interestingly, the chrcteristion of pushdown utomt colgebriclly (Section 3.2) nd the mteril in Section 5 re originl. 1. Bckground In this section we introduce the preliminries on colgebrs nd lgebrs. First, we fix some nottion on sets. We will denote sets by cpitl letters X, Y,... nd functions by lower cse letters f, g,... Given sets X nd Y, X Y is the crtesin product of X nd Y (with the usul projection mps π 1 nd π 2 ), X + Y is the disjoint union (with injection mps κ 1 nd κ 2 ) nd X Y is the set of functions f : Y X. The collection of finite subsets of X is denoted by P ω (X), while the collection of full-probbility distributions with finite support is D ω (X) = f : X [0, 1] f finite support nd x X f(x) = 1}. For set of letters A, A denotes the set of ll words over A; ɛ the empty word; nd w 1 w 2 (nd w 1 w 2 ) the conctention of words w 1, w 2 A Colgebrs. A colgebr is pir (X, f : X F (X)), where X is set of sttes nd F : Set Set is functor. The functor F, together with the function f, determines the trnsition structure (or dynmics) of the F -colgebr [37]. An F -homomorphism from n F -colgebr (X, f) to n F -colgebr (Y, g) is function h: X Y preserving the trnsition structure, i.e., g h = F (h) f.

5 4 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN An F -colgebr (Ω, ω) is sid to be finl if for ny F -colgebr (X, f) there exists unique F -homomorphism [[ ]] X : X Ω. All the functors considered in exmples in this pper hve finl colgebr. Let (X, f) nd (Y, g) be two F -colgebrs. We sy tht the sttes x X nd y Y re behviourlly equivlent, written x F y, if nd only if they re mpped into the sme element in the finl colgebr, tht is [[x]] X = [[y ] Y. For wek pullbck preserving functors, behviourl equivlence coincides with the usul notion of bisimilrity [37] Algebrs. Monds cn be thought of s generliztion of lgebric theories. A mond T = (T, µ, η) is triple consisting of n endofunctor T on Set nd two nturl trnsformtions: unit η : Id T nd multipliction µ: T 2 T. They stisfy the following commuttive lws µ η T = id T = µ T η nd µ µ T = µ T µ. Sometimes it is more convenient to represent mond T, equivlently, s Kleisli triple (T, ( ), η) [31], where T ssigns set T (X) to ech set X, the unit η ssigns function η X : X T (X) to ech set X, nd the extension opertion ( ) ssigns to ech f : X T (Y ) function f : T (X) T (Y ), such tht, f η X = f (η X ) = id T (X) (g f) = g f, for g : Y T (Z). Monds re frequently referred to s computtionl types [32]. We list now few exmples. In wht follows, f : X T (Y ) nd c T (X). Nondeterminism. T (X) = P ω (X); η X is the singleton mp x x}; f (c) = x c f(x). Prtility. T (X) = 1 + X where 1 = } represents terminting (or diverging) computtion; η X is the injection mp κ 2 : X 1 + X; f (κ 1 ( )) = κ 1 ( ) nd f (κ 2 (x)) = f(x). Further exmples of monds include: exceptions (T (X) = E + X), side-effects (T (X) = (S X) S ), interctive output (T (X) = µv.x + (O v) = O X) nd full-probbility (T (X) = D ω (X)). We will use ll these monds in our exmples nd we will define η X nd f for ech lter in Section 3.1. A T-lgebr of mond T is pir (X, h) consisting of set X, clled crrier, nd function h: T (X) X such tht h µ X = h T h nd h η X = id X. A T -homomorphism between two T-lgebrs (X, h) nd (Y, k) is function f : X Y such tht f h = k T f. T-lgebrs nd their homomorphisms form the so-clled Eilenberg-Moore ctegory Set T. There is forgetful functor U T : Set T Set defined by U T ((X, h)) = X nd U T (f : (X, h) (Y, k)) = f : X Y. The forgetful functor U T hs left djoint X (T (X), µ X : T T (X) T (X)), mpping set X to its free T-lgebr. If f : X Y with (Y, h) T-lgebr, the unique T-homomorphism f : (T (X), µ X ) (Y, h) with f η X = f is given by f : T (X) T f T (Y ) h Y. The function f : (T (X), µ X ) (T (Y ), µ Y ) coincides with function extension for Kleisli triple. For the mond P ω the ssocited Eilenberg-Moore ctegory is the ctegory of join semi-lttices, wheres for the mond 1 + is the ctegory of pointed sets.

6 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS 5 2. Motivting exmples In this section, we introduce two motivting exmples. We will present two constructions, the determiniztion of non-deterministic utomton nd the totliztion of prtil utomton, which we will lter show to be n instnce of the sme, more generl, construction Non-deterministic utomt. A deterministic utomton (DA) over the input lphbet A is pir (X, o, t ), where X is set of sttes nd o, t : X 2 X A is function with two components: o, the output function, determines if stte x is finl (o(x) = 1) or not (o(x) = 0); nd t, the trnsition function, returns for ech input letter the next stte. DA s re colgebrs for the functor 2 Id A. The finl colgebr of this functor is (2 A, ɛ, ( ) ) where 2 A is the set of lnguges over A nd ɛ, ( ), given lnguge L, determines whether or not the empty word is in the lnguge (ɛ(l) = 1 or ɛ(l) = 0, resp.) nd, for ech input letter, returns the derivtive of L: L = w A w L}. From ny DA, there is unique mp l into 2 A which ssigns to ech stte its behviour (tht is, the lnguge tht the stte recognizes). X l 2 A o,t ɛ,( ) 2 X A 2 (2 A ) A id l A A non-deterministic utomton (NDA) is similr to DA but the trnsition function gives set of next-sttes for ech input letter insted of single stte. Thus, n NDA over the input lphbet A is pir (X, o, δ ), where X is set of sttes nd o, δ : X 2 (P ω (X)) A is pir of functions with o s before nd where δ determines for ech input letter set of possible next sttes. In order to compute the lnguge recognized by stte x of n NDA A, it is usul to first determinize it, constructing DA det(a) where the stte spce is P ω (X), nd then compute the lnguge recognized by the stte x} of det(a). Next, we describe in colgebric terms how to construct the utomton det(a). Given n NDA A = (X, o, δ ), we construct det(a) = (P ω (X), o, t ), where, for ll Y P ω (X), A, the functions o: P ω (X) 2 nd t: P ω (X) P ω (X) A re 1 y Y o(y) = 1 o(y ) = t(y )() = δ(y)(). 0 otherwise (Observe tht these definitions exploit the join-semilttice structures of 2 nd P ω (X) A ). The utomton det(a) is such tht the lnguge l(x}) recognized by x} is the sme s the one recognized by x in the originl NDA A (more generlly, the lnguge recognized by stte X of det(a) is the union of the lnguges recognized by ech stte x of A). y Y

7 6 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN We summrize the sitution bove with the following commuting digrm: } X P ω (X) 2 A o,δ o,t 2 P ω (X) A 2 (2 A ) A id l A l ɛ,( ) We note tht the lnguge semntics of NDA s, presented in the bove digrm, cn lso be obtined s n instnce of the bstrct definition scheme of λ-coinduction [4, 22] Prtil utomt. A prtil utomton (PA) over the input lphbet A is pir (X, o, ) consisting of set of sttes X nd pir of functions o, : X 2 (1 + X) A. Here o: X 2 is the sme s with DA. The second function : X (1+X) A is trnsition function tht sends ny stte x X to function (x): A 1 + X, which for ny input letter A is either undefined (no -lbelled trnsition tkes plce) or specifies the next stte tht is reched. PA s re colgebrs for the functor 2 (1 + Id) A. Given PA A, we cn construct totl (deterministic) utomton tot(a) by dding n extr sink stte to the stte spce: every undefined -trnsition from stte x is then replced by -lbelled trnsition from x to the sink stte. More precisely, given PA A = (X, o, ), we construct tot(a) = (1 + X, o, t ), where o(κ 1 ( )) = 0 o(κ 2 (x)) = o(x) t(κ 1 ( ))() = κ 1 ( ) t(κ 2 (x))() = (x)() (Observe tht these definitions exploit the pointed-set structures of 2 nd 1 + X). The lnguge l(x) recognized by stte x will be precisely the lnguge recognized by x in the originl prtil utomton. Moreover, the new sink stte recognizes the empty lnguge. Agin we summrize the sitution bove with the help of following commuting digrm, which illustrtes the similrities between both constructions: κ 2 X 1 + X 2 A o, o,t 2 (1 + X) A 2 (2 A ) A id l A l ɛ,( ) 3. Algebriclly structured colgebrs In this section we present generl frmework where both motivting exmples cn be embedded nd uniformly studied. We will consider colgebrs for which the functor type F T cn be decomposed into trnsition type F specifying the relevnt dynmics of system nd mond T providing the stte spce with n lgebric structure. For simplicity, we fix our bse ctegory to be Set. We study colgebrs f : X F T (X) for functor F nd mond T such tht F T (X) is T-lgebr, tht is F T (X) is the crrier of T-lgebr (F T (X), h). In the motivting

8 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS 7 exmples, F would be instntited to 2 Id A (in both) nd T to P ω, for NDAs, nd to 1 + for PAs. The condition tht F T (X) is T-lgebr would mount to require tht 2 P ω (X) A is join-semilttice, for NDAs, nd tht 2 (1 + X) A is pointed set, for PAs. This is indeed the cse, since the set 2 cn be regrded both s join-semilttice (2 = P ω (1)) or s pointed set (2 = 1 + 1) nd, moreover, products nd exponentils preserve the lgebr structure. The inter-ply between the trnsition type F nd the computtionl type T (more precisely, the fct tht F T (X) is T-lgebr) llows ech colgebr f : X F T (X) to be extended uniquely to T -lgebr morphism f : (T (X), µ X ) (F T (X), h) which mkes the following digrm commute. η X X T (X) f f F T (X) f η X = f Intuitively, η X : X T (X) is the inclusion of the stte spce of the colgebr f : X F T (X) into the structured stte spce T (X), nd f : T (X) F T (X) is the extension of the colgebr f to T (X). Next, we study the behviour of given stte or, more generlly, we would like to sy when two sttes x 1 nd x 2 re equivlent. The obvious choice for n equivlence would be F T -behviourl equivlence. However, this equivlence is not exctly wht we re looking for. In the motivting exmple of non-deterministic utomt we wnted two sttes to be equivlent if they recognize the sme lnguge. If we would tke the equivlence rising from the functor 2 P ω (Id) A we would be distinguishing sttes tht recognize the sme lnguge but hve difference brnching types, s in the following exmple. b c b We now define new equivlence, which bsorbs the effect of the mond T. We sy tht two elements x 1 nd x 2 in X re F -equivlent with respect to mond T, written x 1 T F x 2, if nd only if η X (x 1 ) F η X (x 2 ). The equivlence F is just F - behviourl equivlence for the F -colgebr f : T (X) F T (X). If the functor F hs finl colgebr (Ω, ω), we cn cpture the semntic equivlence bove in the following commuting digrm η X [[ ]] X T (X) Ω f ω f F T (X) F (Ω) F [[ ]] c (3.1)

9 8 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN Returning to our first exmple, two sttes x 1 nd x 2 of n NDA (in which T is instntited to P ω nd F to 2 Id A ) would stisfy x 1 T F x 2 if nd only if they recognize the sme lnguge (recll tht the finl colgebr of the functor 2 Id A is 2 A ). It is lso interesting to remrk the difference between the two equivlences in the cse of prtil utomt. The colgebric semntics of PAs [39] is given in terms of pirs of prefix-closed lnguges V, W where V contins the words tht re ccepted (tht is, re the lbel of pth leding to finl stte) nd W contins ll words tht lbel ny pth (tht is ll tht re in V plus the words lbeling pths leding to non-finl sttes). We describe V nd W in the following two exmples, for the sttes s 0 nd q 0 : W = c + c b + c b V = c b c s 0 b s 2 s 1 b c q 0 q 1 b W = c + c b V = c b Thus, the sttes s 0 nd q 0 would be distinguished by F T -equivlence (for F = 2 Id A nd T = 1 + ) but they re equivlent with respect to the mond 1 +, s 0 T F q 0, since they ccept the sme lnguge. We will show in Section 4 tht the equivlence F T is lwys contined in T F Exmples. In this section we show more exmples of pplictions of the frmework bove Prtil Mely mchines. A prtil Mely mchine is set of sttes X together with function t: X (B (1 + X)) A, where A is set of inputs nd B is set of output vlues. We ssume tht B hs distinguished element B. For ech stte x nd for ech input the utomton produces n output vlue nd either termintes or continues to next stte. Applying the frmework bove we will be totlizing the utomton, similrly to wht hppened in the exmple of prtil utomt, by dding n extr stte to the stte spce which will ct s sink stte. The behviour of the totlized utomton is given by the set of cusl functions from A ω (infinite sequences of A) to B ω, which we denote by Γ(A ω, B ω ) [38]. A function f : A ω B ω is cusl if, for σ A ω, the n-th vlue of the output strem f(σ) depends only on the first n vlues of the input strem σ. In the digrm below, we define the finl mp [[ ]: 1 + X Γ(A ω, B ω ): κ 2 X 1 + X Γ(A ω, B ω ) [[κ 1 ( )]](σ) = (,,...) t [[κ 2 (x)]]( : τ) = b : ([[z]](τ)) t where t(x)() = b, z (B (1 + X)) A (B Γ(A ω, B ω )) A Here 1, x X, A, b B, σ A ω, z 1 + X, nd :τ denotes the prefixing of the strem τ A ω with the element. [[ ]]

10 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS Structured Moore utomt. In the following exmples we look t the functor F (X) = T (B) X A for rbitrry sets A nd B nd n rbitrry mond T = (T, η, ( ) ). The colgebrs of F represents Moore utomt with outputs in T (B) nd inputs in A. Since T (B) is T-lgebr, T (X) A is T-lgebr nd the product of T-lgebrs is still T-lgebr, then F T (X) is T-lgebr. For this reson, the (pir of) functions o: X T (B) nd t: X T (X) A lift to (pir of) functions o : T (X) T (B) t : T (X) T (X) A The finl colgebr of F is T (B) A. We cn chrcterize the finl mp [[ ]: T (X) T (B) A, for ll m T (X), A nd w A, by η X X T (X) T (B) A o,t [[m]](ɛ) = o (m) o,t [[m]]( w) = [[t ɛ,( ) (m)()]](w) T (B) T (X) A T (B) (T (B) A ) A Below we shll look t vrious concrete instnces of this scheme, for different choices of the mond T. Moore utomt with exceptions. Let E be n rbitrry set, the elements of which we think of s exceptions. We consider the exception mond T (X) = E + X which hs the function η(x) = κ 2 (x) s its unit. We define the lifting f : T (X) T (Y ), for ny function f : X T (Y ), by f = [id, f]. An F T -colgebr o, t : X (E + B) (E + X) A will ssocite with every stte x n output vlue (either in B or n exception in E) nd, for ech input, next stte or n exception. The behviour of stte x, given by [[η(x)]], will be forml power series over A with output vlues in E + B; tht is, function from A to E + B. The finl mp is defined s follows, for ll e E, x X, A, nd w A : κ 2 X E + X (E + B) A o,t [κ 1 (e)]](w) = κ 1 (e) o,t [κ 2 (x)]](ɛ) = o(x) [κ 2 (x)]]( w) = [[t(x)()](w) (E + B) (E + X) A (E + B) ((E + B) A ) A [[ ]] [[ ]] Moore utomt with side effects. Let S be n rbitrry set of so-clled side-effects. We consider the mond T (X) = (S X) S, with unit η defined, for ll x X nd s S, by η(x)(s) = s, x. We define the lifting f : T (X) T (Y ) of function f : X T (Y ) by f (g)(s) = f(x)(s ), for ny g T (X) nd s S, nd with g(s) = s, x. Consider n F T -colgebr o, t : X (B S) S ((S X) S ) A nd let us explin the intuition behind this type of utomton type. The set S X cn be interpreted s the configurtions of the utomton, where S contins informtion bout the stte of the system nd X bout the control of the system. Using the isomorphism X (S B) S =

11 10 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN S X S B, we cn think of o: X (S B) S s function tht for ech configurtion in S X provides n output in B nd the new stte of the system in S. The trnsition function t: X ((S X) S ) A gives new configurtion for ech input letter nd current configurtion, using gin the fct tht X ((S X) S ) A = S X (S X) A. In ll of this, concrete instnce of the set of side-effects could be, for exmple, the set S = V L of functions ssociting memory loctions to vlues. The behviour of stte x X will be given by [[η(x)]], where the finl mpping is s follows. For ll g (S X) S, s S, A nd w A, nd with g(s) = s, x, we hve η X (S X) S ((B S) S ) A o,t [g]](ɛ)(s) = o(x)(s ) o,t [g]]( w) = [[λs.t(x)()(s )](w) (B S) S ((S X) S ) A (B S) S (((B S) S ) A ) A Moore utomt with interctive output. Let O be n rbitrry set of outputs. Consider the interctive output mond defined by the functor T (X) = µv.x +(O v) = O X together with the nturl trnsformtion η X = λx X. ɛ, x, nd for which the lifting f : T (X) T (Y ) of function f : X T (Y ) is given by f ( w, x ) = ww, y with f(x) = w, y. We consider F T -colgebrs [[ ]] o, t : X (O B) (O X) A For B = 1, the bove colgebrs coincide with (totl) subsequentil trnsducers [17]: o: X O is the finl output function; t: X (O X) A is the piring of the output function nd the next stte-function. The behviour of stte x will be given by [[η(x)]] = [[ ɛ, x ]], where, for every w, x O X, [[ w, x ]]: A O, is given by [[ w, x ]](ɛ) = w o(x) [[ w, x ]](w 1 ) = w ([[t(x)()](w 1 )) Probbilistic Moore utomt. Consider the mond of probbility distributions defined, for ny set X, by T (X) = D ω (X) Its unit is given by the Dirc distribution, defined for x, x X by η(x)(x 1 x = x ) = 0 otherwise The lifting f : T (X) T (Y ) of function f : X T (Y ) is given, for ny distribution c D ω (X) nd ny y Y, by f (c)(y) = c(x) d(y) We will consider F T -colgebrs d D ω(y ) x f 1 (d) o, t : X D ω (B) D ω (X) A

12 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS 11 More specificlly, we tke B = 2 which implies D ω (2) = [0, 1]. For this choice of B, the bove F T -colgebrs re precisely the (Rbin) probbilistic utomt [36]. Ech stte x hs n output vlue in o(x) [0, 1] nd, for ech input, t(x)() is probbility distribution of next sttes. The behviour of stte x is given by [[η(x)]]: A [0, 1], defined below. Intuitively, one cn think of [[η(x)]] s probbilistic lnguge: ech word is ssocited with vlue p [0, 1]. The finl mpping η X o,t D ω (X) [0, 1] A o,t [0, 1] D ω (X) A [0, 1] ([0, 1] A ) A is given, for ny d D ω (X), x X, A, nd w A, by [[d]](ɛ) = ( d(x)) b b [0,1] o(x)=b [[d]](w) = [[λx. ( b=t(x)() d(x)) c(x )]](w) c D ω (X) It is worth noting tht this exctly cptures the semntics of [36], while the ordinry F T coincides with probbilistic bisimilrity of [28]. Moreover T F coincides with the trce semntics of probbilistic trnsition systems defined in [19] (see Section 7.2 of [23]) Pushdown utomt, colgebriclly. Recursive functions in computer progrm led nturlly to stck of recursive function clls during the execution of the progrm. In this section, we provide colgebric model of utomt equipped with stck memory. A pushdown mchine is tuple (Q, A, B, δ), where Q is set of control loctions (sttes), A is set of input symbols, B is set of stck symbols, nd δ is finite subset of Q A B Q B, clled the set of trnsition rules. Note tht we do not insist on the sets Q, A nd B to be finite nd consider only reltime pushdown mchines, i.e. without internl trnsitions (lso clled ɛ-trnsitions) [21]. A configurtion k of pushdown mchine is pir q, β denoting the current control stte q Q nd the current content of the stck β B. In denoting the stck s string of stck symbols we ssume tht the topmost symbol is written first. There is trnsition q, bβ q, αβ if q, α δ(q,, b). A convenient nottion is to introduce for ny string w A the trnsition reltion on configurtions s the lest reltion such tht (1) k ɛ k (2) k w k if nd only if k k nd k w k. A pushdown utomton (pd) is pushdown mchine together with n initil configurtion k 0 nd set K of ccepting configurtions. The sets of ccepting configurtions usully considered re (1) the set F B, where F Q is clled the set of ccepting sttes, or (2) Q ɛ}, but lso (3) F ɛ} for F Q, or (4) Q B B for B subset of B. A word w A w is sid to be ccepted by pd (Q, A, B, δ, k 0, K) if k 0 k for some k K. A pd with ccepting configurtions s in (1) is sid to be with ccepting sttes, wheres, when they re s in (2) then the pd is sid to be ccepting by empty stck. They both [[ ]]

13 12 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN ccept exctly proper context free lnguges (i.e. context free lnguges without the empty word) [3]. Computtions in pushdown mchine re generlly non-deterministic nd cn cuse chnge in the control stte of the utomton s well s in its stck. For this reson we will model the effects of the computtions by mens of the so-clled non-deterministic side-effect mond [5]. For set of sttes S, let T be the functor P ω ( S) S. It is mond when equipped with the unit η X : X T (X), defined by η(x)(s) = x, s }, nd the multipliction µ X : T (T (X)) T (X) given by µ X (k)(s) = c(s ) c,s k(s) Note tht, for function f : X T (Y ), the extension f : T (X) T (Y ) is defined by f (c)(s) = f(x )(s ). x,s c(s) Exmples of lgebrs for this mond re T (1) = P ω (S) S nd 2 S. The ltter cn in fct be obtined s quotient of the former by equting those functions k 1, k 2 : S P ω (S) such tht for ll s S, k 1 (s) = if nd only if k 2 (s) =. Every pushdown mchine (Q, A, B, δ) together with set of ccepting configurtions K induces function o, t : Q F T Q where F is the functor 2 B id A nd T is the mond defined bove specilized for S = B (intuitively, side effects in pushdown mchine re chnges in its stck). The functions o: Q 2 B nd t: Q P ω (Q B ) B A re defined s o(q)(β) = 1 if nd only if q, β K t(q)()(ɛ) = t(q)()(bβ) = q, αβ q, α δ(q,, b)} The trnsition function t describes the steps between pd configurtions nd it is specified in terms of the trnsition instructions δ of the originl mchine. From the bove is cler tht not every function o, t : Q F T Q defines pushdown mchine with ccepting configurtions, s, for exmple, t(q) my depend on the whole stck β nd not just on the top element b. Therefore we restrict our ttention to consider functions o, t : Q F T Q such tht (1) t(q)()(ɛ) = (2) t(q)()(bβ) = q, αβ q, α t(q)()(b)}, Every o, t stisfying (1) nd (2) bove defines the pushdown mchine (Q, A, B, δ) with δ(q,, b) = t(q)()(b) nd with ccepting configurtion K = q, β o(q)(β) = 1}. The first condition is sserting tht mchine is in dedlock configurtion when the stck is empty, while the lst condition ensures tht trnsition steps depend only on the control stte nd the top element of the stck. For this reson we will write q,b α q for q, αβ t(q)()(b) indicting tht the pushdown mchine in the stte q by reding n input symbol nd popping b off the stck, cn move to control stte q pushing the string α B on the current stck (here denoted by β). Similrly to wht we hve shown in the exmples of structured Moore utomt, for every function o, t : Q F T Q there is unique F -colgebr mp [[ ]]: T (Q) 2 B A, which is lso T -lgebr homomorphism. It is defined for ll c P ω (Q B ) B nd β B

14 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS 13 s η Q P ω (Q B ) B 2 B A o,t o,t 2 B P ω (Q B ) B A 2 B 2 B A A [[ ]] [[η(q)](ɛ) = o(q) [[η(q)](w) = [[λβ.t(q)()(β)]](w) [[c]](β) = [[η(q)]](α). q,α c(β) We then hve tht word w A is ccepted by the pd (Q, A, B, δ, k 0, K) with k 0 = q, β if nd only if [[η(q)](w)(β) = 1. The bove definition implies tht for given word w A we cn decide if it is ccepted by o, t : Q F T Q from n initil configurtion k 0 = q, β in exctly w steps (ssuming there is procedure to decide whether o(q)(β) = 1). As consequence, we cnnot use structured Moore utomt to model Turing mchines, for which the hlting problem is undecidble: in generl terms, for Turing mchines, we would need internl trnsitions tht do not consume input symbols. We conclude with n exmple of our construction using pushdown mchine with control sttes Q = q 0, q 1 }, over n input lphbet A =, b} nd using stck symbols B = x, s}. The trnsitions rules δ re given below:,s x q 0 b,x ɛ,x xx We tke K = q 0, ɛ, q 1, ɛ }, mening tht o(q 0 )(ɛ) = 1, o(q 1 )(ɛ) = 1 nd o(q i )(β) = 0 in ll other cses. By considering k 0 = q 0, s s initil configurtion, we then hve q 1 b,x ɛ [[η(q 0 )]](ɛ)(s) = o(q 0 )(s) = 0 mening tht the empty word is not ccepted by the pd (Q, A, B, δ, k 0, K). However, the word b is ccepted: [[η(q 0 )](b)(s) = [[λβ.t(q 0 )()(β)]](b)(s) = [[η(p)](b)(β) p,β t(q 0 )()(s) = [[η(q 1 )]](b)(x) = [[λβ.t(q 1 )(b)(β)]](ɛ)(x) = [[η(p)]](ɛ)(β) p,β t(q 1 )(b)(x) = [[η(q 1 )]](ɛ)(ɛ) = o(q 1 )(ɛ) = 1. In fct, the lnguge ccepted by the bove pushdown utomton is n b n n 1}. The structured sttes c i T Q, their trnsitions nd their outputs of (prt of) the ssocited Moore utomton re given in Figure 1. Context-free grmmrs generting proper lnguges (i.e. not contining the empty word ɛ) re equivlent to reltime pd s [11, 13, 42]. Given n input lphbet A, nd set

15 14 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN,s x q 0 b,x ɛ q 1,x xx b,x ɛ c 4 c b 2 b c 5 c 0 c 1 c 6,b b b c b 3 b c 7 q 0, xβ } if β = sβ c 2 = λβ. q 0, xxβ } if β = xβ otherwise q 0, xxβ } if β = sβ c 4 = λβ. q 0, xxxβ } if β = xβ otherwise c 6 = λβ. b 1 if β = ɛ o (c 0 ) = o (c 1 ) = λβ. 0 otherwise o (c 2 ) = o (c 4 ) = o (c 6 ) = λβ.0 1 if β = x o (c 3 ) = o (c 5 ) = λβ. 0 otherwise 1 if β = xxs o (c 7 ) = λβ. 0 otherwise c 0 = η(q 0 ) c 1 = η(q 1 ) q1, β c 3 = λβ. } if β = xβ otherwise q 1, β } if β = sβ c 5 = λβ. q 1, xβ } if β = xβ otherwise q1, β c 7 = λβ. } if β = xxβ otherwise Figure 1: The structured sttes c i T Q, their trnsitions nd their output of (prt of) the Moore utomton ssocited to the pd (Q, A, B, δ, k 0, K) where Q = q 0, q 1 }, A =, b}, B = x, s}, δ is depicted on the left top, k 0 = q 0, s nd K = q 0, ɛ, q 1, ɛ }. of vribles B, let G = (A, B, s, P ) be context-free grmmr in Greibch norml form [15], i.e. with productions in P of the form b α with b B, A nd α B. We cn construct function o, t : 1 F T 1 (where 1 = }) by setting o( )(β) = 1 if nd only if β = ɛ nd t( )()(bβ) =, αβ b α P }. Clerly this function stisfies conditions (1) nd (2) bove, nd thus, together with the initil configurtion, s defines pd. Furthermore, [[η( )]](w)(s) = 1 if nd only if there exists derivtion for w A in the grmmr G. As n exmple, let us consider the grmmr (, b}, s, x}, s, P ) with productions P = s sx, s x, x b} generting the lnguge n b n n 1}. The ssocited colgebr o, t : 1 F T 1 is given by,s sx *,s x with o( )(β) = 1 iff β = ɛ b,x ɛ

16 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS 15,s sx *,s x b,x ɛ c 3 b c 4 c 0 = η( ), sxβ c 1 = λβ.,, xβ } if β = sβ otherwise c 1, β c 2 = λβ. } if β = xβ otherwise c 0 b c 2 b c 5 c 6, sxxβ c 3 = λβ.,, xβ } if β = sβ otherwise, β c 4 = λβ. } if β = sβ otherwise c 7, sxβ c 5 = λβ.,, xβ } if β = ssβ otherwise 1 if β = ɛ o (c 0 ) = λβ. 0 otherwise 1 if β = x o (c 2 ) = λβ. 0 otherwise, sxβ c 6 = λβ.,, xβ } if β = xsβ otherwise, β c 7 = λβ. } if β = xxβ otherwise o (c 1 ) = o (c 3 ) = o (c 5 ) = o (c 6 ) = λβ.0 1 if β = xx o (c 7 ) = λβ. 0 otherwise 1 if β = s o (c 4 ) = λβ. 0 otherwise Figure 2: The structured sttes c i T Q, their trnsitions nd their output of (prt of) the Moore utomton ssocited to the pd (Q, A, B, δ, k 0, K) where Q = }, A =, b}, B = x, s}, δ is depicted on the left top, k 0 =, s nd K =, ɛ }. Even if the lnguge ccepted by the bove pd is the sme s the one ccepted by the pd in the previous exmple (i.e., [[η( )](w)(s) = [[η(q 0 )]](w)(s) for ll w A ), the two ssocited Moore utomton re not in T F (tht is [[η( )]] [[η(q 0)]]). In fct, the Moore utomton ssocited to the bove colgebr (see below) ccepts the string bb when strting from the configurtion, ss, while the one in the previous exmple does not (in symbols, [[η( )]](bb)(ss) = 1 while [[η(q 0 )]](bb)(ss) = 0). The bove chrcteriztion of context free lnguges over n lphbet A is different nd complementry to the colgebric ccount of context-free lnguges presented in [44]. The ltter, in fct, uses the functor D(X) = 2 X A for deterministic utomt (insted of the Moore utomt with output in 2 B bove, for B set of vribles), nd the idempotent semiring mond T (X) = P ω ((X + A) ) (insted of our side effect mond) to study different

17 16 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN but equivlent wys to present context-free lnguges: using grmmrs, behviourl differentil equtions nd generlized regulr expressions in which the Kleene str is replced by unique fixed point opertor. 4. Colgebrs nd T-Algebrs In the previous section we presented frmework, prmeterized by functor F nd mond T, in which systems of type F T (tht is, F T -colgebrs) cn be studied using novel equivlence T F insted of the clssicl F T. The only requirement we imposed ws tht F T (X) hs to be T-lgebr. In this section, we will present functors F for which the requirement of F T (X) being T-lgebr is gurnteed becuse they cn be lifted to functor F on T-lgebr. For these functors, the equivlence T F coincides with F. In other words, working on F T - colgebrs in Set under the novel T F equivlence is the sme s working on F -colgebrs on T-lgebrs under the ordinry F equivlence. Next, we will prove tht for this clss of functors nd n rbitrry mond T the equivlence F T is contined in T F. Instntiting this result for our first motivting exmple of non-deterministic utomt will yield the well known fct tht bisimilrity implies trce equivlence. Let T be mond. An endofunctor F : Set T Set T is sid to be the T-lgebr lifting of functor F : Set Set if the following squre commutes 1 : Set T F Set T U T Set F If the functor F hs T-lgebr lifting F then F T (X) is the crrier of the lgebr F (T (X), µ). Functors tht hve T-lgebr lifting re given, for exmple, by those endofunctors on Set constructed inductively by the following grmmr Set U T F :: = Id B F F F A T G where A is n rbitrry set, B is the constnt functor mpping every set X to the crrier of T-lgebr (B, h), nd G is n rbitrry functor. Since the forgetful functor U T : Set T Set cretes nd preserves limits, both F 1 F 2 nd F A hve T-lgebr lifting if F, F 1, nd F 2 hve. Finlly, T G hs T-lgebr lifting for every endofunctor G given by the ssignment (X, h) (T GX, µ GX ). Note tht we do not llow tking coproducts in the bove grmmr, becuse coproducts of T-lgebrs re not preserved in generl by the forgetful functor U T. Insted, one could resort to extending the grmmr with the crrier of the coproduct tken directly in Set T. For instnce, if T is the (finite) powerset mond, then we could extend the bove grmmr with the functor F 1 F 2 = F 1 + F 2 +, }. All the functors of the exmples in Sections 2 nd 3, s well s those in Section 5, cn be generted by the bove grmmr nd, therefore, they hve T-lgebr lifting. Now, let F be functor with T-lgebr lifting nd for which finl colgebr Ω exists. If Ω cn be constructed s the limit of the finl sequence (for exmple ssuming the functor 1 This is equivlent to the existence of distributive lw λ: T F F T [24].

18 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS 17 ccessible [1]), then, becuse the forgetful functor U T : Set T Set preserves nd cretes limits, Ω is the crrier of T-lgebr, nd it is the finl colgebr of the lifted functor F. Further, for ny F T -colgebr f : X F T (X), the unique F -colgebr homomorphism [[ ]] s in digrm (3.1) is T -lgebr homomorphism between T (X) nd Ω. Conversely, the crrier of the finl F -colgebr (in Set T ) is the finl F -colgebr (in Set). Intuitively, the bove mens tht for n ccessible functor F with T-lgebr lifting F, F -equivlence in Set T coincides with F -equivlence with respect to T in Set. The ltter equivlence is corser thn the F T -equivlence in Set, s stted in the following theorem. Theorem 4.1. Let T be mond. If F is n endofunctor on Set for which finl colgebr exists nd with T-lgebr lifting, then F T implies T F. Proof. We first show tht there exists functor from the ctegory of F T -colgebrs to the ctegory of F -colgebrs. This functor mps ech F T -colgebr (X, f) into the F -colgebr (T (X), f ) nd ech F T -homomorphism h: (X, f) (Y, g) into the F -homomorphism T (h): (T (X), f ) (T (Y ), g ). In order to prove tht this is functor we just hve to show tht T (h) is n F -homomorphism (i.e., the bckwrd fce of the following digrm commutes). T (X) η X h X f F T (X) f F T (h) T (h) T (Y ) η Y Y g F T (Y ) Note tht the front fce of the bove digrm commutes becuse h is n F T -homomorphism. Also the top fce commutes becuse η is nturl trnsformtion. Thus F T (h) f η X = F T (h) f = g h nd lso g T (h) η X = g η Y h = g h. Since η is the unit of the djunction, then there exists unique j : T (X) F T (Y ) in Set T such tht g h = j η X. Since both F T (h) f nd g T (h) re (by construction) morphisms in Set T, then F T (h) f = g T (h). Let (X, f) nd (Y, g) be two F T -colgebrs nd [[ ]] X nd [[ ]] Y their morphisms into the finl F T -colgebr (Ω, ω). Let (T (X), f ), (T (Y ), g ) nd (T (Ω), ω ) be the corresponding F -colgebrs nd [[ ]] T X, [[ ]] T Y nd [[ ]] T Ω their morphisms into the finl F -colgebr (Ω, ω ). Since T ([ ] X ): (T (X), f ) (T (Ω), ω ) is n F -homomorphism, then by uniqueness, [[ ]] T X = [[ ]] T Ω T ([[ ]] X ). g

19 18 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN [[ ]] T X T (X) η X X f F T (X) f [[ ]] X F T ([[ ]] X ) T ([[ ]] X ) T (Ω) η Ω Ω ω F T (Ω) F ([[ ]] T X ) ω F ([[ ]] T Ω ) [[ ]] T Ω Ω F (Ω ) ω With the sme proof, we obtin [[ ]] T Y = [[ ]] T Ω T ([[ ]] Y ). Recll tht for ll x X nd y Y, by definition, x F T y iff [[x]] X = [[y]] Y nd x T F y iff [[η X (x)]] T X = [[η Y (y)]] T Y. Suppose tht [[x]] X = [[y]] Y. Then, T ([[η X (x)]] X ) = η Ω [[x]] X = η Ω [[y]] Y = T ([[η Y (y)]] Y ) nd, finlly, [[η X (x)] T X = [[ ]] T Ω T ([[η X (x)]] X ) = [[ ]] T Ω T ([η Y (y)]] Y ) = [[η Y (y)]] T Y. The bove theorem instntites to the well-known fcts: for NDA, where F (X) = 2 X A nd T = P ω, tht bisimilrity implies lnguge equivlence; for prtil utomt, where F (X) = 2 X A nd T = 1 +, tht equivlence of pirs of lnguges, consisting of defined pths nd ccepted words, implies equivlence of ccepted words; for probbilistic utomt, where F (X) = [0, 1] X A nd T = D ω, tht probbilistic bisimilrity implies probbilistic/weighted lnguge equivlence. Note tht, in generl, the bove inclusion is strict. Remrk. Let (X, f) be n F T -colgebr for mond T nd functor F. If η : id T is pointwise injective, then F T on the F T -colgebr (X, f) coincides with T F T on the extended T F T -colgebr (X, η F T (X) f) [37, 4]. If moreover F hs T-lgebr lifting then, by the bove theorem (on the extended T F T -colgebr), T F T implies T T F. Combining the two implictions, it follows tht ht F T on the F T -colgebr (X, f) implies T T F on the extended T F T -colgebr (X, η F T (X) f). Finlly, under the ssumption tht F hs T- lgebr lifting, we lso hve tht T F the F T -colgebr (X, f) implies T T F on the extended T F T -colgebr (X, η F T (X) f). This yields the following hierrchy of equivlences. T T F T F T = F T T F

20 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS Beyond Bisimilrity nd Trces The opertionl semntics of interctive systems is usully specified by lbeled trnsition systems (LTS s). The denottionl semntics is given in terms of behviourl equivlences, which depend the mount of brnching structure considered. Bisimilrity (full brnching) is sometimes considered too strict, while trce equivlence (no brnching) is often considered too corse. The liner time / brnching time spectrum [14] shows txonomy of mny interesting equivlences lying in between bisimilrity nd trces. Lbeled trnsition system re colgebrs for the functor P ω (Id) A nd the colgebric equivlence Pω(Id) A coincides with the stndrd notion of Prk-Milner bisimilrity. In [35], it is shown colgebric chrcteriztion of trces semntics (for LTS s) employing Kleisli ctegories. More recently, [33] hve provided chrcteriztion of trce, filure nd redy semntics by men of behviour objects. Another colgebric pproch [26] relies on test-suite tht, intuitively, re frgments of Hennessy-Milner logic. In this section, we show tht (finite) trce equivlence [20], complete trce equivlence [14], filures [9] nd redy semntics [34] cn be seen s specil cses of T F. Before introducing these semntics, we fix some nottions. A lbeled trnsition system is pir (X, δ) where X is set of sttes nd δ : X P ω (X) A is function ssigning to ech stte x X nd to ech lbel A finite set of possible successors sttes: x y mens tht y δ(x)(). Given word w A, we write x w y for x 1... n y nd w = 1... n. When w = ɛ, x ɛ y iff y = x. For function ϕ P ω (X) A, I(ϕ) denotes the set of ll lbels enbled by ϕ, i.e., A ϕ() }, while Fil(ϕ) denotes the set Z A Z I(ϕ) = }. Let X, δ be LTS nd x X be stte. A trce of x is word w A such tht x w y for some y. A trce w of x is complete if x w y nd y stops, i.e., I(δ(y)) =. A filure pir of x is pir w, Z A P ω (A) such tht x w y nd Z Fil(δ(y)). A redy pir of x is pir w, Z A P ω (A) such tht x w y nd Z = I(δ(y)). We use T (x), CT (x), F(x) nd R(x) to denote, respectively, the set of ll trces, complete trces, filure pirs nd redy pirs of x. For I rnging over T, CT, F nd R, two sttes x nd y re I-equivlent iff I(x) = I(y). For n exmple, consider the following trnsition systems lbeled over A =, b, c}. They re ll trce equivlent becuse their trces re, b, c. The trce is lso complete for p, but not for the others. Only r nd s re filure equivlent, since, bc} is filure pir only of p, while, b} nd, c} re filure pirs of p, r nd s, but not of q. Finlly they re ll redy different, since, is redy pir only of p,, b, c} is redy pir of q nd s but not of r, nd, b} nd, c} re redy pirs only of r nd s. b p c b q c b r c s b c c b We cn now show tht these equivlences re instnces of T F. We first show redy equivlence in detils nd then, briefly, the others.

21 20 ALEXANDRA SILVA, FILIPPO BONCHI, MARCELLO BONSANGUE, AND JAN RUTTEN Tke T = P ω nd F = P ω (P ω (A)) id A. For ech set X, consider the function π R X : P ω(x) A F T (X) defined for ll ϕ P ω (X) A by π R X(ϕ) = I(ϕ)}, ϕ. This function llows to trnsform ech LTS (X, δ) into the F T -colgebr (X, πx R δ). The ltter hs the sme trnsitions of X, δ, but ech stte x is decorted with the set I(ϕ)}. Now, by employing the powerset construction, we trnsform X, πx R δ into the F - colgebr (P ω (X), o, t ), where, for ll Y P ω (X), A, the functions o: P ω (X) P ω (P ω (A)) nd t: P ω (X) P ω (X) A re o(y ) = I(δ(y))} t(y )() = δ(y)(). y Y The finl F -colgebr is (P ω (P ω (A)) A, ɛ, ( ) ) where ɛ, ( ) is defined s usul. } X P ω (X) P ω (P ω (A)) A δ [[Y ](ɛ) = o(y ) (P ω (X)) A ɛ,( ) o,t [[Y ](w) = [[t(y )()]](w) πx R P ω (P ω (A)) (P ω (X)) A P ω (P ω (A)) (P ω (P ω (P ω (A)) A )) A Summrizing, the finl mp [[ ]]: P ω (X) P ω (P ω (A)) A mps ech x} into function ssigning to ech word w, the set Z A x w y nd Z = I(δ(y))}. In other terms, Z [[x}]](w) iff w, Z R(x). For the stte s depicted bove, [[s}]](ɛ) = }}, [[s}]]() = b}, b, c}, c}}, [[s}]](b) = [[s}](c) = } nd for ll the other words w, [[s}]](w) =. The other semntics cn be chrcterized in the sme wy, by choosing different functors F nd different functions π X : P ω (X) A F T. For filure semntics, tke the sme functor s for the redy semntics, tht is F = P ω (P ω (A)) id A nd new function π F X : P ω(x) A F T (X) defined ϕ P ω (X) A by [[ ]] π F X(ϕ) = Fil(ϕ), ϕ. The F T -colgebr (X, πx F δ) hs the sme trnsitions of the LTS X, δ, but ech stte x is decorted with the set Fil(ϕ). For both trce nd complete trce equivlence, tke F = 2 id A (s for NDA). For trce equivlence, πx T : P ω(x) A F T (X) mps ϕ P ω (X) A into 1, ϕ. Intuitively, (X, πx T δ) is n NDA where ll the sttes re ccepting. For complete trces, πx CT : P ω(x) A F T (X) mps ϕ in 1, ϕ if I(ϕ) = (nd in 0, ϕ otherwise). By tking T = D ω insted of T = P ω, we hope to be ble to chrcterize probbilistic trce, complete trce, redy nd filure s defined in [25]. y Y

22 GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS Discussion In this pper, we lifted the powerset construction on utomt to the more generl frmework of F T -colgebrs. Our results led to uniform tretment of severl kinds of existing nd new vritions of utomt (tht is, F T -colgebrs) by n lgebric structuring of their stte spce through mond T. We showed s exmples prtil Mely mchines, structured Moore utomt, nondeterministic, prtil nd probbilistic utomt. Furthermore, we hve presented n interesting colgebric chrcteriztion of pushdown utomt nd showed how severl behviourl equivlences stemming from concurrency theory cn be retrieved from the generl frmework. It is worth mentioning tht the frmework instntites to mny other exmples, mong which re weighted utomt [41]. These re simply structured Moore utomt for B = 1 nd T = S ω (for semiring S) [16]. It is esy to see tht F T coincides with weighted bisimilrity [10], while T F coincides with weighted lnguge equivlence [41]. Some of the forementioned exmples cn lso be colgebriclly chrcterized in the frmework of [19, 18]. There, insted of considering F T -colgebrs on Set nd F - colgebrs on Set T (the Eilenberg-Moore ctegory), T G-colgebrs on Set nd G-colgebrs on Set T (the Kleisli ctegory) re studied. The min theorem of [19] sttes tht under certin ssumptions, the initil G-lgebr is the finl G-colgebr tht chrcterizes (generlized) trce equivlence. The exct reltionship between these two pproches hs been studied in [23] (nd, indirectly, it could be deduced from [6] nd [27]). It is worth to remrk tht mny of our exmples do not fit the frmework in [19]: for instnce, the exception, the side effect, the full-probbility nd the interctive output monds do not fulfill their requirements (the first three do not hve bottom element nd the ltter is not commuttive). Moreover, we lso note tht the exmple of prtil Mely mchines is not purely trce-like, s ll the exmples in [19]. The ide of using monds for modeling utomt with non-determinism, probbilism or side-effects dtes bck to the λ-mchines of [2] tht, rther thn colgebrs, rely on lgebrs. More precisely, the dynmic of λ-mchine is morphism δ : F X T X, where F is functor nd T is mond (for instnce the trnsitions of T -structured Moore utomt re function δ : X A T X mpping stte nd n input symbol into n element of T X). Anlogously to our pproch, ech λ-mchine induces n implicit λ-mchine hving T X s stte spce. Mny exmples of this pper (like Moore utomt) cn be seen s λ-mchines, but those systems tht re essentilly colgebric (like Mely mchines) do not fit the frmework in [2]. There re severl directions for future reserch. On the one hnd, we will try to exploit F -bisimultions up to T [29, 30] s sound nd complete proof technique for T F. On the other hnd, we would like to lift mny of those colgebric tools tht hve been developed for brnching equivlences (such s colgebric modl logic [12, 40] nd (xiomtiztion for) regulr expressions [8]) to work with the liner equivlences induced by T F. We hve pursued further the pplictions to decorted trces nd the chllenging modeling of the full liner-time spectrum in seprte pper [7], work which we lso pln to extend to probbilistic trces.