Kleene Coalgebra an overview

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1 Kleene Colgebr n overview Alexndr Silv 1, Mrcello Bonsngue 1,2, nd Jn Rutten 1,3 1 Centrum Wiskunde & Informtic 2 LIACS Leiden Institute of Advnced Computer Science 3 Rdboud Universiteit Nijmegen Abstrct. Colgebrs provide uniform frmework for the study of dynmicl systems, including severl types of utomt. The colgebric view on systems hs recently been proved relevnt by the development of number of expression clculi which generlize clssicl results by Kleene, on regulr expressions, nd by Kozen, on Kleene lgebr. This note contins n overview of the motivtion nd results of the generic frmework we developed Kleene Colgebr to uniformly derive the forementioned clculi. We present n historicl overview of work on regulr expressions nd xiomtiztions, s well discussion of relted work. We show pplictions of the frmework to three types of probbilistic systems: simple Segl, strtified nd Pnueli-Zuck. 1 Introduction Computer systems hve widely spred since their ppernce, nd they now ply crucil role in mny dily ctivities, with their deployment rnging from smll home pplinces to sfety criticl components, such s irplne or utomobile control systems. Accidents cused by either hrdwre or softwre filure cn hve disstrous consequences, leding to the loss of humn lives or cusing enormous finncil drwbcks. One of the gretest chllenges of computer science is to cope with the fst evolution of computer systems nd to develop forml techniques which fcilitte the construction of softwre nd hrdwre systems. Since the erly dys of computer science, mny scientists hve serched for suitble models of computtion nd for specifiction lnguges tht re pproprite for resoning bout such models. When devising model for prticulr system, there is need to encode different fetures which cpture the inherent behvior of the system. For instnce, some systems hve deterministic behvior ( clcultor or n elevtor), wheres others hve inherently non-deterministic or probbilistic behvior (think of csino slot mchine). The rpidly incresing complexity of systems demnds for compositionl nd unifying models of computtion, s well s generl methods nd guidelines to derive specifiction lnguges. In the lst decdes, colgebr hs risen s prominent cndidte for mthemticl frmework to specify nd reson bout computer systems. Colgebric modeling works, on the surfce, s follows: the bsic fetures of system, such s non-determinism or probbility, re collected nd combined in the pproprite

2 wy, determining the type of the system. This type (formlly, functor) is then used to derive suitble equivlence reltion nd universl domin of behviors, which llow to reson bout equivlence of systems. The strength of colgebric modeling lies in the fct tht mny importnt notions re prmetrized by the type of the system. On the one hnd, the colgebric frmework is unifying, llowing for uniform study of different systems nd mking precise the connection between them. On the other hnd, it cn serve s guideline for the development of bsic notions for new models of computtion. One of the simplest models of computtion is tht of deterministic finite utomton. In his seminl pper in 1956, Kleene [32] described finite deterministic utomt (which he clled nerve nets), together with specifiction lnguge: regulr expressions. One of his most importnt results is the theorem which sttes tht ny finite deterministic utomton cn be chrcterized by regulr expression nd tht, conversely, every regulr expression cn be relized by such utomton. This theorem, which is tody referred to s Kleene s theorem, becme one of the cornerstones of theoreticl computer science. In his pper, Kleene left open the question of whether there would exist finite, sound nd complete, xiomtiztion of the equivlence of regulr expressions, which would enble lgebric resoning. The first nswer to this question ws given in 1966 by Slom [47], who presented two complete xiomtiztions. The 1971 monogrph of Conwy [21] presents n extended overview of results on regulr expressions nd their xiomtiztions. Lter, in 1990, Kozen [33] showed tht Slom s xiomtiztion is non-lgebric, in the sense tht it is unsound under substitution of lphbet symbols by rbitrry regulr expressions, nd presented n lgebric xiomtiztion: Kleene lgebrs. McNughton nd Ymd [36] gve lgorithms to build non-deterministic utomton from regulr expression nd bck, nd introduced notion of extended regulr expression with intersection nd complementtion opertors. This enrichment of the lnguge of regulr expressions ws relevnt in the context of the most importnt ppliction of regulr expressions t tht time, in the design of digitl circuits, llowing more esy conversion of nturl lnguge specifiction of problems into regulr expression. Brzozowski [17,18] introduced the notion of derivtive of regulr expressions, which llowed him to prove Kleene s theorem for the extended set of regulr expressions without hving to recur to non-deterministic utomt. Efficient lgorithms to compile regulr expressions to deterministic nd nondeterministic utomt becme crucil when regulr expressions strted to be widely used for pttern mtching. One of the fstest (nd most beutiful) lgorithms to trnslte regulr expressions into utomt ws devised by Berry nd Sethi [8]. This lgorithm becme the bsis of one of the first compilers of the lnguge Esterel [7], synchronous progrmming lnguge, bsed on regulr expressions, dedicted to embedded systems. Esterel is one of the most successful follow ups of Kleene s work: it is used s specifiction lnguge of control-intensive pplictions, such s the ones running in the centrl units of

3 crs or irplnes. In such systems gurnteeing importnt sfety properties is of the utmost importnce nd forml models of computtion ply centrl role. In 1981, Milner dpted Kleene nd Slom s results to lbeled trnsition systems: model of computtion in which non-determinism is llowed [39]. He introduced lnguge for finitely presented behviors, which cn be seen s frgment of the clculus of communicting systems (CCS) [38], nd sound nd complete xiomtiztion with respect to bisimilrity. The pper of Milner served s inspirtion for mny reserchers in the concurrency community. Probbilistic extensions of CCS, together with sound nd complete xiomtiztions (with respect to the pproprite notion of equivlence) hve been presented for instnce in [22,23,51]. In ddition to the models lredy mentioned, other importnt models of computtion include Mely mchines [37] (utomt with input nd output), utomt on gurded strings [34] nd weighted utomt [50]. These three models hve pplictions, for instnce, in digitl circuit design, compiler optimiztion nd imge recognition, respectively. The im of our work is to mke use of the colgebric view on systems to devise frmework where lnguges of specifiction nd xiomtiztions cn be uniformly derived for lrge clss of systems, which include ll the models mentioned bove. As snity check, it should be possible to derive from the generl frmework known results. More importntly, we should be ble to derive new lnguges nd xiomtiztions. We combine the work of Kleene with colgebr, hence the nme Kleene colgebr. The theory of universl colgebr [45] provides stndrd equivlence nd universl domin of behviors, uniquely bsed on the type of the system, given by functor F. It is our min im to show how the type of the system lso llows for uniform derivtion of both set of expressions describing the system s behvior nd corresponding xiomtiztion, sound nd complete with respect to the equivlence induced by F, which enbles lgebric resoning on the specifictions. Furthermore, we wnt to show the correspondence of the behviors denoted by the expressions in the lnguge nd the systems under study, formulting the colgebric nlogue of Kleene s theorem. The clss of systems we study (or in other words, the clss of functors F) is lrge enough to cover finite deterministic utomt nd lbeled trnsition systems. It includes lso other models, such s Mely mchines, utomt on gurded strings, weighted utomt nd severl types of probbilistic utomt, such s Segl, strtified nd Pnueli-Zuck systems. From this generl frmework we will recover known lnguges nd xiomtiztions but, more interestingly, we will lso derive new ones, for the so-clled strtified nd Pnueli-Zuck systems. To give the reder feeling of the type of expressions nd xiomtiztions we will derive, we show in Figure 1 few exmples of systems, together with their type nd exmples of vlid expressions nd xioms.

4 Deterministic utomt Mely mchines b s 1 s 2 b b 0 s s 2 b 0 F = 2 Id A F = (2 Id) A µx.b(x) (µy.b(y) (x) 1) ε ε µx.b(x) (µy.(x) b(y)) 1 µx.ε ε[µx.ε/x] Lbeled trnsition systems s 1 s 2 s 3 b c c s 4 s 5 s 6 1/2 s 2 Segl systems s 1 1/2 s 3 2/3 s 4 1/3 s 5 F = (P ωid) A F = (P ω(d ωid)) A ({b( )} {c({ } { })}) ε ε ε ({1/2 1/2 }) ({1/3 2/3 }) p ε p ε (p + p ) ε Fig. 1. For ech of the systems we show concrete exmple, the corresponding functor type, n expression describing the behvior of s 1, nd n exmple of vlid xiom. 1.1 Relted work The connection between Kleene s regulr expressions, deterministic utomt nd colgebr ws first explored in [44,46]. Rutten studied the colgebric structure of the set of regulr expressions, given by Brzozowski derivtives [18], in order to show tht the colgebric semntics coincides with the stndrd inductive semntics of regulr expressions. He proved the usefulness of the pproch by proving equlities by coinduction. The coinductive proofs turned out to be, in mny cses, more concise nd intuitive thn the lterntive lgebric proof using the xioms of Kleene lgebr. Lter, Jcobs [31] presented bilgebric review on deterministic utomt nd regulr expressions, which llowed him to present n lterntive colgebric proof of Kozen s result on the completeness of Kleene lgebrs for lnguge equivlence [33,34]. We took inspirtion from ll of these ppers: the work of Brzozowski nd Rutten led us to the definition of the colgebric structure on the set of expressions, wheres the work by Jcobs nd Kozen served s guideline to the proof of soundness nd completeness of the xiomtiztion we will introduce for the set of generlized regulr expressions. In the lst few yers severl proposls for specifiction lnguges for colgebrs ppered [40,43,30,29,20,11,12,49,35]. The lnguges derived in our frmework (for detils see the thesis of the first uthor [52] or the joint ppers with the

5 other uthors [13,15,14,54,10,53]) re similr in spirit to tht of Rössiger [43], Jcobs [30], Pttinson nd Schröder [49] in tht we use the ingredients of functor for typing expressions. They differ from the logics presented in [43,30] becuse we do not need n explicit next-stte opertor, s we cn deduce it from the type informtion. Aprt from the logics introduced by Kupke nd Venem in [35], the lnguges mentioned bove do not include fixed point opertors. Our lnguge of generlized regulr expressions is similr to frgment of the logic presented in [35] nd cn be seen s n extension of the colgebric logic of [11] with fixed point opertors, s well s the multi-sorted logics of [49]. However, our gol is rther different: we wnt (1) finitry lnguge tht chrcterizes exctly ll loclly finite colgebrs; (2) Kleene like theorem for the lnguge or, in other words, mp (nd not reltion) from expressions to colgebrs nd vice-vers. Similr to mny of the works bove, we lso derive modulr xiomtiztion, sound nd complete with respect to the equivlence induced by the functor. The lnguges studied in the relm of our work llow for recursive specifictions nd therefore formlize potentilly infinite computtions. This type of computtions were studied lso in the context of itertive theories, which hve been introduced by Elgot [25]. The min exmple of n itertive theory is the theory of regulr trees, tht is trees which hve finitely mny distinct subtrees. Adámek, Milius nd Velebil hve presented Elgot s work from colgebric perspective [1,2], simplified some of his originl proofs, nd generlized the notion of free itertive theory to ny finitry endofunctor of every loclly presentble ctegory. The lnguge ssocited with ech functor, which we introduce in our work, modulo the xioms is closely relted to the work bove: it is n initil itertive lgebr. This lso shows the connection of our work with the work by Bloom nd Ésik on itertive lgebrs/theories [9]. Kleene s theorem hs been extended in vrious wys. Büchi [19] extended it to infinite words nd ω-utomt, introducing n ω opertor on lnguges. Ochmnski [41] introduced concurrent version of the Kleene str opertor, which led him to define notion of co-rtionl lnguges, obtined s the rtionl ones by simply replcing the str by the concurrent itertion. He then generlized Kleene s theorem showing tht the recognizble trce lnguges re exctly the co-rtionl lnguges. Gstin, Petit nd Zielonk [26,27] extended Ochmnski s results to infinite trce lnguges. For weighted utomt, Schützenberger [50] hs shown tht the set of recognizble forml power series (corresponding to the behvior of weighted utomt) coincides with the set of rtionl forml power series. For timed utomt, there were severl proposls, including the ppers by Bouyer nd Petit [16], Asrin, Cspi nd Mler [4], nd Asrin nd Dim [5]. Recently, the results of Bouyer nd Petit s well s those of Schützenberger hve been extended to the clss of weighted timed utomt by Droste nd Qus [24]. Furthermore, Kozen hs extended Kleene s lnguge with Boolen tests s finitry representtion of regulr sets of gurded strings nd proved n nlogue of Kleene s theorem for utomt on gurded strings [34].

6 b 1/2 1/2 2/3 1/3 1 1/2 1/2 1/3 2/3 b 1/3 2/3 b b (i) (ii) (iii) 1 Fig. 2. (i) A simple Segl system, (ii) strtified system nd (iii) Pnueli-Zuck system From the forementioned extensions, only the weighted utomt exmple of Schützenberger nd the utomt on gurded strings of Kozen would fit in the frmework we developed. The lnguge we will derive is, however, different from the ones they proposed. Schützenberger s nd Kozen s lnguges hd full sequentil composition nd str in their syntx, insted of the ction prefixing nd unique fixed point opertors tht we will use in our lnguge. 2 Probbilistic Systems Mny different types of probbilistic systems hve been defined in literture: exmples include rective, genertive, strtified, lternting, (simple) Segl, bundle nd Pnueli-Zuck. Ech type corresponds to functor, nd the systems of certin type re colgebrs of the corresponding functor. A systemtic study of ll these systems s colgebrs ws mde in [6]. In prticulr, Figure 1 of [6] provides full correspondence between types of systems nd functors. By employing this correspondence, we will show the derived expressions nd xioms for three different types of probbilistic systems: simple Segl, strtified, nd Pnueli-Zuck 4. Simple Segl systems Simple Segl systems re trnsition systems where both probbility nd non determinism re present. They re colgebrs of the functor P ω (D ω (Id)) A. Ech lbelled trnsition leds, non-deterministiclly, to probbility distribution of sttes insted of single stte. An exmple is shown in Figure 2(i). For simple Segl systems we derived the following syntx nd xioms. ε:: = ε ε µx.ε x ({ε P }) where A, p i (0, 1] nd p i = 1 i 1...n ε :: = L p i ε i i 1 n (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 ε ε ε ε ε (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 (p 1 ε) (p 2 ε) (p 1 + p 2) ε ε[µx.ε/x] µx.ε γ[ε/x] ε µx.γ ε 4 This is prt of our joint work with Filippo Bonchi.

7 We show next n exmple of ppliction of the xioms. The expression ({1/2 1/2 }) ({1/3 2/3 }) b({1 }) is bisimilr to the top-most stte in the simple Segl system depicted in Figure 2(i). Using the xiomtiztion, we cn derive: ({1/2 1/2 }) ({1/3 2/3 }) b({1 }) ({(1/2 + 1/2) }) ({(1/3 + 2/3) }) b({1 }) ({1 }) ({1 }) b({1 }) ({1 }) b({1 }) Thus, we cn conclude tht the system presented in Figure 2(i) is bisimilr to the following one: b 1 1 The lnguge nd xiomtiztion we presented bove re the sme s the one presented in [23] (with the difference tht in [23] prllel composition opertor ws lso considered). This is of course ressuring for the correctness of the generl frmework we presented. In the next two exmples, we will present new results (tht is syntx/xiomtiztions which did not exist). This is where the generlity strts pying off: not only one recovers known results but lso derives new ones, ll of this inside the sme uniform frmework. Strtified systems Strtified systems re colgebrs of the functor D ω (Id) + (B Id) + 1. Ech stte of these systems either performs unlbelled probbilistic trnsitions or one B-lbelled trnsition or it termintes. We first derive expressions nd xioms for R Id + (B P ω (Id)) + 1 nd then we restrict the syntx to chrcterize only D ω (Id) + (B Id) + 1-behviours. This, together with the introduction of some syntctic sugr, leds to the following syntx nd xioms. ε:: = µx.ε x b, ε L i 1 n p i ε i where b B, p i (0, 1] nd P i 1...n p i = 1 (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 (p 1 ε) (p 2 ε) (p 1 + p 2) ε ε[µx.ε/x] µx.ε γ[ε/x] ε µx.γ ε Here denotes termintion nd b, ε denotes stte which cn mke trnsition lbeled by b to nother stte specified by ε. We cn use these xioms (together with Kleene s theorem) to reson bout the system presented in Figure 2(ii). The topmost stte of this system is bisimilr to the expression 1/2 (1/3, 2/3, ) 1/2 b, which in turn is provbly equivlent to 1/2 (1, ) 1/2 b,. Tht leds us to conclude tht the forementioned system is equivlent to the following

8 simpler one. 1/2 1/2 1 The lnguge of expressions we propose for these systems is subset of the lnguge originlly proposed in [28] (there prllel composition opertor is lso considered). More interestingly, there ws no xiomtiztion of the lnguge in [28] nd thus the xiomtiztion we present here is completely new. Pnueli-Zuck systems These systems re colgebrs of the functor P ω D ω (P ω (Id) A ). Intuitively, the ingredient P ω (Id) A denotes A-lbelled trnsitions to other sttes. Then, D ω (P ω (Id) A ) corresponds to probbility distribution of lbelled trnsitions nd then, ech stte of P ω D ω (P ω (Id) A )-colgebr performs non-deterministic choice mongst probbility distributions of lbelled trnsitions. For n exmple, consider the system depicted in Figure 2(iii). The expressions nd xioms we derive for these systems re the following. ε:: = ε ε µx.ε x {ε P } where A, p i (0, 1] nd p i = 1 ε :: = L i 1 n p i ε i ε :: = ε ε ({ε}) b i 1...n (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 ε ε ε ε ε (ε 1 ε 2) ε 3 ε 1 (ε 2 ε 3) ε 1 ε 2 ε 2 ε 1 (p 1 ε ) (p 2 ε ) (p 1 + p 2) ε (ε 1 ε 2 ) ε 3 ε 1 (ε 2 ε 3 ) ε 1 ε 2 ε 2 ε 1 ε ε ε ε ε ε[µx.ε/x] µx.ε γ[ε/x] ε µx.γ ε The expression {1/3 (({ }) ({ })) 2/3 (b({ }) ({ }))} {1 b({ })} specifies the Pnueli-Zuck system in Figure 2(iii). Note tht we use the sme symbol for denoting two different kinds of non-deterministic choice. This is sfe, since they stisfy exctly the sme xioms. Both the syntx nd the xioms we propose here for these systems re to the best of our knowledge new. In the pst, these systems were studied using temporl logic [42]. References 1. J. Adámek, S. Milius, nd J. Velebil. Free itertive theories: A colgebric view. Mthemticl Structures in Computer Science, 13(2): , J. Adámek, S. Milius, nd J. Velebil. Itertive lgebrs t work. Mthemticl Structures in Computer Science, 16(6): , R. Amdio, ed, Proceedings of FOSSACS 2008, vol of LNCS. Springer, E. Asrin, P. Cspi, nd O. Mler. A Kleene theorem for timed utomt. In Proceedings of LICS 1997, pp , E. Asrin nd C. Dim. Blnced timed regulr expressions. ENTCS 68(5), 2002.

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