Universal coalgebra: a theory of systems

Transcription

1 Theoreticl Computer cience 249 (2000) Fundmentl tudy Universl colgebr: theory of systems J.J.M.M. Rutten CWI, P.O. Box 94079, 1090 GB Amsterdm, Netherlnds Communicted by M:W: Mislove Abstrct In the semntics of progrmming, nite dt types such s nite lists, hve trditionlly been modelled by initil lgebrs. Lter nl colgebrs were used in order to del with innite dt types. Colgebrs, which re the dul of lgebrs, turned out to be suited, moreover, s models for certin types of utomt nd more generlly, for (trnsition nd dynmicl) systems. An importnt property of initil lgebrs is tht they stisfy the fmilir principle of induction. uch principle ws missing for colgebrs until the work of Aczel (Non-Well-Founded sets, CLI Leethre Notes, Vol. 14, center for the study of Lnguges nd informtion, tnford, 1988) on theory of non-wellfounded sets, in which he introduced proof principle nowdys clled coinduction. It ws formulted in terms of bisimultion, notion originlly stemming from the world of concurrent progrmming lnguges. Using the notion of colgebr homomorphism, the denition of bisimultion on colgebrs cn be shown to be formlly dul to tht of congruence on lgebrs. Thus, the three bsic notions of universl lgebr: lgebr, homomorphism of lgebrs, nd congruence, turn out to correspond to colgebr, homomorphism of colgebrs, nd bisimultion, respectively. In this pper, the ltter re tken s the bsic ingredients of theory clled universl colgebr. ome stndrd results from universl lgebr re reformulted (using the forementioned correspondence) nd proved for lrge clss of colgebrs, leding to series of results on, e.g., the lttices of subcolgebrs nd bisimultions, simple colgebrs nd coinduction, nd covriety theorem for colgebrs similr to Birkho s vriety theorem. c 2000 Elsevier cience B.V. All rights reserved. MC: 68Q10; 68Q55 PAC: D.3; F.1; F.3 Keywords: Colgebr; Algebr; Dynmicl system; Trnsition system; Bisimultion; Universl colgebr; Universl lgebr; Congruence; Homomorphism; Induction; Coinduction; Vriety; Covriety E-mil ddress: jnr@cwi.nl (J.J.M.M. Rutten) /00/$ - see front mtter c 2000 Elsevier cience B.V. All rights reserved. PII: (00)

2 4 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 Contents 1. Introduction::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::4 2. Colgebrs, homomorphisms, nd bisimultions ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::8 3. ystems, systems, systems, ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::13 4. Limits nd colimits of systems ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::24 5. Bsic fcts on bisimultions :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::31 6. ubsystems :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::34 7. Three isomorphism theorems ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::37 8. imple systems nd coinduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::39 9. Finl systems::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Existence of nl systems ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Exmples of coinductive denitions ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Exmples of proofs by coinduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Induction nd coinduction:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Gretest nd lest xed points ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Nturl trnsformtions of systems :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: A unique xed point theorem:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Cofreeness nd covrieties of systems ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Dynmicl systems nd symbolic dynmics ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Notes nd relted work ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::68 Acknowledgements :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::70 Appendix :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::70 References :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::77 1. Introduction In the semntics of progrmming, dt types re usully presented s lgebrs (cf. [24, 47]). For instnce, the collection of nite words A over some lphbet A is n lgebr A ;:(1+(A A )) A ; where mps (the sole element of the singleton set 1 = { }) to the empty word nd pir ; w to w. This exmple is typicl in tht A is n initil lgebr. Initil lgebrs re generliztions of lest xed points, nd stisfy fmilir inductive proof nd denition principles. For innite dt structures, the dul notion of colgebr hs been used s n lterntive to the lgebric pproch [6]. For instnce, the set A of nite nd innite words over A cn be described by the pir A ;: A (1+(A A )) ; where mps the empty word to nd non-empty word to the pir consisting of its hed (the rst letter) nd til (the reminder). It is colgebr becuse is function from the crrier set A to n expression involving A, tht is, 1 + (A A ), s opposed to the lgebr bove, where ws function into the crrier set A. Agin the exmple is typicl becuse A is nl colgebr, which generlizes the notion of gretest xed point. Colgebrs re lso suitble for the description of the dynmics of systems such s deterministic utomt (cf. [5, 52]). Trditionlly, these re represented s tuples Q; A; B; : Q A Q; : Q B ;

3 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) consisting of set of sttes Q, n input lphbet A, n output lphbet B, next stte function, nd n output function (in ddition n initil stte is often specied s well). Alterntively, such n utomton cn be represented s colgebr of the form Q; : Q (Q A B) ; where Q A is the set of ll functions from A to Q, nd cn be dened in n obvious mnner from nd (nd vice vers). Another, more recent use of colgebrs is mde in the speciction of the behviour of clsses in object-oriented lnguges [35, 36, 68]. imilrly, Peter Aczel uses colgebric description of (nondeterministic trnsition) systems in constructing model for theory of non-wellfounded sets [2]. Mybe more importntly, he lso introduces proof principle for nl colgebrs clled strong extensionlity. It is formulted in terms of the notion of bisimultion reltion, originlly stemming from the eld of concurrency semntics [56, 60]. Using the notion of colgebr homomorphism, the denition of bisimultion hs been generlized in [4], s forml dul to the notion of congruence on lgebrs (see lso [76]). This bstrct formultion of bisimultion gives rise to denition nd proof principles for nl colgebrs (generlizing Aczel s principle of strong extensionlity), which re the colgebric counterprt of the inductive principles for initil lgebrs, nd which therefore re clled coinductive [75]. These observtions, then, hve led to the development in the present pper of generl theory of colgebrs clled universl colgebr, long the lines of universl lgebr. Universl lgebr (cf. [16, 54]) dels with the fetures common to the mny well-known exmples of lgebrs such s groups, rings, etc. The centrl concepts re -lgebr, homomorphism of -lgebrs, nd congruence reltion. The corresponding notions [76] on the colgebr side re: colgebr, homomorphism of colgebrs, nd bisimultion equivlence. These notions constitute the bsic ingredients of our theory of universl colgebr. (More generlly, the notion of substitutive reltion corresponds to tht of bisimultion reltion; hence congruences, which re substitutive equivlence reltions, correspond to bisimultion equivlences.) Adding to this the bove-mentioned observtion tht vrious dynmicl systems (utomt, trnsition systems, nd mny others s we shll see) cn be represented s colgebrs, mkes us spek of universl colgebr s theory of systems. We shll go even s fr s, t lest for the context of the present pper, to consider colgebr nd system s synonyms. The correspondence between the bsic elements of the theories of lgebr nd colgebr re summrized in the following tble: Universl lgebr: (-)lgebr lgebr homomorphism substitutive reltion (congruence reltion) Universl colgebr: colgebr = system system homomorphism bisimultion reltion (bisimultion equivlence)

4 6 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 As mentioned bove, universl lgebr plys guiding role in the development of universl lgebr s theory of colgebrs (= systems). Much of this involves replcing the centrl notions from universl lgebr by the corresponding colgebric notions, nd see whether the resulting sttements cn ctully be proved. Often, fcts on -lgebrs turn out to be vlid (in their trnslted version) for systems s well. Exmples re bsic observtions on quotients nd subsystems, nd the so-clled three isomorphism theorems. In other cses, more cn be sid in the world of colgebrs bout the dul of n lgebriclly importnt notion thn bout tht notion itself. For instnce, initil lgebrs ply role of centrl importnce. Initil colgebrs re usully trivil but nl colgebrs re most relevnt. A relted exmple: initil lgebrs re miniml: they hve no proper sublgebrs. This property is equivlent to the fmilir induction proof principle. Dully, nl colgebrs re simple: they hve no proper quotients, which cn be interpreted s so-clled coinductive proof principle. In previous pper [71], the bove progrmme hs been crried out for one prticulr fmily of systems: unlbelled nondeterministic trnsition systems (lso clled frmes). As it turns out, ll observtions on such systems pply to mny other kinds of systems s well, such s deterministic nd nondeterministic utomt, binry systems, nd hypersystems. Also the fore-mentioned innite dt structures, which cn be interpreted s dynmicl systems s well, re exmples to which the theory pplies. All these dierent exmples cn be conveniently described in one single frmework, using some bsic ctegory theory. Ech of these clsses of systems turns out to be the collection of colgebrs of prticulr functor, nd dierent functors correspond to dierent types of systems. (In tht respect, the world of universl lgebr is simpler becuse of the existence of generl, nonctegoricl wy of describing ll -lgebrs t the sme time, nmely s sets with opertions, the type of which is specied by the signture. A ctegoricl tretment is lso fesible in the lgebric cse, though; see [51].) The generlity of the colgebric theory presented here thus lies in the fct tht ll results re formulted for colgebrs of collection of well-behved functors on the ctegory of sets nd functions, nd thereby pply to gret number of dierent systems. This number cn be seen to be lrger still by vrying the ctegory involved. Tking, for instnce, the ctegory of complete metric spces rther thn simply sets llows us to del with (discrete time) dynmicl systems (ection 18). ome fmilirity with the bsic elements of ctegory theory, therefore, will be of use when reding this pper. The notions of ctegory nd functor will be ssumed to be known. ection 20 hs been included to provide some bckground informtion. It contins some bsic denitions, fcts, nd nottion both on sets nd functors on the ctegory of sets, nd is to be consulted when needed. The fmily of (nondeterministic lbelled) trnsition systems [43, 65] will be used s running exmple throughout the rst sections of the pper. The reder might wnt to refer to [71], where mny of the present observtions re proved in less bstrct wy for trnsition systems; to [39], for n introduction to colgebr nd coinduction;

5 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) or to [73], where deterministic utomt re treted colgebriclly, but without ny reference to ctegory theory. A synopsis of the contents of the present pper is given by the second column of the following tble, which extends the one bove. Its rst column shows the corresponding lgebric notions. (ee ection 13 for discussion on the forml reltionship between the lgebric nd the colgebric notions.) Universl lgebr (-)lgebr lgebr homomorphism substitutive reltion congruence sublgebr miniml lgebr (no proper sublgebrs) induction proof principle simple lgebr (no proper quotients) initil lgebr (is miniml, plus: induction denition principle) nl lgebr (often trivil) free lgebr (used in lgebric speciction) cofree lgebr (often trivil) vriety (closed under sublgebrs, quotients, nd products) denble by quotient of free lgebr covriety (closed under sublgebrs, quotients, nd coproducts) Universl colgebr colgebr=system system homomorphism bisimultion reltion bisimultion equivlence subsystem miniml system (no proper subsystems) simple system (no proper quotients) coinduction proof principle initil system (often trivil) nl system (is simple, plus: coinduction denition principle) free system (often trivil) cofree system (used in colgebric speciction) vriety (closed under subsystems, quotients, nd products) covriety (closed under subsystems, quotients, nd coproducts) denble by subsystem of cofree system Note tht this tble is not to suggest tht the theory of systems is dul to tht of lgebrs. (If so the pper would end here.) It is true tht certin fcts bout lgebrs cn be dulized nd then pply to systems. For instnce, the fct tht the quotient of system with respect to bisimultion equivlence is gin system is dul to

6 8 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 the fct tht the quotient of n lgebr with respect to congruence yields gin n lgebr. However, mny notions tht re dened in both worlds in the sme wy, hve entirely dierent properties. Exmples re the fore-mentioned initil lgebrs nd nl colgebrs. Deep insights bout groups re not obtined by studying universl lgebr. Nor will universl colgebr led to dicult theorems bout (specic types of) systems. Like universl lgebr, its possible merit consists of the fct tht it ::: tidies up mss of rther trivil detil, llowing us to concentrte our powers on the hrd core of the problem [16]. There re mybe two spects tht we might wnt to dd to this. Firstly, induction principles re well-known nd much used. The coinductive denition nd proof principles for colgebrs re less well-known by fr, nd often even not very clerly formulted. Universl colgebr oers simple context for good understnding of coinduction. econdly, mny fmilies of systems look rther dierent from the outside, nd so do the corresponding notions of bisimultion. A systemtic study of colgebrs brings to light mny, sometimes unexpected similrities. This pper both gives n overview of some of the existing insights in the theory of colgebrs, nd, in ddition, presents some new mteril. ection 19 contins brief description per section of which results hve been tken from the literture, s well s discussion of relted work. In summry, the present theory ws preceded by [71], which t its turn builds on previous joint work with Turi [75, 76], from which number of results on nl systems is tken. Mny observtions tht re folklore in the context of prticulr exmples (such s trnsition systems) re generlized to rbitrry systems. The section on the existence of nl systems is bsed on results from Brr [9]. The work of Jcobs on colgebric semntics for object-oriented progrmming [35] nd colgebric speciction [33] hs gretly inuenced the section on cofree systems. The present pper is reworking of [72]. ince the ppernce of the ltter report, much new theory on colgebr hs been developed. Mny of these recent developments cn be found in [38, 40]. 2. Colgebrs, homomorphisms, nd bisimultions The bsic notions of colgebr, homomorphism, nd bisimultion reltion re introduced. A running exmple for this section will be the fmily of lbelled trnsition systems. Mny more exmples will follow in ection 3. Let F : et et be functor. An F-colgebr or F-system is pir (; ) consisting of set nd function : F(). The set is clled the crrier of the system, lso to be clled the set of sttes; the function is clled the F-trnsition structure (or dynmics) of the system. When no explicit reference to the functor (i.e., the type of the system) is needed, we shll simply spek of system nd trnsition structure. Moreover, when no explicit reference to the trnsition structure is needed, we shll often use insted of (; ).

7 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) Exmple 2.1. Consider lbelled trnsition systems (; ;A), consisting of set of sttes, trnsition reltion A, nd set A of lbels [29, 43, 65]. As lwys, s s is used to denote s; ; s. Dene B(X )=P(A X )={V V A X }; for ny set X. We shll see below tht B is functor from et to et. A lbelled trnsition system (; ;A) cn be represented s B-system (; ) by dening : B(); s { ; s s s }: And conversely, ny B-system (; ) corresponds to trnsition system (; A; )by dening s s ; s (s): In other words, the clss of ll lbelled trnsition systems coincides with the clss of ll B-systems. Let (; ) nd (T; T ) be two F-systems, where F is gin n rbitrry functor. A function f : T is homomorphism of F-systems, or F-homomorphism, if F(f) = T f: F() f F(f) T T F(T ): Intuitively, homomorphisms re functions tht preserve nd reect F-trnsition structures (see the exmple below). We sometimes write f :(; ) (T; T ) to express tht f is homomorphism. The identity function on n F-system (; ) is lwys homomorphism, nd the composition of two homomorphisms is gin homomorphism. Thus the collection of ll F-systems together with F-system homomorphisms is ctegory, which we denote by et F. Exmple 2.1 (continued). Let (; A; ) nd (T; A; T ) be two lbelled trnsition systems with the sme set A of lbels, nd let (; ) nd (T; T ) be the corresponding representtions s B-systems. Per denition, B-homomorphism f :(; ) (T; T ) is function f : T such tht B(f) = T f, where the function B(f), lso denoted by P(A f), is dened by B(f)(V )=P(A f)(v )={ ; f(s) ; s V }:

8 10 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 Note tht B is dened both on sets nd on functions. Moreover, B cn be shown to preserve identities: B(1 )=1 B(), nd compositions: B(f g)=b(f) B(g). In other words, B is indeed functor. One cn esily prove tht the equlity B(f) = T f is equivlent to the following two conditions: for ll s in, 1. for ll s in, ifs s then f(s) T f(s ); 2. for ll t in T,iff(s) T t then there is s in with s T s nd f(s )=t. Thus homomorphism is function tht is trnsition preserving nd reecting. An F-homomorphism f : T with n inverse f 1 : T which is lso homomorphism is clled n isomorphism between nd T. As usul, = T mens tht there exists n isomorphism between nd T. An injective homomorphism is clled monomorphism. Dully, surjective homomorphism is clled epimorphism. Given systems nd T, we sy tht cn be embedded into T if there is monomorphism from to T. If there exists n epimorphism from to T, T is clled homomorphic imge of. In tht cse, T is lso clled quotient of. Remrk 2.2. The bove denitions re sucient for our purposes but, more generlly, monomorphisms could be dened s homomorphism tht re mono in the ctegory et F : tht is, homomorphisms f : T such tht for ll homomorphisms k : U nd l : U : iff k = f l then k = l. Clerly injective homomorphisms re mono. One cn show tht for lrge clss of functors, the converse of this sttement holds s well. A dul remrk pplies to epimorphisms. Further detils re given in Proposition 4.7. The following properties will be useful. Proposition 2.3. Every bijective homomorphism is necessrily n isomorphism. Proof. If f :(; ) (T; T )isnf-homomorphism nd g : T is n inverse of f then g = F(g) F(f) g = F(g) T f g = F(g) T ; thus g is homomorphism. Lemm 2.4. Let ; T; nd U be systems; nd f : T; g : U; nd h : U T ny functions. 1: If f = h g; g is surjective; nd f nd g re homomorphisms; then h is homomorphism. 2: If f = h g; h is injective; nd f nd h re homomorphisms; then g is homomorphism.

9 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) Proof. We prove 1, the proof of 2 is similr. Consider u U nd let s be such tht g(s)=u. Then F(h) U (u) = F(h) U g(s) = F(h) F(g) (s) = F(f) (s) = T f(s) = T h g(s) = T h(u): We now come to the third fundmentl notion of universl colgebr. A bisimultion between two systems is intuitively trnsition structure respecting reltion between sets of sttes. Formlly, it is dened, for n rbitrry set functor F : et et, s follows [4]: Let (; ) nd (T; T )bef-systems. A subset R T of the Crtesin product of nd T is clled n F-bisimultion between nd T if there exists n F-trnsition structure R : R F(R) such tht the projections from R to nd T re F-homomorphisms: We shll lso sy, mking explicit reference to the trnsition structures, tht (R; R ) is bisimultion between (; ) nd (T; T ). If (T; T )=(; ) then (R; R ) is clled bisimultion on (; ). A bisimultion equivlence is bisimultion tht is lso n equivlence reltion. Two sttes s nd t re clled bisimilr if there exists bisimultion R with s; t R. (ee ection 19 for some references to lterntive ctegoricl pproches to bisimultion.) Exmple 2.1 (continued). Consider gin two (lbelled trnsition systems represented s) B-systems (; ) nd (T; T ). We show tht B-bisimultion between nd T is reltion R T stisfying, for ll s; t R,

10 12 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) for ll s in, ifs s then there is t in T with t T t nd s ;t R, 2. for ll t in T,ift T t then there is s in with s s nd s ;t R, which is the fmilir denition of bisimultion from concurrency theory [56, 60]. For let R be B-bisimultion with trnsition structure R : R B(R). As before, R induces reltion R R A R. Let s; t R, nd suppose s s. Becuse s = 1 s; t this implies 1 s; t s, nd becuse 1 is homomorphism, it follows tht there is s ;t R with s; t R s ;t nd 1 s ;t = s. Thus s ;t R. Becuse 2 is homomorphism it follows tht t T t, which concludes the proof of cluse 1. Cluse 2 is proved similrly. Conversely, suppose R stises cluses 1 nd 2. Dene R : R B(R), for s; t R, by R s; t = { ; s ;t s s nd t T t nd s ;t R}: It is immedite from cluses 1 nd 2 tht the projections re homomorphisms from (R; R )to(; ) nd (T; T ). (Note tht in generl R is not the only trnsition structure on R hving this property.) A concrete exmple of bisimultion reltion between two trnsition systems is the following. Consider two systems nd T : Then { s i ;s j i; j 0} { s i;s j i; j 0} is bisimultion on. And { s i ;t i 0} { s i;t i 0} is bisimultion between nd T. Note tht the function f : T dened by f(s i )=t nd f(s i)=t is homomorphism, nd tht there exists no homomorphism from T to. The lst observtion of the exmple bove (tht f is homomorphism) is n immedite consequence of the following fundmentl reltionship between homomorphisms nd bisimultions. Theorem 2.5. Let (; ) nd (T; T ) be two systems. A function f : T is homomorphism if nd only if its grph G(f) is bisimultion between (; ) nd (T; T ). Proof. Let : G(f) F(G(f)) be such tht (G(f);) is bisimultion between (; ) nd (T; T ). Let 1 nd 2 be the projections from G(f) to nd T. Becuse 1 is

11 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) bijective it hs n inverse 1 1 :(; ) (G(f);) which is lso homomorphism. Becuse f = 2 1 1, lso f is homomorphism. Conversely, suppose f is homomorphism. We cn tke F( 1 ) 1 1 s trnsition structure on G(f). This clerly turns 1 into homomorphism. The sme holds for 2 : F( 2 ) (F( 1 ) 1 1 ) = F( ) 1 = F(f) 1 = T f 1 = T 2 : (Becuse F( 1 ) is mono, there is only one trnsition structure on G(f).) Therefore homomorphisms re sometimes clled functionl bisimultions. Remrk 2.6. The chrcteriztion of B-bisimultion in the exmple of trnsition systems bove is n instnce of the following more generl result. Let gin (; ) nd (T; T )betwof-systems. A reltion R T is n F-bisimultion if nd only if, for ll s in nd t in T, s; t R (s);(t) G(F( 1 )) 1 G(F( 2 )); where the ltter expression denotes the reltionl composition of the inverse of the grph of F( 1 ) followed by the grph of F( 2 ). If set functor preserves wek pullbcks, then this composition cn be tken s the denition of the ction of F on the reltion R, thus extending F from the ctegory of sets nd functions to the ctegory of sets nd reltions. uch extensions re sometimes clled reltors. In [74], the connection between reltors, colgebrs nd bisimultions is further investigted. 3. ystems, systems, systems, ::: The colgebrs, homomorphisms, nd bisimultions of number of functors tht cn be considered s the bsic building blocks for most systems re described. (All functors tht re used re described in the Appendix.)

12 14 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) Deterministic systems Deterministic systems exist in mny dierent forms. The simplest ones re colgebrs of the identity functor I()=: s s (s)=s : ; The nottion s s for (s)=s is used s shorthnd, which puts emphsis on the fct tht ctully gives the dynmics of the system (; ), nd should be red s: in stte s the system cn mke trnsition step to stte s. The rrow nottion will turn out to be prticulrly useful for the chrcteriztion of homomorphisms nd bisimultions. Formlly, the bove equivlence is simply stting tht ny function is lso (functionl) reltion. Conversely, it is often convenient to dene the dynmics of system by specifying its trnsitions (in prticulr when deling with nondeterministic systems, see below). For instnce, specifying for the set of nturl numbers trnsitions ; denes the deterministic system (N; succ), where succ is the successor function. A homomorphism between two deterministic systems (; ) nd (T; T ) is function f : T stisfying for ll s in, s s f(s) f(s ): (Note tht we hve dropped the subscripts from nd T, convention we shll often pply.) Thus, homomorphisms between deterministic systems re trnsition preserving functions. A bisimultion between deterministic systems nd T is ny reltion R T such tht, for ll s nd t T, s; t R nd s s nd t t s ;t R: Thus, bisimultions between deterministic systems re trnsition respecting reltions. For instnce, there is n obvious bisimultion reltion between the bove system (N; succ), nd the system Not only re there mny deterministic systems (tke ny set nd ny function from the set to itself), mny of them hve more interesting dynmics thn one would expect t rst sight, in spite of the fct tht the functor t stke is trivil. For instnce, let A be ny set (lphbet) nd let A Z be the set of ll so-clled bi-innite sequences

13 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) (words) over A. (Here Z is the set of ll integers.) It cn be given the following dynmics: shift A Z shift()=m:(m +1): A Z ; This exmple is of centrl importnce in the theory of symbolic dynmics (cf. [12]). There the set of bi-innite words is supplied with metric, by which the shift exmple becomes even more interesting. ee ection 18 for some observtions bout such metric systems Termintion Any set crries colgebr structure of the constnt functor F()=1: 1; s (s)= ; where 1 = { }. Thus cn be viewed s system with trivil dynmics, in which no stte cn tke step nd every stte s termintes, s expressed by the rrow nottion s. Any function between such systems trivilly is homomorphism nd ny reltion bisimultion. Thus the ctegory et 1 of ll systems of the constnt functor is just (isomorphic to) the ctegory of sets. Deterministic systems with termintion re colgebrs of the functor F()=1 + : 1+; s s (s)=s ; s (s)= : uch system cn in stte s either mke trnsition to stte s or terminte. Homomorphisms (nd bisimultions) re s before, with the dditionl property tht

14 16 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 terminting sttes re mpped to (relted to) terminting sttes. Note tht homomorphisms not only preserve but lso reect trnsitions: if f : T is homomorphism nd f(s) t, for s nd t T, then there exists s with s s nd f(s )=t. An exmple of deterministic system with termintion is the system of the extended nturl numbers [6] N = {0; 1; 2;:::} { }, with dynmics ; which, equivlently, cn be dened s N 1+ N; pred if n =0; pred(n)= n 1 if 0 n ; if n = : In this system, nturl number n cn tke precisely n trnsition steps nd then termintes, nd the dditionl number only tkes step to itself nd hence never termintes Input ystems in which stte trnsitions my depend on input re colgebrs of the functor F()= A (here A = {f f : A }): s s (s)()=s ; A ; where A is ny set (to be thought of s n input lphbet) nd the rrow cn be red s: in stte s nd given input, the system cn mke trnsition to stte s. Typicl exmples of deterministic systems with input re deterministic utomt, which trditionlly re represented s pirs (Q; :(Q A) Q), consisting of set Q of sttes nd stte trnsition function tht for every stte q nd input symbol in A determines the next stte q;. (Often n initil stte nd set of nl sttes is specied s well, but they cn be delt with seprtely.) As observed in the introduction, in [66, 52], such utomt re precisely the deterministic systems with input mentioned bove, becuse of the following bijection: {f f : Q A Q} = {f f : Q Q A }:

15 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) A homomorphism between (; ) nd (T; T ) is ny function f : T stisfying for ll s in, in A, s s f(s) f(s ): A bisimultion between systems nd T is now reltion R T such tht, for ll in A, s; t R nd s s nd t t s ;t R: For instnce, ll sttes in the following two systems re bisimilr: 3.4. Output Trnsitions my lso yield n output. Thus we consider colgebrs of the functor F()=A : A ; s s (s)= ; s ; where A is n rbitrry set nd the rrow cn be red s: in stte s, one cn observe the output, nd the system cn mke trnsition to the stte s. An intuition tht often pplies is to consider the output s the observble eect of the stte trnsition. (Note tht the sme rrow nottion is used both for trnsitions with input nd with output. In generl, the right interprettion follows from the context.) uch systems re lso clled deterministic lbelled trnsition systems [65]. Homomorphisms nd bisimultions cn be chrcterized by n obvious vrition on the descriptions bove.

16 18 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 A concrete exmple is the set A! of innite sequences over A, with h; t A! A A! ; 0 ; 1 ;::: 0 1 ; 2 ;::: : The pir h; t ssigns to n innite sequence its hed (the rst element) nd til (the reminder). Adding the possibility of termintion yields, for instnce, the following two vritions, where the functors involved re F()=1+(A ) nd F()=A +(A ): 1+(A ); A +(A ): An exmple of the rst type is the set A of nite nd innite strems, with A 1+(A A ); ; 0 ; 1 ;::: 0 1 ; 2 ;::: : imilrly, the set A + of non-empty nite nd innite strems over A is n exmple of the lst type, A +(A ) Binry systems Binry systems re colgebrs of the functor F()=. Now trnsition yields two new sttes: ; s s 1 ;s 2 (s)= s 1 ;s 2 :

17 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) A homomorphism between binry systems nd T is ny function f : T stisfying for ll s in, s s 1 ;s 2 f(s) f(s 1 );f(s 2 ) : imilrly for bisimultions. A concrete exmple of binry system is the set Z of integers with trnsitions Note tht the fct tht there re two outgoing trnsitions from ech stte should in this context not be interpreted s form of nondeterminism (see below): the system is perfectly deterministic in tht for ech stte one trnsition is possible, leding to pir of new sttes. The system cn equivlently be dened by pred; succ Z Z Z; m m 1;m+1 : Vritions of binry systems cn be obtined by dding lbels (output) nd the possibility of termintion: A ; (A ) (A ); 1+((A ) (A )): Exmples of such systems re, respectively: the set of innite node-lbelled binry trees, where ech node is ssigned its lbel in A, together with the nodes of the two subtrees; the set of innite rc-lbelled binry trees, where node is mpped to the two nodes of its subtrees, ech together with the lbel of the corresponding rc; nd the set of ll rc-lbelled binry trees with nite nd innite brnches.

18 20 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) Nondeterministic systems From one stte, severl trnsitions my be possible: P(); s s s (s): A vrition of this type of systems is obtined by dding lbels, thus considering colgebrs of the functor F()=P(A ): P(A ); s s ; s (s): These re the nondeterministic lbelled trnsition systems of Exmple 2.1, where homomorphisms nd bisimultions hve been chrcterized s trnsition-preserving nd reecting functions nd reltions. Often one wishes to consider systems with bounded nondeterminism, in which from n rbitrry stte only nite number of trnsitions is possible. uch systems cn be modelled using the nite powerset functor: P f (A ); nd re clled nitely brnching. Yet nother clss of systems re the colgebrs of the functor F()=P f () A : P f () A ; which re clled imge nite: for every s in nd in A, the number of rechble sttes {s s s } is nite.

19 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) Hyper systems The contrvrint powerset functor cn be used to model hyper systems, in which stte cn mke nondeterministiclly step to set of sttes: s V V (s): P( P()); Here P()=2 nd thus P( P())=2 (2) ; see Appendix. Note tht we re describing the elements of P( P()) s subsets rther thn chrcteristic functions. The rrow s V should be red s: from stte s the system cn rech the set V of sttes (but not necessrily ech individul element of V ). Using the denition of the contrvrint powerset functor, one cn show tht homomorphism between hyper systems nd T is ny function f : T stisfying, for ll s in nd W T, s f 1 (W ) f(s) W: Bisimultions re generlly not so esy to chrcterize. For the specil cse of bisimultion equivlence R on hypersystem, the following holds: 1 for ll s nd s in, s; s R (for every R-equivlence clss V ; s V s V ): The reder is invited to try nd model hyper systems using the covrint powerset functor, to nd tht the notions of homomorphism nd bisimultion re rther dierent in tht cse. This exmple illustrtes the importnce of functors, which operte both on sets nd on functions, in theory of colgebrs More exmples ome further exmples re given, using functors tht combine some of the bsic constructions mentioned bove. Automt: re systems with input nd output, possibly with termintion, such s (B ) A ; B A ; C +(B A ): 1 This type of bisimultion seems to be underlying mny of the recently proposed probbilistic bisimultions [46, 85]. It ws found in joint work with Erik de Vink [21].

20 22 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 ystems of the rst nd second type re known s Mely mchines nd Moore mchines, respectively, the min dierence being tht with the ltter the output does not depend on the input. For the cse of B =2={0; 1}, Moore mchines re known s deterministic utomt: o; t s s t(s)()=s ; s o(s)=1: 2 A ; The output function o indictes whether stte s in is ccepting (lso clled nl): o(s) = 1, or not: o(s) = 0. The trnsition function t ssigns to stte s function t(s):a, which species the stte t(s)() tht is reched fter n input symbol hs been consumed. Even though we re using the sme nottion s, note tht n ccepting stte is not terminting in the sense used t the beginning of this section, since ny stte s cn, for ny input, progress to next stte t(s)(). Trditionlly (but isomorphiclly), deterministic utomt re represented s sets together with trnsition function of type ( A) (corresponding to t : A ), together with set of ccepting sttes F (corresponding to o : 2). The colgebric theory of this clssicl exmple of utomt is described in ll detil in [73]. Grphs: A directed (1-)grph (V; E) consists of set V of points (vertices) nd n edge reltion E V V, representing the rcs of the grph. Grphs re in one-to-one correspondence with nondeterministic systems becuse of the bijection {f f : V P(V )} = P(V V ): Note tht the stndrd notion of grph homomorphism is function preserving the rc reltion [77], without necessrily reecting it. In contrst, homomorphism of (grphs s) nondeterministic systems both preserves nd reects the rcs, s consequence of the ctegoricl denition of homomorphism of F-colgebrs. Nevertheless, the trditionl wy of representing grphs nd rc-preserving homomorphisms between them cn be modelled in the present frmework by considering the following, so to spek mny-sorted colgebric denition: 2 Consider the functor F :(et et) (et et); X; Y 1; X X : 2 This denition ws suggested by Andre Corrdini.

21 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) A grph (V; E) cn be represented s colgebr of F by dening (V; E) (1; s; t ) (1;V V ); where s : E V nd t : E V re the projections from E to V, which we cll source nd trget. An F-homomorphism (1; s; t ) (V; E) (f; g) (V ;E ) (1; s ;t ) (1;V V ) (1;f f) (1;V V ) is pir of functions f : V V nd g : E E such tht f(s(e)) = s (g(e)); f(t(e)) = t (g(e)); which is the usul denition of grph homomorphism. Frmes nd models: Afrme in the world of modl logic (cf. [25]) is directed grph, nd thus (s we hve seen bove) cn be represented s nondeterministic system. A model (V; E; f) is frme (V; E) together with function f : P(V ), where is collection of tomic formuls in some given modl logic. Intuitively, f species for ech formul in which sttes v in V it holds. Becuse of the isomorphism {f f : P(V )} = {f f : V P()}; it is esily veried tht models correspond to systems of type V P() P(V ): As it turns out, homomorphisms nd bisimultions for these systems correspond precisely to the so-clled p-morphisms nd zig-zg reltions of modl logic.

22 24 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 Resumptions: re systems of type (P(B )) A : In other words, resumptions re nondeterministic systems with input nd output. They ply centrl role in the semntics of (nondeterministic nd prllel) progrmming lnguges (cf. [8, 29]). 4. Limits nd colimits of systems We wnt to prove sttements like: the union of collection of bisimultions is gin bisimultion; the quotient of system with respect to bisimultion equivlence is gin system; nd the kernel of homomorphism is bisimultion equivlence. These fcts re well-known for certin systems such s nondeterministic lbelled trnsition systems. As it turns out, they do not depend on prticulr properties of such exmples, nd ctully pply to (lmost) ll systems we hve seen sofr. Therefore, this section presents number of bsic ctegoricl constructions tht will enble us, in the subsequent sections, to prove ll these sttements for ll systems t the sme time. There re three bsic constructions in the ctegory et F of F-systems tht re needed: the formtion of coproducts (sums), coequlizers, nd pullbcks (cf. Appendix). In this section, they re discussed in some detil for rbitrry F-systems. The fmily of lbelled trnsition systems is used gin s running exmple. (We shll see tht in et F coproducts nd coequlizers exist, for rbitrry functors F. If the functor F preserves pullbcks, then lso pullbcks exist in et F nd they cn be constructed s in et. For completeness, generl description of limits nd colimits of systems is presented t the end of this section.) 4.1. Coproducts Coproducts (s well s coequlizers nd, more generlly ny type of colimit) in et F re s esy s they re in the ctegory et. The coproduct (or sum) of two F-systems (; ) nd (T; T ) cn be constructed s follows. Let i : ( + T ) nd i T : T ( + T ) be the injections of nd T into their disjoint union. It is esy to see tht there is unique function :( + T ) F( + T ) such tht both i nd i T re

23 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) homomorphisms: The function cts on s F(i ) nd on T s F(i T ) T. The system ( + T; ) hs the following universl property: for ny system (U; U ) nd homomorphisms k :(; ) (U; U ) nd l :(T; T ) (U; U ) there exists unique homomorphism h :( + T; ) (U; U ) mking the following digrm commute: Tht is ( + T; ) is the coproduct of (; ) nd (T; T ). imilrly, the coproduct of n indexed fmily { i } i I of systems cn be constructed. Exmple 4.1. Recll from Exmple 2.1 tht lbelled trnsition systems (lts) re B- systems where B(X )=P(A X ). The coproduct of two lts s (; ) nd (T; T ) consists of the disjoint union + T of the sets of sttes together with B-trnsition structure : + T B( + T ), dened for s in nd t in T by (s)= (s); (t)= T (t): Becuse A A ( + T) nd A T A ( + T ) (identifying for convenience + T nd T ), this denes indeed function from + T into B( + T ) Coequlizers Next we show how in et F coequlizer of two homomorphisms cn be constructed. Consider two homomorphisms f :(; ) (T; T ) nd g :(; ) (T; T ). We hve to nd system (U; U ) nd homomorphism h :(T; T ) (U; U ) such tht 1. h f = h g; 2. for every homomorphism h :(T; T ) (U ; U ) such tht h f = h g, there exists unique homomorphism l :(U; U ) (U ; U ) with the property tht l h = h. ince (per denition) f nd g re functions f : T nd g : T in et, there exists coequlizer h : T U in et (see Appendix). Consider F(h) T : T F(U). Becuse F(h) T f = F(h) F(f)

24 26 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 = F(h f) = F(h g) = F(h) F(g) = F(h) T g; nd h : T U is coequlizer, there exists unique function U : U F(U) mking the following digrm commute: Thus (U; U )isnf-system nd h is homomorphism. One esily checks tht the universl property (2) is stised. Exmple 4.1 (continued). Let (; ) nd (T; T ) be gin two lts s nd consider homomorphisms f; g :(; ) (T; T ). Let R be the smllest equivlence reltion on T tht contins the set { f(s);g(s) s }; nd let q : T T=R be the function tht mps t in T to its R-equivlence clss [t] R. Then T=R cn be supplied with B-trnsition structure R : T=R B(T=R) by specifying trnsitions [t] R [t ] R t [t ] R ; t T t : It is moreover the only possible choice for R mking q : T T=R into homomorphism. A specil instnce of this exmple is obtined by tking bisimultion equivlence on B-system, sy 1 ; 2 :(R; R ) (T; T ): Then the coequlizer of 1 nd 2 is the quotient T=R, showing tht the quotient of n lts with respect to bisimultion equivlence yields gin n lts. This observtion will be generlized in Proposition 5.8. The results bove re summrized for future reference in the following. Theorem 4.2. Let F : et et be ny functor. In the ctegory et F of F-systems; ll coproducts nd ll coequlizers exist; nd re constructed s in et.

25 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) (Wek) pullbcks The construction of pullbcks in et F depends on the functor F. More speciclly, if F : et et preserves pullbcks then pullbcks in et F cn be constructed s in et: Let f :(; ) (T; T ) nd g :(U; U ) (T; T ) be homomorphisms. Let 2 P U 1 g f T be the pullbck of f nd g in et, with P = { s; u f(s)=g(u)}. Becuse F preserves pullbcks, F( 2) F(P) F(U) F( 1) F(g) F() F(f) F(T ) is pullbck of F(f) nd F(g) inet. Consider 1 : P F() nd U 2 : P F(U). Becuse F(f) 1 = T f 1 = T g 2 = F(g) U 2 ; there exists, by the fct tht F(P) is pullbck, unique function P : P F(P) such tht F( 1 ) P = 1 nd F( 2 ) P = U 2. Thus (P; P )isnf-system, nd 1 nd 2 re homomorphisms. It is esily veried tht (P; P ) is pullbck of f nd g in et F. Note tht s consequence, the pullbck (P; P ) is bisimultion on nd U : P U nd the projections 1 nd 2 re homomorphisms.

26 28 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) 3 80 As it turns out, the pullbck of two homomorphisms is bisimultion even in the cse tht F only preserves wek pullbcks (cf. Appendix). Theorem 4.3. Let F : et et be functor tht preserves wek pullbcks; nd let f :(; ) (T; T ) nd g :(U; U ) (T; T ) be homomorphisms of F-systems. Then the pullbck (P; 1 ; 2 ) of f nd g in et is bisimultion on nd T. Proof. The proof is essentilly the sme s the proof of the existence of pullbcks in et F in cse F preserves pullbcks. The only dierence is tht F(P) is now, by ssumption, wek pullbck. As consequence, there exists gin (no longer necessrily unique) trnsition structure P : P F(P) on P such tht 1 nd 2 re homomorphisms. Exmple 4.1 (continued). Let f :(; ) (T; T ) nd g :(U; U ) (T; T ) be homomorphisms of lts s. Becuse lts s re B-systems nd the functor B preserves wek pullbcks (cf. Appendix), the bove rgumenttion pplies. The following gives more direct construction. As bove, let P = { s; u f(s)=g(u)}. It cn be supplied with B-trnsition structure by specifying trnsitions s; u s ;u f(s )=g(u ) nd s s nd u U u : It is strightforwrd to prove tht the projections from P to nd U re homomorphisms. Thus P is bisimultion. A specil cse is obtined by tking only one homomorphism f :(; ) (T; T ) nd considering the pullbck of f nd f. The resulting set is P = { s; s f(s)=f(s )}, which is the kernel of f. It follows tht it is bisimultion (equivlence). Agin, this will be proved in greter generlity in Proposition 5:7. Becuse Theorem 4.3 will be clled upon time nd gin, nd becuse ll functors we hve seen in the exmples sofr do preserve wek pullbcks (but for the contrvrint powerset functor, cf. Appendix), we shll ssume in the sequel tht when tlking bout n rbitrry functor F, it preserves wek pullbcks: Convention In the rest of this pper; set functors F : et et re ssumed to preserve wek pullbcks. 4 If (the proof of) lemm; proposition; or theorem ctully mkes use of this ssumption; then it is mrked with n sterisk. 3 Functors F : et et tht preserve wek pullbcks re reltors (cf. Remrk 2.6 nd [74]). 4 ometimes notbly in Theorem 6:4 we shll ssume F to preserve generlized wek pullbcks; i.e.; pullbcks of more thn two, possibly innitely mny functions t the sme time. It ws pointed out to us by H.Peter Gumm tht this is in fct stronger requirement.

27 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) Limits nd colimits, generlly This section is concluded with the observtion tht the bove constructions of coproducts, coequlizers, nd pullbcks cn be generlized by mens of the so-clled forgetful functor U : et F et, which sends systems to their crrier: U(; )=, nd homomorphisms f :(; ) (T; T ) to the function f : T. (see, e.g., [9]). Theorem 4.5. The functor U : et F et cretes colimits. This mens tht ny type of colimit in et F exists; nd is obtined by rst constructing the colimit in et nd next supplying it (in unique wy) with n F-trnsition structure. imilrly, there is the following generl sttement bout limits in et F. Theorem 4.6. If F : et et preserves (certin type of) limit; then the functor U : et F et cretes tht (type of) limit. This mens tht ny type of limit in et tht is preserved by F lso exists in et F ; nd is obtined by rst constructing the limit in et nd next supplying it (in unique wy) with n F-trnsition structure. Recently, it hs been shown tht in et F ll limits exist, independently of the question whether they re preserved by the functor F or not [67]. Wht Theorem 4.6 sys is tht in cse F does preserve certin limit, then the crrier of the corresponding limit in et F is precisely the limit in et. In generl, however, limits in et F look quite bit more complicted thn the corresponding limits in et of the underlying crriers. The interested reder is invited to compute, for instnce, the product of the following nondeterministic trnsition system with itself Epi s nd mono s in et F Using the results of this section, we re now in position to supply the detils nnounced in Remrk 2.2 bout epi s nd mono s in the ctegory et F of F-systems. Proposition 4.7. Let F : et et be functor nd f :(; ) (T; T ) n F-homomorphism. 1. The homomorphism f is n epimorphism (i.e.; surjective) if nd only if f is epi in the ctegory et F.

28 30 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) If the homomorphism f is monomorphism (i.e.; injective) then it is mono in the ctegory et F. If the functor F preserves wek pullbcks then the converse is lso true: if f is mono then it is injective. Proof. We use the following ctegoricl chrcteriztion of epi s [13, Proposition 2:5:6]. Let C be n rbitrry ctegory. An rrow : A B in C is epi if nd only if the following digrm is pushout in C: A B B 1B 1 B B: By Theorem 4.5, the forgetful functor U : et F et cretes colimits nd hence pushouts. Moreover, it is esily veried tht U preserves ny colimit tht it cretes. o in prticulr U preserves pushouts. Thus we obtin the following equivlence: (; ) f (T; T ) f (T; T ) 1T 1 T (T; T ) is pushout in et F f T f T 1T 1 T T is pushout in et: As consequence, the homomorphism f is epi in et F if nd only if the function f is epi, nd hence surjective, in et. Injective homomorphisms re redily seen to be mono s in et F. For the converse, there is the following elementry proof (suggested to us by Tobis chroeder). Let f : T be mono in the ctegory et F. We shll see lter tht if F preserves wek pullbcks then the kernel K(f) is bisimultion (Proposition 5.7). Let 1 ; 2 : K(f) be the projections. Then f 1 = f 2, by the denition of K(f), implying 1 = 2 since f is mono. This proves tht f is injective.

29 J.J.M.M. Rutten / Theoreticl Computer cience 249 (2000) Bsic fcts on bisimultions This section dels with rbitrry F-systems. For ech prticulr choice of F, ll the results of this section re strightforwrd. In fct, some of them hve lredy been proved for the specil cse of lbelled trnsition systems in Exmple 4:1. The point is to prove such properties for ll F-systems t the sme time. Let, T nd U be three F-systems with trnsition structures, T nd U, respectively. Proposition 5.1. The digonl of system is bisimultion. Proof. Follows from Theorem 2.5 nd the observtion tht equls the grph of the identity 1 :. The inverse of bisimultion is bisimultion. Theorem 5.2. Let (R; R ) be bisimultion between systems nd T. The inverse R 1 of R is bisimultion between T nd. Proof. Let i : R R 1 be the isomorphism sending s; t R to t; s R 1. Then (R 1 ; F(i) R i 1 ) is bisimultion between T nd. Consider two homomorphisms with common domin T, T g U: f uch pir is sometimes clled spn. The following lemm sys tht the imge of spn is bisimultion. The lemm will be used to prove tht the composition nd union of bisimultions is gin bisimultion. Lemm 5.3. The imge f; g (T )={ f(t);g(t) t T } of two homomorphisms f : T nd g : T U is bisimultion between nd U. Proof. Consider the following digrm: