Draft. Draft. Introduction to Coalgebra. Towards Mathematics of States and Observations. Bart Jacobs. Draft Copy.

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1 1 Introduction to Colgebr. Towrds Mthemtics of Sttes nd Observtions Brt Jcobs Institute for Computing nd Informtion Sciences, Rdboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlnds. brt Copy. Comments / bugs / improvements etc. re welcome t brt@cs.ru.nl (Plese check the ltest version on the web first, to see if the issue you wish to report hs lredy been dderessed) Version 1.00, 22nd April 2005

2 ii Prefce Mthemtics is bout the forml structures underlying counting, mesuring, trnsforming etc. It hs developed fundmentl notions like number systems, groups, vector spces, see e.g. [168], nd hs studied their properties. In more recent decdes lso dynmicl fetures hve become subject of reserch. The emergence of computers hs contributed to this development. Typiclly, dynmics involves stte of ffirs, which cn possibly be observed nd modified. For instnce, the contents of tpe of Turing mchine contribute to its stte. Such mchine my thus hve mny possible sttes, nd cn move from one stte to nother. Also, the combined contents of ll memory cells of computer cn be understood s the computers stte. A user cn observe prt of this stte vi the screen (or vi the printer), nd modify this stte by typing commnds. In rection, the computer cn disply certin behviour. Describing the behviour of such computer system is non-trivl mtter. However, forml descriptions of such complicted systems re needed if we wish to reson formlly bout their behviour. Such resoning is required for the correcteness or security of these systems. It involves specifiction describing the required behviour, together with correctness proof demonstrting tht given implementtion stisfies the specifiction. Mthemticins nd computer scientists hve introduced vrious forml structures to cpture the essence of stte-bsed dynmics, such s utomt (in vrious forms), trnsitions systems, Petri nets, event systems, etc. The re of colgebrs hs emerged with unifying clim. It ims to be the mthemtics of computtionl dynmics. It combines notions nd ides from the mthemticl theory of dynmicl systems nd from the theory of stte-bsed computtion. The re of colgebr is still in its infncy, but promises perspective on uniting, sy, the theory of differentil equtions with utomt nd process theory, by providing n pproprite semnticl bsis with ssocited logic. The theory of colgebrs my be seen s one of the originl contributions stemming from the re of theoreticl computer science. The spn of pplictions of colgebrs is still firly limited, but my in the future be extended to include dynmicl phenomen in res like physics, biology or economics bsed for instnce on the clim of Adlemn (the fther of DNAcomputing) tht biologicl life cn be equted with computtion [12]; or on [186] which gives colgebric description of type spces used in economics [110]. Colgebrs re of surprising simplicity. They consist of stte spce, or set of sttes, sy X, together with structure mp of the form X F (X). The symbol F describes some expression involving X ( functor), cpturing the possible outcomes of the structure mp pplied to stte. The mp X F (X) cptures the dynmics in the form of function cting on sttes. For instnce, one cn hve F (X) = P(X) for non-deterministic computtion X P(X), or F (X) = { } X for possibly non-terminting computtions X { } X. At this level of generlity, lgebrs re described s the duls of colgebrs (or the other wy round), nmely s mps of the form F (X) X. Computer science is bout generted behviour Wht is the essence of computing? Wht is the topic of the discipline of computer science? Answers tht re often herd re dt processing or symbol mnipultion. Here iii

3 iv we follow more behviouristic pproch nd describe the subject of computer science s generted behviour. This is the behviour tht cn be observed on the outside of computer, for instnce vi screen or printer. It rises in interction with the environment, s result of the computer executing instructions, lyed down in computer progrm. The im of computer progrmming is to mke computer do certin things, i.e. to generte behviour. By executing progrm computer displys behviour tht is ultimtely produced by humns, s progrmmers. This behviouristic view llows us to understnd the reltion between computer science nd the nturl sciences: biology is bout spontneous behviour, nd physics concentrtes on lifeless nturl phenomen, without utonomous behviour. Behviour of system in biology or physics is often described s evolution, where evolutions in physics re trnsformtionl chnges ccording to the lws of physics. Evolutions in biology seem to lck inherent directionlity nd predictbility [98]. Does this men tht behviour is deterministic in (clssicl) physics, nd non-deterministic in biology? And tht colgebrs of corresponding kinds cpture the sitution? At this stge the colgebric theory of modeling hs not yet demonstrted its usefulness in those res. Therefore this text concentrtes on colgebrs in mthemtics nd computer science. The behviouristic view does help in nswering questions like: cn computer think? Or: does computer feel pin? All computer cn do is disply thinking behviour, or pin behviour, nd tht is it. But it is good enough in interctions think of the fmous Turing test becuse in the end we never know for sure if other people ctully feel pin. We only see pin behviour, nd re conditioned to ssocite such behviour with certin internl sttes. But this ssocition my not lwys work, for instnce not in different culture: in Jpn it is common to touch one s er fter burning finger; for Europens this is non-stndrd pin behviour. This issue of externl behviour versus internl sttes is nicely demonstrted in [177] where it turns out to be surprisingly difficult for humn to kill Mrk III Best robot, once it strts displying desprte survivl behviour with corresponding sounds, so tht people esily ttribute feelings to the mchine nd strt to feel pity. These wide-rnging considertions form the bckground for theory bout computtionl behviour in which the reltion between observbles nd internl sttes is of centrl importnce. The generted behviour tht we clim to be the subject of computer science rises by computer executing progrm ccording to strict opertionl rules. The behviour is typiclly observed vi the computer s input & output (I/O). More techniclly, the progrm cn be understood s n element in n inductively defined set P of terms. This set forms suitble (initil) lgebr F (P ) P, where the expression (or functor) F cptures the signture of the opertions for forming progrms. The opertionl rules for the behviour of progrms re described by colgebr P G(P ), where the functor G cptures the kind of behviour tht cn be displyed such s deterministic, or with exceptions. In bstrct form, generted computer behviour mounts to the repeted evlution of n (inductively defined) colgebr structure on n lgebr of terms. Hence the lgebrs nd colgebrs tht re studied systemticlly in this text form the bsic structures t the hert of computer science. One of the big chllenges of computer science is to develop techniques for effectively estblishing properties of generted behviour. Often such properties re formulted positively s wnted, functionl behviour. But these properties my lso be negtive, like in computer security, where unwnted behviour must be excluded. However, n pproprite logicl view bout progrm properties within the combined lgebric/colgebric setting hs not been fully elborted yet. Algebrs nd colgebrs The dulity with lgebrs forms source of inspirtion nd of opposition: there is htelove reltionship between lgebr nd colgebr. First, there is fundmentl divide. Think of the difference between n inductively defined dt type in functionl progrmming lnguge (n lgebr) nd clss in n object-oriented progrmming lnguge ( colgebr). The dt type is completely determined by its constructors : lgebric opertions of the form F (X) X going into the dt type. The clss however involves n internl stte, given by the vlues of ll the public nd privte fields of the clss. This stte cn be observed (vi the public fields) nd cn be modified (vi the public methods). These opertions of clss ct on stte (or object) nd re nturlly described s destructors pointing out of the clss: they re of the colgebric form X F (X). Next, besides these differences between lgebrs nd colgebrs there re lso mny correspondences, nlogies, nd dulities, for instnce between bisimultions nd congruences, or between initility nd finlity. Whenever possible, these connections will be mde explicit nd will be exploited in the course of this work. As lredy mentioned, ultimtely, stripped to its bre minimum, progrmming lnguge involves both colgebr nd n lgebr. A progrm is n element of the lgebr tht rises (s so-clled initil lgebr) from the progrmming lnguge tht is being used. Ech lnguge construct corresponds to certin dynmics, cptured vi colgebr. The progrm s behviour is thus described by colgebr cting on the stte spce of the computer. This is the view underlying the so-clled structurl opertionl semntics. Colgebric behviour is generted by n lgebric progrm. This is simple, cler nd ppeling view. It turns out tht in such situtions one needs certin level of comptibility between the lgebrs nd colgebrs involved. It is expressed in terms of so-clled distributive lws connecting lgebr-colgebr pirs. These lws pper towrds the end of this text. Colgebrs hve blck box stte spce Colgebr is thus the study of sttes nd their opertions nd properties. The set of sttes is best seen s blck box, to which one hs limited ccess like with the sttes of computer mentioned bove. As lredy mentioned, the tension between wht is ctully inside nd wht cn be observed externlly is t the hert of the theory of colgebrs. Such tension lso rises for instnce in quntum mechnics where the reltion between observbles nd sttes is n issue. Similrly, it is n essentil element of cryptogrphy tht prts of dt re not observble vi encryption or hshing. In colgebr it my very well be the cse tht two sttes re internlly different, but re indistinguishble s fr s one cn see with the vilble opertions. In tht cse one clls the two sttes bisimilr. Bisimilrity is indeed one of the fundmentl notions of the theory of colgebrs, see Chpter 3. Also importnt re invrint properties of sttes: once such property holds, it continues to hold no mtter which of the vilble opertions is pplied, see Chpter 4. Sfety properties of systems re typiclly expressed s invrints. Finlly, specifictions of the behviour of systems re conveniently expressed using ssertions nd modl opertors like: for ll direct successor sttes (nexttime), for ll future sttes (henceforth), for some future stte (eventully), see Chpter 5. This text describes these bsic elements of the theory of colgebrs bisimilrity, invrints nd ssertions. It is ment s n introduction to this new nd fscinting field within theoreticl computer science. The text is too limited in both size nd ims to justify the grnd unifying clims mentioned bove. But hopefully, it does inspire nd generte much further reserch in the re. Brief historicl perspective Colgebr does not come out of the blue. Below we shll sketch severl, reltively independent, developments during the lst few decdes tht ppered to hve common v

4 vi colgebric bsis, nd tht hve contributed to the re of colgebr s it stnds tody. This short sketch is of course fr from complete. 1. The ctegoricl pproch to mthemticl system theory. During the 1970s Arbib, Mnes nd Goguen nlysed Klmn s [153] work on liner dynmicl systems, in reltion to utomt theory. They relised tht linerity does not relly ply role in Klmn s fmous results bout miniml relistion nd dulity, nd tht these results could be reformulted nd proved more bstrctly using elementry ctegoricl constructions. Their im ws to plce sequentil mchines nd control systems in unified frmework (bstrct of [20]), by developing notion of mchine in ctegory (see lso [8]). This led to generl notions of stte, behviour, rechbility, observbility, nd relistion of behviour. However, the notion of colgebr did not emerge explicitly in this pproch, probbly becuse the setting of modules nd vector spces from which this work rose provided too little ctegoricl infrstructure (especilly: no Crtesin closure) to express these results purely colgebriclly. 2. Non-well-founded sets. Aczel [4] formed next crucil step with his specil set theory tht llows infinitely descending -chins, becuse it used colgebric terminology right from the beginning. The development of this theory ws motivted by the desire to provide mening to Milner s theory CCS of concurrent processes with potentilly infinite behviour. Therefore, the notion of bisimultion from process theory plyed crucil role. An importnt contribution of Aczel is tht he showed how to tret bisimultion in colgebric setting, especilly by estblishing the first link between proofs by bisimultions nd finlity of colgebrs, see lso [7, 5]. 3. Dt types of infinite objects. The first systemtic pproch to dt types in computing [93] relied on initility of lgebrs. The elements of such lgebric structures re finitely generted objects. However, mny dt types of interest in computing (nd mthemtics) consist of infinite objects, like infinite lists or trees (or even rel numbers). The use of (finl) colgebrs in [237, 21, 107, 193] to cpture such structures provided next importnt step. 4. Initil nd finl semntics. In the semntics of progrm nd process lnguges it ppered tht the relevnt semnticl domins crry the structure of finl colgebr (sometimes in combintion with initil lgebr structure [80, 72]). Especilly in the metric spce bsed trdition (see e.g. [26]) this insight ws combined with Aczel s techniques by Rutten nd Turi. It culminted in the recognition tht comptible lgebr-colgebr pirs (clled bilgebrs) re highly relevnt structures, described vi distributive lws. The bsic observtion of [232, 231], further elborted in [33], is tht such lws correspond to specifiction formts for opertionl rules on (inductively defined) progrms. These bilgebrs stisfy elementry properties like: observtionl equivlence (i.e. bisimultion wrt. the colgebr) is congruence (wrt. the lgebr). 5. Behviourl pproches in specifiction. Reichel [204] ws the first to use soclled behviourl vlidity of equtions in the specifiction of lgebric structures tht re computtionlly relevnt. The bsic ide is to divide one s types (lso clled sorts) into visible nd hidden ones. The ltter re supposed to cpture sttes, nd re not directly ccessible. Equlity is only used for the observble elements of visible types. For elements of hidden types (or sttes) one uses behviourl equlity insted: two elements x 1 nd x 2 of hidden type re behviourlly equivlent if t(x 1 ) = t(x 2 ) for ech term t of visible type. This mens tht they re equl s fr s cn be observed. The ide is further elborted in wht hs become known s hidden lgebr [92], see for instnce lso [84, 223, 39], nd hs been pplied to describe clsses in object-oriented progrmming lnguges, which hve n encpsulted stte spce. But it ws lter relised tht behviourl equlity is essentilly bisimilrity in colgebric context (see e.g. [172]), nd it ws gin Reichel [206] who first used colgebrs for the semntics of object-oriented lnguges. Lter on they hve been pplied lso to ctul progrmming lnguges like Jv [141]. 6. Modl logic. A more recent development is the connection between colgebrs nd modl logics. In generl, such logics qulify the truth conditions of sttements, concerning knowledge, belief nd time. In computer science such logics re used to reson bout the wy progrms behve, nd to express dynmicl properties of trnsitions between sttes. Temporl logic is prt of modl logic which is prticulrly suitble for resoning bout (rective) stte-bsed systems, s rgued for exmple in [200, 201], vi its nexttime nd lsttime opertors. Since colgebrs give bstrct formlistions of such stte-bsed systems one expects connection. It ws Moss [183] who first ssocited suitble modl logic to colgebrs which inspired much subsequent work [209, 211, 165, 123, 136, 190, 163]. The ide is tht the role of equtionl formuls in lgebr is plyed by modl formuls in colgebr. Position of this text There re severl recent texts presenting synthesis of severl of the developments in the re of colgebr [144, 233, 100, 216, 164, 191]. This text is first systemtic presenttion of the subject in the form of book. Key phrses re: colgebrs re generl dynmicl systems, finl colgebrs describe behviour of such systems (often s infinite objects) in which sttes nd observtions coincide, bisimilrity expresses observtionl indistinguishbility, the nturl logic of colgebrs is modl logic, etc. During the lst decde colgebric community hs emerged, centered round the workshops Colgebric Methods in Computer Science, see the proceedings [139, 145, 207, 57, 185, 101, 11] nd ssocited specil journl issues [140, 146, 58, 102]. This text is specificlly not focused on tht community, but tries to rech wider udience. This mens tht the emphsis lies on explining the theory vi concrete exmples, nd on motivtion rther thn on generlity nd bstrction. Colgebr nd ctegory theory The field of colgebr requires the theory of ctegories lredy in the definition of the notion of colgebr itself since it requires the concept of functor. However, the reder is not ssumed to know ctegory theory: in this text the intention is not to describe the theory of colgebrs in its highest form of generlity, mking systemtic use of ctegory theory right from the beginning. After ll, this is only n introduction. Rther, the text strts from concrete exmples nd introduces the bsics of ctegory theory s it proceeds. Ctegories will thus be introduced grdully, without mking it proper subject mtter. Hopefully, reders unfmilir with ctegory theory cn thus pick up the bsics long the wy, seeing directly how it is used. Anywy, most of the exmples tht re discussed live in the fmilir stndrd setting of sets nd functions, so tht it should be reltively esy to see the underlying ctegoricl structures in concrete setting. Thus, more or less fmilir set-theoretic lnguge is used most of the time, but with perspective on the greter generlity offered by the theory of ctegories. In this wy we hope to serve the reders without bckground in ctegory theory, nd t the sme time offer the more experienced cognoscienti n ide of wht is going on t more bstrct level which they cn find to limited extent in the exercises, but to greter extent in the literture. Clerly, this is compromise which runs the risk of stisfying no-one: the description my be too bstrct for some, nd too concrete for others. The hope is tht it does hve something to offer for everyone. Often the theory of ctegories is seen s very bstrct prt of mthemtics, tht is not very ccessible. However, it is essentil in this text, for severl good resons. vii

5 viii 1. It gretly helps to properly orgnise the relevnt mteril on colgebrs. 2. Only by using ctegoricl lnguge the dulity between colgebr nd lgebr cn be fully seen nd exploited. 3. Almost ll of the literture on colgebr uses ctegory theory in one wy or nother. Therefore, n introductory text tht wishes to properly prepre the reder for further study cnnot void the lnguge of ctegories. Probbly the most controversil spect of this text within the colgebric / ctegoricl community is its restriction to so-clled polynomil functors, nd its emphsis on the ssocited opertions of predicte nd reltion lifting. Agin, this is motivted by our wish to produce n introduction tht is ccessible to non-specilists. Certinly, the generl perspective is lwys right round the corner, nd will hopefully be pprecited once this more introductory mteril hs been digested. In the end, we think tht colgebrs form very bsic nd nturl mthemticl concept, nd tht their identifiction is rel step forwrd. Mny people seem to be using colgebrs in vrious situtions, without being wre of it. Hopefully this text cn mke them wre, nd cn contribute to better understnding nd exploittion of these situtions. And hopefully mny more such ppliction res will be identified, further enriching the theory of colgebrs. Intended udience This text is written for everyone with n interest in the mthemticl spects of computtionl behviour. This probbly includes primrily mthemticins, logicins nd (theoreticl) computer scientists, but hopefully lso n udience with different bckground such s for instnce mthemticl physics or biology, or even economics. A bsic level of mthemticl mturity is ssumed, for instnce vi fmilirity with elementry set theory nd logic (nd its nottion). The exmples in the text re tken from vrious res. Ech section is ccompgnied by series of exercises, to fcilitte teching typiclly t lte bchelor or erly mster level. Acknowledgements [To be written.] Contents Prefce 1 Motivtion Nturlness of colgebric representtions The power of the coinduction Generlity of temporl logic of colgebrs Temporl opertors for sequences Temporl opertors for clsses Abstrctness of the colgebric notions Preliminries on colgebrs nd lgebrs Constructions on sets Polynomil functors nd their colgebrs Sttements nd sequences Trees Deterministic utomt Non-deterministic utomt nd trnsition systems Context-free grmmrs Non-well-founded sets Finl colgebrs Beyond sets Algebrs Bilgebrs Dilgebrs Hidden lgebrs Colgebrs s lgebrs Adjunctions, cofree colgebrs, behviour-relistion Bisimultions Reltion lifting, bisimultions nd congruences Properties of bisimultions Bisimultions s spns Compring definitions of bisimultion Bisimultions nd the coinduction proof principle Process semntics Process descriptions A simple process lgebr Invrints Predicte lifting Predicte lowering s liftings left djoint Invrints iii ix

6 x CONTENTS Gretest invrints nd limits of colgebrs Temporl logic of colgebrs Bckwrd resoning Existence of finl colgebrs Finl colgebrs for ω-continuous functors Finl colgebrs for ω-ccessible functors Trce semntics Assertions [Not included] Arities Algebric specifiction Monds Colgebric specifiction Comonds Modl logic of colgebrs A bounded stck specifiction Modl lgebrs nd colgebrs Colgebric clss specifictions Solving recursive equtions vi finlity Finite nd infinite terms, nd substitution Recursive equtions Algebr meets colgebr 181 References 181 Subject Index 197 Definition nd Symbol Index 205 Chpter 1 Motivtion This chpter tries to explin why colgebrs re interesting structures in mthemtics nd computer science. It does so vi severl exmples. The nottion used for these exmples will be explined informlly, s we proceed. The emphsis is t this stge not so much on precision in explntion, but on trnsfer of ides nd intuitions. Therefore, for the time being we define colgebr very informlly to be function of the form: S c S (1.1) Wht we men is: colgebr is given by set S nd function c with S s domin nd with structured codomin (result, output, the box ), in which the domin S my occur gin. The precise form of these codomin boxes is not of immedite concern. Some terminology: We often cll S the stte spce or set of sttes, nd sy tht the colgebr cts on S. The function c is sometimes clled the trnsition function or lso trnsition structure. The ide tht will be developed is tht colgebrs describe generl stte-bsed systems provided with dynmics given by the function c. For stte x S, the result c(x) tells us wht the successor sttes of x re, if ny. The codomin is often clled the type or interfce of the colgebr. Lter we shll see tht it is functor. A simple exmple of colgebr is the function, Z n (n 1, n + 1) Z Z with stte spce Z occurring twice on the right hnd side. Thus the box or type of this colgebr is: ( ) ( ). The trnsition function n (n 1, n + 1) my lso be written using λ-nottion s λn. (n 1, n + 1) or s λn Z. (n 1, n + 1). Another exmple of colgebr, this time with stte spce the set A N of functions from N to some given set A, is: A N σ (σ(0), λn. σ(n + 1)) A A N In this cse the box is A ( ). If we write σ s n infinite sequence (σ n ) n N we my write this colgebr s pir of functions hed, til where hed ( ) (σ n ) n N = σ 0 nd til ( ) (σ n ) n N = (σ n+1 ) n N. Mny more exmples of colgebrs will occur throughout this text. This chpter is devoted to selling nd promoting colgebrs. It does so by focusing on the following topics. 1

7 2 CHAPTER 1. MOTIVATION 1. A representtion s colgebr (1.1) is often very nturl. 2. There re poweful coinductive definition nd proof principles for colgebrs. 3. There is very nturl (nd generl) temporl logic ssocited with colgebrs. 4. The colgebric notions re on suitble level of bstrction, so tht they cn be recognised nd used in vrious settings. Full pprecition of this lst point requires some fmilirity with bsic ctegory theory. It will be provided in Section Nturlness of colgebric representtions We turn to first re where colgebric representions s in (1.1) occur nturlly nd my be useful, nmely progrmming lnguges used for writing computer progrms. Wht re progrms, nd wht do they do? Well, progrms re lists of instructions telling computer wht to do. Fir enough. But wht re progrms from mthemticl point of view? Put differently, wht do progrms men 1? One view is tht progrms re certin functions tht tke n input nd use it to compute certin result. This view does not cover ll progrms: certin progrms, often clled processes, re ment to be running forever, like operting systems, without relly producing result. But we shll follow the view of progrms s functions for now. The progrms we hve in mind do not only work on input, but lso on wht is usully clled stte, for exmple for storing intermedite results. The effect of progrm on stte is not immeditely visible, nd is therefore often clled the side-effect of the progrm. One my think of the stte s given by the contents of the memory in the computer tht is executing the progrm. This is not directly observble. Our progrms should thus be ble to modify stte, typiclly vi n ssignment like i = 5 in so-clled impertive progrmming lnguge 2. Such n ssignment sttement is interpreted s function tht turns stte x into new, successor stte x in which the vlue of the identifier i is equl to 5. Sttements in such lnguges re thus described vi suitble stte trnsformer functions. In simplest form, ignoring input nd output, they mp stte to successor stte, s in: S stt where we hve written S for the set of sttes. Its precise structure is not relevnt. Often the set S of sttes is considered to be blck box to which we do not hve direct ccess, so tht we cn only observe certin spects. For instnce vi function i: S Z representing the bove integer i. The vlue i(x ) should be 5 in the result stte x fter evluting the ssignment i = 5, considered s function S S. This description of sttements s functions S S is fine s first pproximtion, but one quickly relises tht sttements do not lwys terminte normlly nd produce successor stte. Sometimes they cn hng nd continue to compute without ever producing successor stte. This typiclly hppens becuse of n infinite loop, for exmple in while sttement, or becuse of recursive cll without exit. There re two obvious wys to incorporte such non-termintion. S def 1. Adjust the stte spce. In this cse one extends the stte spce S to spce S = { } S, where is new bottom element not occurring in S tht is especilly 1 This question comes up frequently when confronted with two progrms one possibly s trnsformtion from the other which perform the sme tsk in different mnner, nd which could thus be seen s the sme progrms. But how cn one mke precise tht they re the sme? 2 Thus, purely functionl progrmming lnguges re not included in our investigtions NATURALNESS OF COALGEBRAIC REPRESENTATIONS 3 used to signl non-termintion. Sttements then become functions: S stt S with the requirement stt( ) = The side-condition expresses the ide tht once sttement hngs it will continue to hng. The disdvntge of this pproch is tht the stte spce becomes more complicted, nd tht we hve to mke sure tht ll sttements stisfy the side-condition, nmely tht they preserve the bottom element. But the dvntge is tht composition of sttements is just function composition. 2. Adjust the codomin. The second pproch keeps the stte spce S s it is, but dpts the codomin of sttements, s in: stt S S where, recll, S = { } S In this representtion we esily see tht in ech stte x S the sttement cn either hng, when stt(x) =, or terminte normlly, nmely when stt(x) = x for some successor stte x S. Wht is lso good is tht there re no side-conditions nymore. But composition of sttements cnnot be defined vi function composition, becuse the types do not mtch. Thus the types force us to del explicitly with the propgtion of non-termintion: for these kind of sttements s 1, s 2 : S S the composition s 1 ; s 2, s function S S, is defined vi cse distinction (or pttern mtch) s: { if s 1 (x) = s 1 ; s 2 = λx S. s 2 (x ) if s 1 (x) = x This definition is more difficult thn function composition (s used in 1. bove), but it explicitly dels with the cse distinction tht is of interest, nmely between nontermintion nd norml termintion. Hence being forced to mke these distinctions explicitly is mybe not so bd t ll. We push these sme ides bit further. In mny progrmming lnguges (like Jv [23]) progrms my not only hng, but my lso terminte bruptly becuse of n exception. An exception rises when some constrint is violted, such s division by zero or n ccess [i] in n rry which is null-reference. Abrupt termintion is fundmentllty different from non-termintion: non-termintion is definitive nd irrevocble, wheres progrm cn recover from brupt termintion vi suitble exception hndler tht restores nornl termintion. In Jv this is done vi try-ctch sttement, see for instnce [23, 97, 132]. Let us write E for the set of exceptions tht cn be thrown. Then there re gin two obvious representtion of sttements tht cn terminte normlly or bruptly, or cn hng. 1. Adjust the stte spce. Sttements then remin endofunctions 3 on n extended stte spce: ( ) ( ) stt { } S (S E) { } S (S E) The entire stte spce clerly becomes complicted now. But lso the side-conditions re becoming non-trivil: we still wnt stt( ) =, nd lso stt(x, e) = (x, e), for x S nd e E, but the ltter only for non-ctch sttements. Keeping trck of such side-conditions my esily led to mistkes. But on the positive side, composition of sttements is still function composition in this representtion. 3 An endofunction is function A A from set A to itself.

8 4 CHAPTER 1. MOTIVATION 2. Adjust the codomin. The lterntive pproch is gin to keep the stte spce S s it is, but to dpt the codomin types of sttement, nmely s: S stt ( ) { } S (S E) Now we do not hve side-conditions nd we cn clerly distinguish the three possible termintion modes of sttements. This structured output type in fct forces us to mke these distinctions in the definition of the composition s 1 ; s 2 of two such sttements s 1, s 2 : S { } S (S E), s in: if s 1 (x) = s 1 ; s 2 = λx S. s 2 (x ) if s 1 (x) = x (x, e) if s 1 (x) = (x, e) Thus, if s 1 hngs or termintes bruptly, then the subsequent sttement s 2 is not executed. This is very cler in this second colgebric representtion. When such colgebric representtion is formlised within the typed lnguge of theorem prover (like in [142]), the typechecker of the theorem prover will mke sure tht cse distinctions re mde. See lso [132] where Jv s exception mechnism is described vi such cse distinctions, closely following the officil lnguge definition [97]. These exmples illustrte tht colgebrs s functions with structured codomins, like in (1.1), rise nturlly, nd tht the structure of the codomin indictes the kind of computtions tht cn be performed. This ide will be developed further, nd pplied to vrious forms of computtion. For instnce, non-deterministic sttements my be represented vi the powerset P s colgebric stte trnsformers S P(S) with multiple result sttes. But there re mny more such exmples. (Reders fmilir with computtionl monds [182] my recognise similrities. Indeed, in computtionl setting there is close connection between colgebric nd mondic representtions. Briefly, the mond introduces the computtionl structure, like composition nd extension, wheres the colgebric view leds to n pproprite progrm logic. This is elborted for Jv in [141].) Exercises (i) Prove tht the composition opertion ; s defined for colgebrs S { } S is ssocitive, i.e. stisfies s 1 ;(s 2 ; s 3) = (s 1 ; s 2) ; s 3, for ll sttements s 1, s 2, s 3: S { } S. Define sttement skip: S { } S which is unit for composition ;, i.e. which stisfies skip ; s = s = s ; skip, for ll s: S { } S. (ii) Do the sme for ; defined on colgebrs S { } S (S E). [In both cses, sttements with n ssocitive composition opertion nd unit element form monoid.] Define lso composition monoid for colgebrs S P(S). 1.2 The power of the coinduction In this section we shll look t sequences or lists, or words, s they re lso clled. Sequences re bsic dt structures, both in mthemtics nd in computer science. One cn distinguish finite sequences 1,..., n nd infinite 1, 2,... ones. The mthemticl theory of finite sequences is well-understood, nd fundmentl prt of computer science, 1.2. THE POWER OF THE COINDUCTION 5 used in mny progrms. Definition nd resoning with finite lists is commonly done with induction. As we shll see, infinite lists require coinduction. Infinite sequences cn rise in computing s the observble outcomes of progrm tht runs forever. Also, in functionl progrmming, they cn occur s so-clled lzy lists, like in the lnguges Hskell [41] or Clen [196]. In the reminder of this section we shll use n rbitrry but fixed set A, nd wish to look t both finite 1,..., n nd infinite 1, 2,... sequences of elements i of A. The set A my be understood s prmeter, nd our sequences re thus prmetrised by A, or, put differently, re polymorphic in A. We shll develop slightly unusul nd bstrct perspective on sequences. It does not tret sequences s completely given t once, but s rising in locl, step-by-step mnner. This colgebric pproch relies on the following bsic fct. It turns out tht the set of both finite nd infinite sequences enjoys certin universl property, nmely tht it is finl colgebr (of suitble type). We shll explin wht this mens, nd how this specil property cn be exploited to define vrious opertions on sequences nd to prove properties bout them. A specil feture of this universlity of the finl colgebr of sequences is tht it gives wy to void mking the (globl) distinction between finiteness nd infiniteness for sequences. First some nottion. We write A for the set of finite sequences 1,..., n (or lists or words) of elements i A, nd A N for the set of infinite ones: 1, 2,.... The ltter my lso be described s functions ( ) : N A, which explins the exponent nottion in A N. Sometimes, the infinite sequences in A N re clled strems. Finlly, the set of both finite nd infinite sequences A is then the (disjoint) union A A N. The set of sequences A crries colgebr or trnsition structure, which we simply cll next. It tries to decompose sequence into its hed nd til, if ny. Hence one my understnd next s prtil function. But we describe it s totl function which possibly outputs specil element for undefined. A next { } ( A A ) { σ if σ is the empty sequence (1.2) (, σ ) if σ = σ with hed A nd til σ A The type of the colgebr is thus { } (A ( )). The successor of stte σ A, if ny, is its til sequence, obtined by removing the hed. The function next cptures the externl view on sequences: it tells wht cn be observed bout sequence σ, nmely whether or not it is empty, nd if not, wht its hed is. By repeted ppliction of the function next ll observble elements of the sequence pper. This observtionl pproch is fundmentl in colgebr. A first point to note is tht this function next is n isomorphism: its inverse next 1 sends to the empty sequence, nd pir (, τ) A A to the sequence τ obtined by prefixing to τ. The following result describes crucil finlity property of sequences tht cn be used to identify the set A. Indeed, s we shll see lter in Lemm 2.3.3, finl colgebrs re unique, up-to-isomorphism Proposition (Finlity of sequences). The colgebr next: A { } A A from (1.2) is finl mong colgebrs of this type: for n rbitrry colgebr c: S { } (A S) on set S there is unique behviour function beh c : S A which is homomorphism of colgebrs. Tht is, for ech x S, both: if c(x) =, then next(beh c (x)) =. if c(x) = (, x ), then next(beh c (x)) = (, beh c (x )).

9 6 CHAPTER 1. MOTIVATION Both these two points cn be combined in commuting digrm, nmely s, { } (A S) { } (A beh c) { } (A A ) c S beh c = next where the function { } (A beh c ) on top mps to nd (, x) to (, beh c (x)). In the course of this chpter we shll see tht generl notion of homomorphism between colgebrs (of the sme type) cn be defined by such commuting digrms. Proof. The ide is to obtin the required behviour function beh c : S A vi repeted ppliction of the given colgebr c s follows. if c(x) = if c(x) = (, x ) c(x ) = beh c (x) =, if c(x) = (, x ) c(x ) = (, x ) c(x ) =. Doing this formlly requires some cre. We define for n N n iterted version c n : S { } A S of c s: c 0 (x) = c(x) { if c n (x) = c n+1 (x) = c(y) if c n (x) = (, y) Obviously, c n (x) implies c m (x), for m < n. Thus we cn define: 0, 1, 2,... if n N. c n (x), nd c i (x) = ( i, x i ) beh c (x) = if m N is the lest number with c 0,..., m 1 (x) =, nd c i (x) = ( i, x i ), for i < m We check the two conditions for homomorphism from the proposition bove. If c(x) =, then the lest m with c m (x) = is 0, so tht beh c (x) =, nd thus lso next(beh c (x)) =. If c(x) = (, x ), then we distinguish two cses: If n N. c n (x), then n N. c n (x ), nd c i+1 (x) = c i (x ). Let c i (x ) = ( i, x i ), then next(beh c (x)) = next(, 0, 1,... ) = (, 0, 1,... ) = (, beh c (x )). If m is lest with c m (x) =, then m > 0 nd m 1 is the lest k with c k (x ) =. For i < m 1 we hve c i+1 (x) = c i (x ), nd thus by writing c i (x ) = ( i, x i ), we get s before: next(beh c (x)) = next(, 0, 1,..., m 2 ) = (, 0, 1,..., m 2 ) = (, beh c (x )). A 1.2. THE POWER OF THE COINDUCTION 7 Finlly, we still need to prove tht this behviour function beh c is the unique homomorphism from c to next. Thus, ssume lso g: S A is such tht c(x) = next(g(x)) = nd c(x) = (, x ) next(g(x)) = (, g(x )). We then distinguish: g(x) is infinite, sy 0, 1,.... Then one shows by induction tht for ll n N, c n (x) = ( n, x n ), for some x n. This yields beh c (x) = 0, 1,... = g(x). g(x) is finite, sy 0,..., m 1. Then one proves tht for ll n < m, c n (x) = ( n, x n ), for some x n, nd c m (x) =. So lso now, beh c (x) = 0,..., m 1 = g(x). Before exploiting this finlity result we illustrte the behviour function Exmple (Deciml representtions s behviour). So fr we hve considered sequence colgebrs prmetrised by n rbitrry set A. In this exmple we tke specil choice, nmely A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, the set of deciml digits. We wish to define colgebr (or mchine) which genertes deciml representtions of rel numbers in the unit intervl [0, 1) R. Notice tht this my give rise to both finite sequences ( 1 8 should yield the sequence 1, 2, 5, for 0.125) nd infinite ones ( 1 3 should give 3, 3, 3,... for ). The colgebr we re looking for computes the first deciml of rel number r [0, 1). Hence it should be of the form, [0, 1) nextdec { } ( {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} [0, 1) ) with stte spce [0, 1). How to define nextdec? Especilly, when does it stop (i.e. return ), so tht finite sequence is generted? Well, deciml representtion like my be identified with with til of infinitely mny zeros. Clerly, we wish to mp such infinitely mny zeros to. Fir enough, but it does hve s consquence tht the rel number 0 [0, 1) gets represented s the empty sequence. A little thought brings us to the following: nextdec(r) = { if r = 0 (d, 10r d) otherwise, where d is such tht d 10r < d + 1. Notice tht this function is well-defined, becuse in the second cse the successor stte 10r d is within the intervl [0, 1). According to the previous proposition, this nextdec colgebr gives rise to behviour function: beh ( ) nextdec [0, 1) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} In order to understnd wht it does, i.e. which sequences re generted by nextdec, we consider two exmples. Strting from 1 8 [0, 1) we get: nextdec( 1 8 ) = (1, 1 4 ) becuse < 2 nd = 1 4 nextdec( 1 4 ) = (2, 1 2 ) becuse < 3 nd = 1 2 nextdec( 1 2 ) = (5, 0) becuse < 6 nd = 0 nextdec(0) =. Thus the resulting nextdec-behviour on 1 8 is indeed 1, 2, 5, i.e. beh nextdec( 1 8 ) = 1, 2, 5.

10 8 CHAPTER 1. MOTIVATION Next, when we run nextdec on 1 9 [0, 1) we see tht: nextdec( 1 9 ) = (1, 1 9 ) becuse < 2 nd = 1 9 Thus nextdec immeditely loops on 1 9, nd we get n infinite sequence 1, 1, 1,... s behviour. This completes the exmple. One sees in the proof of Proposition tht mnipulting sequences vi their elements is cumbersome nd requires us to distinguish between finite nd infinite sequences. However, the nice thing bout the finlity property is tht we do not hve to work this wy nymore. This property sttes two impornt spects, nmely existence nd uniqueness of homomorphism S A into the set of sequences, provided we hve colgebr structure on S. These two spects give us two principles: A coinductive definition principle. The existence spect tells us how to obtin functions S A into A. A conductive proof principle. The uniqueness spect tells us how to prove tht two functions f, g: S A re equl, nmely by showing tht they re both homomorphisms from one colgebr c: S { } (A S) to the finl colgebr next: A { } (A A ). Coinduction is thus the use of finlity just like induction is the use of initility, s will be illustrted in Section 2.4 in the next Chpter. We shll see severl exmples of the use of these definition nd proof principles for sequences in the reminder of this section. Nottion. One thing the previous proposition shows us is tht colgebrs c: S { } (A S) cn be understood s genertors of sequences, nmely vi the resulting behviour function beh c : S A. Alterntively, these colgebrs cn be understood s certin utomt. The behviour of stte x S of this utomton is then the resulting sequence beh c (x) A. These sequences beh c (x) only show the externl behviour, nd need not tell everything bout sttes. Given this behviour-generting perspective on colgebrs, it will be convenient to use trnsition style nottion. For stte x S of n rbitrry colgebr c: S { } (A S) we shll often write x if c(x) = nd x x if c(x) = (, x ). (1.3) In the first cse there is no trnsition strting from the stte x: the utomton c hlts immeditely t x. In the second cse one cn do c-computtion strting with x; it produces n observble element A nd results in successor stte x. This trnsition nottion pplies in prticulr to the finl colgebr next: A { } (A A ). In tht cse, for σ A, σ mens tht the sequence σ is empty. In the second cse σ σ expresses tht the sequence σ cn do n -step to σ, nd hence tht σ = σ. Given this new nottion we cn reformulte the two homomorphism requirements from Proposition s two implictions: x = beh c (x) ; x x = beh c (x) beh c (x ). In the trdition of opertionl semntics, such implictions cn lso be formulted s rules: x beh c (x) x x beh c (x) beh c (x ) (1.4) 1.2. THE POWER OF THE COINDUCTION 9 Such rules thus describe implictions: (the conjunction of) wht is bove the line implies wht is below. In the reminder or this section we consider exmples of the use of coinductive definition nd proof principles for sequences. Evenly listed elements from sequence Our first im is to tke sequence σ A nd turn it into the new sequence evens(σ) A consisting only of the elements of σ t even positions. Step-by-step we will show how such function evens: A A cn be defined within colgebric frmework, using finlity. Our informl description of evens(σ) cn be turned into three requirements: If σ then evens(σ), i.e. if σ is empty, then evens(σ) should lso be empty. If σ σ nd σ, then evens(σ) σ. Thus if σ is the singleton sequence, then lso evens(σ) =. Notice tht by the previous point we could equivlently require evens(σ) evens(σ ) in this cse. If σ σ nd σ σ, then evens(σ) evens(σ ). This mens tht if σ hs hed nd til σ, which in its turn hs hed nd til σ, i.e. if σ = σ, then evens(σ) should hve hed nd til evens(σ ), i.e. then evens(σ) = evens(σ ). Thus, the intermedite hed t odd position is skipped. And this is repeted coinductively : s long s needed. Like in (1.4) bove we cn write these three requirements s rules: σ evens(σ) σ σ σ evens(σ) evens(σ ) σ σ σ σ (1.5) evens(σ) evens(σ ) One could sy tht these rules give n observtionl description of the sequence evens(σ): they describe wht we cn observe bout evens(σ) in terms of wht we cn observe bout σ. For exmple, if σ = 0, 1, 2, 3, 4 we cn compute: evens(σ) = 0 evens( 2, 3, 4 ) = 0 2 evens( 4 ) = = 0, 2, 4. Now tht we hve resonbly understnding of the function evens: A A we will see how it rises within colgebric setting. In order to define it coinductively, following the finlity mechnism of Proposition 1.2.1, we need to hve suitble colgebr structure e on the domin A of the function evens, like in digrm: Tht is, for σ A, { } (A A ) { } (A beh e) { } (A A ) e A if e(σ) =, then evens(σ) ; evens = beh e if e(σ) = (, σ ), then evens(σ) evens(σ ). = next A

11 10 CHAPTER 1. MOTIVATION Combining these two points with the bove three rules (1.5) we see tht the colgebr e must be: if σ e(σ) = (, σ ) if σ σ with σ (, σ ) if σ σ σ σ This function e thus tells wht cn be observed immeditely, if nything, nd wht will be used in the recursion (or co-recursion, if you like). It contins the sme informtion s the bove three rules. In the terminology used erlier: the colgebr or utomton e genertes the behviour of evens Remrk. The colgebr e: A { } (A A ) illustrtes the difference between sttes nd observbles. Consider n rbitrry sequence σ A nd write σ 1 = 1 σ nd σ 2 = 2 σ, where, 1, 2 A with 1 2. These σ 1, σ 2 A re clerly different sttes of the colgebr e: A { } (A A ), but they hve the sme behviour: evens(σ 1 ) = evens(σ) = evens(σ 2 ), where evens = beh e. Such observtionl induistinguishbility of the sttes σ 1, σ 2 is clled bisimilrity, written s σ 1 σ 2, nd will be studied systemticlly in Chpter 3. Oddly listed elements from sequence Next we would like to hve similr function odds: A A which extrcts the elements t odd positions. We leve formultion of the pproprite rules to the reder, nd clim this function odds cn be defined coinductively vi the behviour-generting colgebr o: A { } (A A ) given by: o(σ) = { if σ or σ σ with σ (, σ ) if σ σ σ σ Thus, we tke odds = beh o to be the behviour function resulting from o following the finlity principle of Proposition Hence o(σ) = odds(σ) nd o(σ) = (, σ ) odds(σ) odds(σ ). This llows us to compute: odds( 0, 1, 2, 3, 4 ) = 1 odds( 2, 3, 4 ) since o( 0, 1, 2, 3, 4 ) = ( 1, 2, 3, 4 ) = 1 3 odds( 4 ) = 1 3 = 1, 3. since o( 2, 3, 4 ) = ( 3, 4 ) since o( 4 ) = At this point the reder my wonder: why not define odds vi evens, using n pproprite til function? We shll prove tht this gives the sme outcome, using coinduction Lemm. One hs odds = evens til, where the function til: A A is given by: { σ if σ til(σ) = σ if σ σ THE POWER OF THE COINDUCTION 11 Proof. In order to prove tht the two functions odds, evens til: A A re equl one needs to show by Proposition tht they re both homomorphisms for the sme colgebr structure on A. Since odds rises by definition from the bove function o, it suffices to show tht evens til is lso homomorphism from o to next. This involves two points: If o(σ) =, there re two subcses, both yielding the sme result: If σ then evens(til(σ)) = evens(σ). If σ σ nd σ, then evens(til(σ)) = evens(σ ). Otherwise, if o(σ) = (, σ ), becuse σ σ nd σ σ, then we hve evens(til(σ)) = evens(σ ) evens(til(σ )) since: If σ, then evens(σ ) evens(σ ) = evens(til(σ )). And if σ σ, then evens(σ ) evens(σ ) = evens(til(σ )). Such equlity proofs using uniqueness my be bit puzzling t first. But they re very common in ctegory theory, nd in mny other res of mthemtics deling with universl properties. Lter, in Section 3.4 we shll see tht such proofs cn lso be done vi bisimultions. This is common proof technique in process theory nd in colgebr, of course. Merging sequences In order to further fmilirise the reder with the wy the coinductive gme is plyed, we consider merging two sequences, vi binry opertion merge: A A A. We wnt merge(σ, τ) to lterntingly tke one element from σ nd from τ, strting with σ. In terms of rules: σ τ merge(σ, τ) σ τ τ merge(σ, τ) merge(σ, τ ) σ σ merge(σ, τ) merge(τ, σ ) Notice the crucil reversl of rguments in the lst rule. Thus, the function merge: A A A is defined coinductively s the behviour beh m of the colgebr given by: ( A A ) m { } (A ( A A )) m(σ, τ) = if σ τ (, (σ, τ )) if σ τ τ (, (τ, σ )) if σ σ. At this stge we cn combine ll of the coinductively defined functions so fr in the following result. It sys tht the merge of the evenly listed nd oddly listed elements in sequence is equl to the originl sequence. At first, this my seem obvious, but recll tht our sequences my be finite or infinite, so there is some work to do. The proof is gin n exercise in coinductive resoning using uniqueness. It does not involve globl distincition between finite nd infinite, but proceeds by locl, single step resoning Lemm. For ech sequence σ A, merge(evens(σ), odds(σ)) = σ.