Decidability, Behavioural Equivalences and Infinite Transition Graphs. Hans Hüttel

Transcription

1 Decidbility, Behviourl Equivlences nd Infinite Trnsition Grphs Hns Hüttel Doctor of Philosophy University of Edinburgh 1991

2 Abstrct This thesis studies behviourl equivlences on lbelled infinite trnsition grphs nd the role tht they cn ply in the context of modl logics nd notions from lnguge theory. A nturl clss of such infinite grphs is tht corresponding to the SnS -inble tree lnguges first studied by bin. We show tht modl mu-clculus with lbel set f0;:::;n,1gcn ine these tree lnguges up to n observtionl equivlence. Another nturl clss of infinite trnsition grphs is tht of normed BPA processes, which correspond to the grphs of leftmost derivtions in context-free grmmrs without useless productions. A remrkble result is tht strong bisimultion is decidble for these grphs. After n outline of the existing proofs due to Beten et l. nd Cucl we present much simpler proof using tbleu system closely relted to the brnching lgorithms employed in lnguge theory following Korenjk nd Hopcroft. We then present result due to Colin Stirling, giving wekly sound nd complete sequent-bsed equtionl theory for bisimultion equivlence for normed BPA processes from the tbleu system. Moreover, we show how to extrct fundmentl reltion (s in the work of Cucl) from successful tbleu. We then introduce silent ctions nd consider clss of normed BPA processes with the restriction tht processes cnnot terminte silently, showing tht the decidbility result for strong bisimilrity cn be extended to vn Glbbeek s brnching bisimultion equivlence for this clss of processes. Wecompletethepictureby estblishing tht ll other known behviourl equivlences nd number of preorders re undecidble for normed BPA processes.

3 Acknowledgements First of ll I wnt to thnk my supervisor Colin Stirling for our fruitful discussions nd his mny useful comments tht gretly influenced the contents nd presenttion of this thesis. Thnks re lso due to Didier Cucl for his importnt insights into the topics of Chpters 4 to 6 tht inspired much of my work; in prticulr thnks for pointing out some serious errors in n erly version of wht ws to become Chpter 6. I lso wnt to thnk him, his wife Ctherine nd olnd Monfort for their immense hospitlity during my visit to IISA in June Hd it not been for Jn Friso Groote nd the discussions we hd during his visit to the LFCS in My of 1991, Chpter 6 would hve been very short nd boring. Mny of the results in tht chpter re due to him. My work on brnching bisimultion in Chpter 5 begn s the result of discussion I hd in Alborg with Kim Lrsen. Exmple is his. The Deprtment of Computer Science t Edinburgh provided me with n interesting work environment nd I wnt to thnk ll the people I got to know there during my sty. Specil thnks to Kees nd Fbio for being such good office-mtes. An importnt spect of going brod is tht you get to mke new friends. My wrm thnks go to Mds nd Chrlotte for providing me with proper perspective of mny things in life, to Bjrne for drinks osv. nd for keeping me in touch with things Dnish, to Dvid for mels nd Jpnese film nd thetre (which will never be the sme) nd to Soni for food, compny nd understnding (she is very specil!). And thnks to Edurdo, Nigel, Hns Jørgen & Nomi nd lst but by no mens 3

4 4 lest Søren for shring ccommodtion with me t vrious times during my yers in Edinburgh nd for generlly tolerting my strnge whims. The constnt emotionl nd prcticl support tht my mother hs given me throughout my self-imposed exile nd whenever I ws in Denmrk hs been ll-importnt; without her, things could hve looked very grim indeed nd I cnnot thnk her enough for being there. My grndmother did not live to see the end of my sty in Edinburgh; it is to the memory of her tht my thesis is dedicted. The work in this thesis ws mde possible by reserch position t the Deprtment of Mthemtics t Ïrhus University, trvel grnts nd pyment of reserch costs by Alborg University Centre nd C.W. Obel Fonden nd finncil support from the Dnish eserch Acdemy.

5 Declrtion This is the revised version of my thesis incorporting the required corrections suggested by my exminers obin Milner nd Mtthew Hennessy. The thesis ws composed by myself, nd the work reported hs not been presented for ny university degree before. The ides nd results tht I do not ttribute to others re my own. Prts of the thesis hve lredy been published elsewhere. Chpter 3 is slightly revised version of [Hüt90]. Chpter 4 contins n expnded version of [HS91]. Chpter 5 is essentilly [Hüt91], nd Chpter 6 is essentilly [GH91]. Hns Hüttel 5

6 Tble of Contents 1 Introduction Determining the qulities of behviourl equivlences : : : : : : : : : : Infinite-stte systems : : : : : : : : : : : : : : : : : : : : : : : : : : : Behviourl equivlences nd progrm logics : : : : : : : : : : : : : : Tbleu techniques : : : : : : : : : : : : : : : : : : : : : : : Expressiveness : : : : : : : : : : : : : : : : : : : : : : : : : : Decidbility of behviourl equivlences : : : : : : : : : : : : : : : : Lyout of the Thesis : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 2 Bckground Infinite trees nd bin utomt : : : : : : : : : : : : : : : : : : : : Infinite trees : : : : : : : : : : : : : : : : : : : : : : : : : : : bin utomt : : : : : : : : : : : : : : : : : : : : : : : : : Normed recursive BPA processes : : : : : : : : : : : : : : : : : : : : Syntx nd semntics : : : : : : : : : : : : : : : : : : : : : : Bisimultion equivlence on BPA processes : : : : : : : : : : : Axiomtiztions of bisimultion equivlence : : : : : : : : : : Normed recursive BPA processes in Greibch Norml Form : : Self-bisimultions : : : : : : : : : : : : : : : : : : : : : : : : The split lemm : : : : : : : : : : : : : : : : : : : : : : : : 38 6

7 7 3 A modl chrcteriztion of SnS Syntx nd semntics of SnS nd CML : : : : : : : : : : : : : : : : SnS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : CML : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : SnS is t lest s expressive s CML : : : : : : : : : : : : : : : : : CML is s expressive s SnS modulo ' A : : : : : : : : : : : : : : : 44 4 Deciding bisimilrity for normed BPA Existing pproches : : : : : : : : : : : : : : : : : : : : : : : : : : : Beten, Bergstr, nd Klop s proof : : : : : : : : : : : : : : : Cucl s proof : : : : : : : : : : : : : : : : : : : : : : : : : : The tbleu decision method : : : : : : : : : : : : : : : : : : : : : : : Constructing subtbleux : : : : : : : : : : : : : : : : : : : : Decidbility, soundness, nd completeness : : : : : : : : : : : An equtionl theory : : : : : : : : : : : : : : : : : : : : : : : : : : : Extrcting fundmentl reltions : : : : : : : : : : : : : : : : : : : : : 72 5 Introducing silent ctions Brnching bisimilrity : : : : : : : : : : : : : : : : : : : : : : : : : : Normed BPA rec : : : : : : : : : : : : : : : : : : : : : : : : : : : : : A tbleu system for brnching bisimultion : : : : : : : : : : : : : : : Building tbleux : : : : : : : : : : : : : : : : : : : : : : : : Termintion, completeness, nd soundness : : : : : : : : : : : Complexity of the tbleu system nd decidbility : : : : : : : 97 6 Negtive results Deterministic BPA processes : : : : : : : : : : : : : : : : : : : : : : : Simultion equivlence : : : : : : : : : : : : : : : : : : : : : : : : : n-nested simultion equivlence : : : : : : : : : : : : : : : : : : : : : 103

8 8 6.4 n-bounded-tr-bisimultion : : : : : : : : : : : : : : : : : : : : : : : : Filures, rediness, filure-trce nd redy-trce equivlences : : : : : : edy-simultion or 2/3-bisimultion : : : : : : : : : : : : : : : : : : Conclusion Summry of the min results : : : : : : : : : : : : : : : : : : : : : : : Vrious kinds of infinite trnsition grphs : : : : : : : : : : : : : : : : SnS-inble tree lnguges : : : : : : : : : : : : : : : : : : Context-free grphs : : : : : : : : : : : : : : : : : : : : : : : Unnormed BPA : : : : : : : : : : : : : : : : : : : : : : : : : Beyond BPA : : : : : : : : : : : : : : : : : : : : : : : : : : : Complexity bounds : : : : : : : : : : : : : : : : : : : : : : : : : : : Wek equivlences : : : : : : : : : : : : : : : : : : : : : : : : : : : : Equtionl theories : : : : : : : : : : : : : : : : : : : : : : : : : : : : 122

9 Chpter 1 Introduction The problem of determining if progrm stisfies given specifiction is one of the centrl motivting problems in theoreticl computer science, nd severl pproches exist. Denottionl semntics cn serve s vluble tool for progrm verifiction. But in the cse of nondeterministic, prllel or non-terminting progrms n opertionl ccount is often preferred. Over the pst decde much ttention hs been devoted to the study of process clculi such s CCS [Mil80,Mil89], ACP [BK84,BK88] nd CSP [Ho88]. Of prticulr interest hs been the study of the behviourl semntics of these clculi s given by trnsition grphs rising from structurl opertionl semntics in the trdition originted by [Plo81]. A prticulrly importnt question is when processes cn be sid to exhibit the sme behviour, nd plethor of behviourl equivlences exist tody. 1.1 Determining the qulities of behviourl equivlences The min rtionle behind the vrious behviourl equivlences tht hve been proposed hs been to cpture behviourl spects tht the lnguge equivlence known from lnguge theory does not tke into ccount. For instnce, 9

10 10 Chpter 1. Introduction b + c = (b + c) holds for lnguge equivlence, but is n identifiction tht these other notions of behviour nd behviourl equivlence do not mke, since these two process expressions do not exhibit the sme dedlock properties: fter n initil -ction the former is only ble to perform one ction (b or c), wheres the ltter hs choice between b nd c. Equivlences re usully clssified ccording to their corseness, i.e. how mny identifictions they mke with respect to the brnching behviour of processes. This liner/brnching time hierrchy is illustrted in Figure 1 1 (fter [vg90]). The corsest equivlences re then the trce equivlence nd the completed trce equivlence; the ltter differs from trce equivlence in tht only the completed trce lnguges re compred nd is thus the usul lnguge equivlence. Directly bove we hve the testing/filures equivlence investigted by Hennessy nd denicol et l. (see e.g. [Hen89]). At the top of the digrm is bisimultion equivlence (or bisimilrity), notion introduced by Prk in [Pr81] nd subsequently used by Milner nd others in the CCS trdition (s exemplified in [Mil89]). It hs lso been rgued tht behviourl equivlences should be judged ccording to how well they obey some computtionlly justifible criterion of observbility. For instnce, while bisimultion equivlence hs mny nice mthemticl properties it fils to hve computtionl justifiction in tht (in)equivlence is not intuitively observble. Indeed, within the frmework of testing some very unintuitive testing opertors must be used. Abrmsky hs shown [Abr87] tht bisimultion cn be chrcterized by test lnguge tht contins testing opertor tht enumertes ll next-sttes of the process subjected to testing. In [Gro89] Groote presents nother test lnguge; here lookhed in combintion with the possibility to check for the bsence of ctivity is needed. Bloom et l. rgue [BIM90] tht only completed trces should count s observtions nd ine n equivlence which is (completed) trce congruence under resonble set of process

11 1.1. Determining the qulities of behviourl equivlences 11 Bisimultion equivlence? ) n-nested simultion equivlence n-bounded-tr-bisimultion? edy simultion equivlence Q QQQ? Q? ) edy trce equivlence QQQ Possible-futures equivlence J J JJ Q QQQ JJ^ J J^ ediness equivlence J JJ Filure trce equivlence J J^ Filures equivlence? Completed trce equivlence?, Trce equivlence,,,,, Q QQs Simultion equivlence,,,,,,,, Figure 1 1: The liner/brnching time hierrchy of equivlences (bsed on [vg90]). constructs. This equivlence is the redy simultion equivlence of Figure 1 1. The study of processes with unobservble ctions leds to nother liner/brnching time hierrchy similr to Figure 1 1 except tht we now hve the corresponding wek equivlences bsed on the wek trnsition reltion =) where unobservble ctions re disregrded. The best-known wek equivlence is Milner s wek bisimultion equivlence [Mil89]; however, it hs been rgued tht this is not the proper wek version of bisimilrity since it does not reflect the chnges in brnching properties tht my hppen s the result of performing n unobservble ction. Moreover, wek bisimultion is not

12 12 Chpter 1. Introduction robust under simple notion of ction refinement. The notion of brnching bisimilrity, put forwrd by vn Glbbeek nd Weijlnd in [vgw89b], reflects these concerns. According to this inition, ll intermedite unobservble steps in wek trnsition must be mtched. The ide is not new, however. De Nicol nd Vndrger hve proved [DNV90] tht brnching bisimultion corresponds to the stuttering equivlence on Kripke structures considered in e.g. [BCG88]. ecently it hs been shown [DNMV90] tht there is nturl connection between brnching bisimultion nd wek bisimultion in tht the former is the so-clled bck nd forth vrint of the ltter. Brnching bisimilrity hs mny plesnt properties; in prticulr complete equtionl theory for finite BPA, the clss of finite processes with choice nd sequentil composition, cn be obtined by dding just two new xioms to those for strong bisimultion for finite BPA processes. Moreover, this xiomtiztion cn esily be turned into complete term rewriting system [vg90], something tht is not the cse for wek bisimultion equivlence. Finlly, there is similr hierrchy of preorders on processes which is yet to be determined in detil. Exmples of such preorders include the simultion preorder [Pr81], the testing/filures preorder [Hen89] nd the redy simultion preorder of [BIM90]. 1.2 Infinite-stte systems Milner [Mil84] hs shown tht the clss of finite trnsition grphs corresponds to tht of the trnsition grphs for regulr processes, i.e. the recursively ined CCS processes over the signture f:; +g where : is n ction prefixing opertor for every in set of tomic ctions Act nd + is nondeterministic choice. egulr CCS processes correspond to the usul finite utomt; their finitry trce lnguges re the finitry regulr lnguges. However, this result lso sys tht s soon s we move beyond these constructs (known s the dynmic process constructs), recursively ined processes cn hve

13 1.3. Behviourl equivlences nd progrm logics 13 trnsition grphs with infinitely mny sttes nd trce lnguges tht re no longer regulr. This includes mny relistic cses; in prticulr processes tht re ined using vrious notions of prllel composition such s the synchronous prllel opertor j of CCS cn hve infinitely mny sttes. In fct, the theory of finite-stte systems nd their equivlences cn now be sid to be well-estblished. One my be led to wonder wht the results will look like for infinite-stte systems. 1.3 Behviourl equivlences nd progrm logics There is striking reltionship between behviourl equivlences nd some progrm logics. A modl chrcteriztion ofbisimilrityexists [HM85] in tht twoprocesses re bisimilr iff they stisfy the sme formule in modl logic now known s Hennessy- Milner logic. Mny other relted modl nd temporl logics lso chrcterize bisimultion equivlence in this wy [BCG88,Sti87,Sti91]. Similr chrcteriztions exist for the other prts in the liner/brnching time hierrchy; in the logics tht chrcterize these equivlences either restrictions of Hennessy-Milner logic re mde or opertors of liner-time temporl logic re introduced in plce of the Hennessy-Milner modlities. Indeed, for the description of progrm properties the tendency is to prefer modl nd temporl logics for describing properties of progrms. One such logic is the modl muclculus [Koz83], which lso chrcterizes bisimilrity in the sense tht two processes re bisimilr iff they stisfy the sme closed formule [Sti91] Tbleu techniques An importnt problem in the context of progrm logics is tht of model checking. Model checking consists in determining whether process stte p stisfies formul F written p j= F. In [SW89] Stirling nd Wlker hve given model checker for the modl muclculus nd finite trnsition grphs in the form of tbleu system, gol-directed proof

14 14 Chpter 1. Introduction system for the reltion p j= F. An dvntge of the tbleu-bsed pproch to model checking is tht it is locl in the sense tht only those sttes relevnt to determining whether or not p j= F need to be exmined. In [BS90] Brdfield nd Stirling hve given tbleu system tht dels with infinite trnsition grphs. In this thesis we use relted tbleu technique pproch to look t decision problems for behviourl equivlences. Our pproch turns out lso to be closely relted to the brnching lgorithms for equivlences studied in forml lnguge theory. The method, introduced by Korenjk nd Hopcroft in [KH66] hs been widely pplied for giving decision procedures for vrious equivlence problems see e.g. [Cou83] Expressiveness The modl mu-clculus is very expressive logic; it incorportes the full expressibility of CTL [Dm90] nd thus serves s nturl brnching time logic for expressing properties of processes. Moreover, the modl mu-clculus is decidble, in fct even elementry [ES84]. Another importnt decidble theory, in fct one of the most generl decidble theories round, is the second-order mondic theory of n successors, SnS, s introduced by bin [b69]. SnS is the generliztion of S1S, the second-order mondic theory of 1 successor, which ws shown to be decidble by Büchi in [Büc60] by utomt-theoretic mens similr to those lter used by bin. Since SnS is very generl theory, severl other theories hve been shown to be decidble by interprettions into SnS [b69,b77]; exmples include the wek second-order theory of linerly ordered sets, the second-order theory of totlly ordered sets with countble domin nd vrious propositionl modl logics [b77]. The clss of SnS-inble sets corresponds to tht of the sets of infinite n-ry node-lbelled trees ccepted by bin utomt [b69] nd is well-known clss of infinite-stte systems. The cceptnce condition of bin utomton cn be seen s describing firness property long tree pths. Thus it would seem tht the SnS-inble sets cn be ined

15 1.3. Behviourl equivlences nd progrm logics 15 through tree property tht cn be described in somewht more nturl wy, nmely through using brnching-time temporl logic, since such logic cn be interpreted on infinite trees in strightforwrd fshion. So since SnS is very powerful decidble theory nd the modl mu-clculus lso is very powerful, subsuming mny well-known modl nd temporl logics, nturl nd interesting question is how these two logics re relted with respect to expressiveness. Some work hs lredy been done in this field. In [VWS83] Vrdi et l. show tht the temporl logic ETL [Wol83] cn ine exctly the clss of!-regulr lnguges, corresponding to the S1S -inble sets. And in [Niw88] it ws shown by Niwinski tht fixed-point clculus whose signture prt from mximl nd miniml fixed points nd disjunction includes the usul opertors on trees cn ine exctly the SnS-inble sets. Finlly, Hfer nd Thoms hve proved [HT87] tht restricted version of SnS with set quntifiction restricted to pths is expressively equivlent to CTL for binry tree models. However, there re certinly bound to be differences. For one thing, the full SnS is non-elementry [Mey75], wheres the modl mu-clculus is elementry [ES84]. Moreover, s ws lso shown in [HT87] the full SnS cn express properties which hve no correlte in brnching time temporl logic which does not hve opertors tht incorporte informtion bout the ordering of nodes in tree. An exmple is counting the nodes in tree which re incomprble to node x w.r.t. to the ncestrl ordering, : A(x) = 9x 1 ;:::;x m :^m i=1 (x 6 x i ^ x i 6 x) ^ ^i;j (i 6= j x i 6= x j ) This problem would not rise if we could somehow refer to the ncestrl ordering in our modl opertors. In fct, in this thesis we show tht modl mu-clculus with lbel set f0;:::;n,1gcn chrcterize SnS up to bisimultion-like equivlence on node-lbelled trees.

16 16 Chpter 1. Introduction 1.4 Decidbility of behviourl equivlences Lnguge equivlence is known to be decidble for finite utomt. However, it is lso well known (see e.g. [HU79]) tht lnguge equivlence becomes undecidble when one moves beyond finite utomt to context-free lnguges. For finite-stte processes ll of the equivlences of Figure 1 1 cn be seen to be decidble. For instnce, the bisimilrity problem p q for processes p nd q is decidble for finite trnsition grphs becuse we cn enumerte ll the finitely mny binry reltions over the stte set nd serch for bisimultion mong them contining the pir (p; q). Moreover, for regulr CCS complete equtionl theories exist for strong bisimilrity [Mil84]. In this thesis we rgue tht decidbility or lck thereof should be thought of s nother criterion for determining the computtionl merits nd iciencies of behviourl equivlences. A nturl question is then whether the decidbility cn be extended beyond the finitestte cse. One limittion tht should be noted is tht strong bisimultion equivlence ( ) in process lnguge with generl sttic constructs nd recursive initions is undecidble. In fct, the generl bisimilrity problem is not even r.e. For, using the opertors communiction (j), restriction (n) nd sequentil composition one cn code ny Turing mchine M nd input string w s process expression p M;w such tht ll moves of M re represented by internl (-)ctions of p M;w nd such tht the possible eventul hlting is represented by specil success ction d. The encoding consists in expressing the tpe of M s two stcks, one of which hs been initilized to hold w. The stcks communicte with the finite control, represented by regulr process. The problem Does M diverge on input w? cn now be expressed s p M;w x:(:x) (1.1) However, (1.1) is undecidble since the bove divergence problem is not r.e.

17 1.4. Decidbility of behviourl equivlences 17 In the setting of process lgebr, n exmple of infinite-stte systems is tht of the trnsition grphs of processes in the clculus BPA (Bsic Process Algebr) [BK88]. These re recursively ined processes over the signture f; +;:gwhere rnges over set of tomic ctions, + is nondeterministic choice nd is sequentil composition. A BPA process is ined by system of recursion equtions of the form X 0 = E 0 (X 0 ;:::;X n ). X n = E n (X 0 ;:::;X n ) where the E i s re BPA expressions. A system of the bove form where every occurrence of vrible in ny expression E i is within the scope of n tomic ction is sid to be gurded. Any gurded system of equtions cn effectively be put in the BPA equivlent of Greibch Norml Form (GNF), i.e. system of equtions where ll equtions re of the form X i = P j ij ij where the ij s re compositions of process vribles. A specil cse is tht of normed BPA processes. The norm of process is ined s the lest number of trnsitions necessry to terminte. A process is sid to be normed if every stte hs finite norm. Even though normed BPA does not incorporte ll regulr processes, systems ined in this clculus cn in generl hve infinitely mny sttes. There is n obvious correspondence between process equtions of the form X i = P j ij ij nd the GNF context-free productions X! i1 i1 j ::: j ik ik, so normed BPA processes correspond to context-free grmmrs without useless or empty productions. It is therefore esy to see tht both trce equivlence nd completed trce equivlence (or lnguge equivlence) re undecidble for normed BPA processes. However, recent result shows tht strong bisimilrity for normed BPA processes is decidble. Two proofs of this result exist, one by Beten, Bergstr nd Klop [BBK87] nd nother due to Cucl [Cu88,Cu90]. These proofs re very different (nd re sketched in Chpter 4). The (lengthy nd impenetrble) proof in [BBK87] consists in showing tht one cn exhibit decomposition of the process grph with certin regulr-

18 18 Chpter 1. Introduction ities. The proof in [Cu88,Cu90] consists in showing tht the mximl bisimultion is finitely representble by confluent nd strongly normlizing Thue system, nd tht there re only finitely mny cndidtes for this Thue system. These proofs do not correspond to one s intuition bout how to determine whether or not two normed BPA processes re bisimilr. And they do not led to complete proof systems for the process lgebr involved. But wht these proofs do tell us is tht we cn go beyond finite-stte systems while mintining the decidbility of bisimultion equivlence. Moreover, somewht unexpectedly, we hve gined something by using n equivlence different from lnguge equivlence in setting tht involves structures from lnguge theory. On the other hnd, Huynh nd Tin [HT90] hve proved the negtive result tht the filures nd rediness equivlences re undecidble for normed BPA. Their proof consists in giving specil clss of normed BPA processes for which these equivlences coincide with lnguge equivlence nd then showing tht the lnguge equivlence problem for rbitrry normed BPA processes reduces to tht for the specil clss. A nturl question in the light of this is now for which equivlences nd for which process signtures we hve tht the equivlence in question is decidble for normed processes. In this thesis we show tht in fct none of the other behviourl equivlences in Figure 1 1 re decidble for normed BPA processes. elted questions re wht hppens when we introduce silent ctions nd wht the sitution looks like for preorders. For the ltter there re bound to be some differences. Friedmn hs shown [Fri76] tht the lnguge inclusion preorder is undecidble for so-clled simple grmmrs, clss of context-free grmmrs tht correspond to tht of deterministic normed BPA processes. However, Korenjk nd Hopcroft hve shown tht the lnguge equivlence problem for this clss of grmmrs/processes is decidble [KH66]. Finlly, nother importnt question is whether we cn find nturl nd convenient method of determining whether or not two normed BPA processes re bisimultion

19 1.5. Lyout of the Thesis 19 equivlent. In this thesis we show tht it is indeed possible; our method is tbleu technique relted to the locl model checking systems of [SW89,BS90]. This method is lso closely relted to the brnching lgorithms for equivlence problems on grmmrs introduced by Korenjk nd Hopcroft in [KH66]. We use the sme tbleu method to prove tht the brnching bisimultion equivlence of Weijlnd nd vn Glbbeek [vgw89,vgw89b] is decidble for clss of normed BPA processes with silent ctions. 1.5 Lyout of the Thesis In this finl section we outline the contents of the rest of this thesis. In Chpter 2 we present the necessry bckground mteril for the chpters tht follow. We give the initions of infinite n-ry trees nd bin utomt (studied in Chpter 3) nd introduce in greter detil the process clculus BPA (studied in Chpters 4 to 6) nd the notion of bisimultion equivlence. We describe the subclss of normed BPA nd show how every system of BPA equtions cn be effectively rewritten into 3-GNF. In Chpter 3 we show tht modl mu-clculus with lbel set f0;:::;n,1g cn ine the SnS-inble n-ry tree lnguges up to n observtionl equivlence. The minide is to use bin s theoremstting thtthe SnS-inble lnguges correspond to the n-ry bin-recognizble tree lnguges, which re the sets of infinite n-ry lbelled trees recognizble by bin utomt. Thus our result lso underpins the ide tht equivlences other thn those normlly used cn be of use in problems relted to lnguge theory. In Chpter 4 we first outline the existing proofs of the decidbility of bisimultion equivlence for normed BPA processes due to Beten, Bergstr nd Klop [BBK87b, BBK87] nd Cucl [Cu88,Cu90] nd then give n lterntive nd much simpler proof of this result. Our decidbility proof uses tbleu system which is similr to

20 20 Chpter 1. Introduction the tbleu systems used for model-checking the modl mu-clculus [SW89,BS90] nd closely relted to the brnching lgorithms of lnguge theory [KH66,Cou83]. If successful tbleu for n eqution = exists, the tbleu provides us with finite witness for bisimultion contining (; ), the witness being self-bisimultion in the sense of [Cu88,Cu90]. We give complexity bound for the tbleu method in terms of the length of the longest possible pth in ny tbleu for given eqution. Then we present result due to Colin Stirling, sequent-bsed equtionl theory for bisimultion equivlence for normed BPA processes in 3-GNF extrcted from the tbleu system. The theory is shown to be strongly sound nd wekly complete. Finlly we show how one cn find fundmentl reltion (s in the work of [Cu88,Cu90]) from successful tbleu. This is done vi nother so-clled uxiliry tbleu system. In Chpter 5 we introduce silent ctions into normed BPA. We consider clss of BPA processes with the restriction tht process termintion must involve performing n observble ction. We then show how the decidbility result of Chpter 4 cn be extended to brnching bisimultion equivlence, giving complexity bounds for the tbleu method. In Chpter 6 we show tht ll equivlences below bisimultion in the liner/brnching time hierrchy re undecidble for normed BPA processes in 3-GNF nd thus tht they re undecidble for BPA processes in generl. The proofs involve reductions to the lnguge inclusion problem for simple grmmrs of [Fri76] nd the lnguge nd trce equivlence problems for normed BPA processes. Chpter 7 sums up the conclusions of this thesis nd give directions for further work.

21 Chpter 2 Bckground In this chpter we give vrious initions tht will be used throughout the rest of this thesis. Section 2.1 introduces the notions of infinite node-lbelled trees, bisimultions on such trees nd bin utomt tht will be used in Chpter 3. In Section 2.2 we introduce the clss of normed BPA processes studied in Chpters 4 to 6 nd the notion of bisimultion equivlence for such processes. 2.1 Infinite trees nd bin utomt In Chpter 3 we shll look t SnS [b69] nd version of the modl mu-clculus [Koz83], both of which re logics interpreted on infinite node-lbelled trees of fixed rity Infinite trees An n-ry infinite tree cn be seen s prefix-closed set, with suffixing representing the successor reltion. Definition The full infinite n-ry tree is the set f0;:::;n,1g with the successor reltion! f0;:::;n,1g f0;:::;n,1g + ined by w! wi for i 2f0;:::;n,1g. wi is clled the ith successor of w. The root is. 21

22 22 Chpter 2. Bckground In other words, ny node wx where x = i 1 i 2 :::i m is the unique node rechble from w vi the pth w! wi 1! wi 1 i 2!wi 1 i 2 :::i m. We shll sometimes need n ordering on the node set. Definition The ncestrl ordering on f0;:::;n,1g is given by w 0 <w 00 if there exists w 000 2f0;:::;n,1g + such tht w 0 w 000 = w 00. A lbelled tree t whose lbels re in the lphbet A is ined s lbelling function on f0;:::;n,1g. Definition An n-ry A-lbelled tree is function t : f0;:::;n,1g! A. The set of ll A-lbelled trees is denoted by T A!.Asetof n-ry A-lbelled trees is clled tree lnguge over A. In the rest of this section we ssume without loss of generlity tht ll trees considered re binry, mening n =2. We now ine notion of equivlence on trees which sttes tht two nodes hve the sme brnching properties with respect to some sublphbet A 0. First we ine the notion of n A 0 -descendnt. Definition For ny tree t 2 T A! nd A 0 A, thea 0 -descendnt reltion =) A 0 f0; 1g (f0; 1g f0;1g )is given by w =) A 0 (u0v 0 ;u1v 1 ) whenever w < u0v 0 nd w < u1v 1 where t(w 0 ) 62 A 0 whenever w < w 0 < u0v 0 or w < w 0 < u1v 1 but t(u0v 0 );t(u1v 1 )2A 0. Thus, the A 0 -descendnts of node w re the first descendnts of w lbelled by elements from A 0 long pir of incomprble pths. For A-lbelled trees =) clerly A reduces to!. If A = A 0 [fgwhere is specil invisible lbel, 62 A 0, =) A 0 cn be seen s the successor reltion modulo invisible lbels. Thus, the reltion is similr to the wek trnsition reltions for edge-lbelled trnsition grphs with silent ctions (cf. Definition 5.1.1) nd gives rise to n equivlence of nodes tht is similr to

23 2.1. Infinite trees nd bin utomt 23 the observtionl equivlence of [Mil89] nd essentilly is the equivlence on trees of [BCG88]: Definition For ny trees t 1 2T! A 1 ;t 2 2T! A 2 nd A 0 A 1 \ A 2,nA-bisimultion is reltion A 0 f0;1g f0;1g such tht whenever (w 0 ;w 00 ) 2 A 0 we hve 1. t 1 (w 0 )=t 2 (w 00 ) nd t 1 (w 0 ) 2 A 0 2. w 0 =) A 0 (w 0 0 ;w0 1 )=)9(w00 0 ;w00 1 ):w00 =) (w 00 0 ;w00 1 ) with w0 0 w00 nd 0 w0 1 w00 1 A 0 3. w 00 =) (w 00 0 ;w00 1 )=)9(w0 0 ;w0 1 ):w0 =) (w 0 0 ;w0)with 1 w0 0 w00 nd 0 w0 1 w00 1 A 0 A 0 We ine' A 0 by' A 0= f(w 0 ;w 00 )jw 0 A 0w 00 for some A 0 -bisimultion A 0g. Ifw 0 ' A 0 w 00 we sy tht w 0 nd w 00 re A 0 -bisimilr. Thus, ' A 0 identifies two trees lbelled by lphbets contining A 0 if their ncestrl informtion w.r.t. A 0 is the sme. We hve Proposition ' A 0 is n equivlence reltion on f0; 1g f0;1g bin utomt The use of bin utomt is crucil to the equi-expressiveness proof in Chpter 3. The bin utomton, introduced in [b69], is n importnt type of utomton on infinite trees, first used s n uxiliry notion in the proof of the decidbility of SnS. It provides generliztion of the Büchi utomt on infinite sequences introduced in Büchi s proof of the decidbility of S1S [Büc60]. Definition A bin utomton on binry A-lbelled trees is qudruple A = (Q; q 0 ; f! j 2 Ag; ), whereq is finite set of sttes, q 0 is the strt stte, f! Q (Q Q) j 2 Ag is finite fmily of finite trnsition reltions nd 2 Q 2 Q is finite collection of finite cceptnce pirs. Whenever (q; (q 1 ;q 2 )) 2! we write q! (q 1 ;q 2 ).

24 24 Chpter 2. Bckground bin utomt re thus nondeterministic. This feture is essentil; deterministic bin utomt cn be shown to be strictly less powerful thn their nondeterministic counterprts [Tho90]. Definition A run of the bin utomton A = (Q; q 0 ; f! j 2 Ag; ) on n A-lbelled tree t is ny Q-lbelled tree r such tht r() =q 0 if t(s) =then r(s) =q; r(s0) = q 0 ;r(s1) = q 00 for some q! (q 0 ;q 00 ) bin cceptnce sttes tht there is run where every pth stisfies firness condition in tht long ny pth there is n cceptnce pir (L i ;U i )such tht sttes in U i occur infinitely often nd sttes in L i do not. We let In(r j ) denote the set of sttes occurring infinitely often long pth in the run r. Definition A run r of A is ccepting if for ll pths in r there is n cceptnce pir (L i ;U i )2 such tht In(r j )\ L i = ; nd In(r j )\ U i 6= ; Definition A tree lnguge L is bin-recognizble if there is bin utomton A such tht t 2 L iff t dmits n ccepting run of A. Exmple For the lphbet f; bg consider the bin utomton A =(fq 1 ;q 2 g;q 1 ;fq 1!(q 1 ;q 1 );q 1 b!(q 2 ;q 2 );q 2!(q 1 ;q 1 );q 2 b!(q 2 ;q 2 )g;f(fq 2 g;fq 1 g)g) The stte q 2 is ssumed exctly when b is encountered, so from the cceptnce condition we see tht the tree lnguge recognized by A is tht of the trees tht do not hve pth contining infinitely mny b s. 2 In [b69] bin proved tht set of trees cn be ined in SnS if nd only it it is bin-recognizble. This result, stted in this thesis s Theorem 3.3.1, provided generliztion of Büchi s result tht S1S-inbility corresponds to the notion of the notion of being Büchi-recognizble nd will ply crucil role in Chpter 3.

25 2.2. Normed recursive BPA processes Normed recursive BPA processes In Chpters 4 to 6 we shll look t (edge)-lbelled trnsition grphs tht rise from the structurl opertionl semntics of process clculi. Definition A lbelled trnsition grph G =(Pr; Act; f!g)is triple consisting of set of sttes or processes Pr,setoftomic ctions Act nd fmily of trnsition reltions f! Pr Pr j 2 Actg. We refer to processes in generl by p;q;::: Whenever (p; q) 2! we write p! q. If there is no q such tht p! q we write p 6 The trnsitive closure f!ju2act u + g of the trnsition reltions is ined for w 2 Act + by p! w q if p! p 0 nd p 0! w q for some p 0.!. In prticulr we look t the trnsition grphs ined by the clss of gurded recursive normed BPA (Bsic Process Algebr) processes (see e.g. [BBK87,BK88]) Syntx nd semntics In the rest of this chpter nd in Chpters 4 to 6 E;F;G:::will be used to denote BPA process expressions. These re given by the bstrct syntx E ::= j X j E 1 + E 2 j E 1 E 2 Here rnges over set of tomic ctions Act, ndx over fmily of vribles. The opertor + is nondeterministic choice while E 1 E 2 is the sequentil composition of E 1 nd E 2 we usully omit the. A BPA process is ined by finite system of recursive process equtions =fx i = E i j 1 i kg where the X i re distinct, nd the E i re BPA expressions with free vribles in Vr = fx 1 ;:::;X m g. One vrible (generlly X 1 ) is singled out s the root. We shll occsionlly write 1 2 for binry reltions ; this should be red s stting tht

26 26 Chpter 2. Bckground the roots of 1 nd 2 re relted by. Often we shll only look t reltions within the trnsition grph for single. We cn do so without loss of generlity, since we cn let be the disjoint union of the 1 nd 2 tht we re compring (with suitble renmings of vribles, if required); its trnsition grph is then the disjoint union of those for 1 nd 2. We restrict our ttention to gurded systems of recursive equtions. Definition A BPA expression is gurded if every vrible occurrence is within the scope of n tomic ction. The system =fx i = E i j 1 i kg is gurded if ll E i re for 1 i k. Here nd in Chpters 4 to 6 we use X;Y;::: to rnge over vribles in Vr nd Greek letters ;;:::to rnge over elements in Vr. In prticulr, denotes the empty vrible sequence. Definition Any system of process equtions ines lbelled trnsition grph. The trnsition reltions re given s the lest reltions stisfying the following rules: E! E 0 E + F! E 0 E! E 0 EF! E 0 F F! F 0 E + F! F 0! 2 Act E! E 0 X! E 0 X = E 2 E! EF! F Finlly, the importnt extr restriction on fmily is normedness. Definition The norm of BPA expression E is ined s jej = minflength(w) j E w! ; w 2 Act + g A system of ining equtions is normed if for ny vrible X 2 VrjXj<1. The mximl norm of ny vrible in is m = mxfjxjjx2vrg.

27 2.2. Normed recursive BPA processes 27 An importnt property of the norm is tht it is dditive under sequentil composition nd for nondeterministic choice corresponds to tking the minimum. This reduces the clcultion of norms of vribles in system of process eqution to solving systems of equtions over the nturl numbers. Proposition For BPA expressions E;F we hve jefj = jej+jfj nd je + Fj = min(jej; jf j). POOF: Let w E be ny shortest string with the property tht E w E! nd let w F be ny shortest string with the property tht F w F!. Clerly, w E w F is shortest string u such tht EF! u. Also it is obvious tht the shorter of w E nd w F is the shortest string u with the property tht E + F u!. 2 As from Section we restrict our ttention to the clss of BPA processes in GNF, whose sttes re ll members of Vr. The following initions nd results re only stted for sttes tht re indeed strings of vribles. Definition The lnguge L() ccepted by 2 Vr is ined by L() =fw2act + j w! g We sy tht nd re lnguge equivlent iff L() =L(). Definition The set of trces Tr()for 2 Vr is given by Tr()=fw2Act + j w!g We sy tht nd re trce equivlent iff Tr()=Tr(). Exmple Consider the system = fx = + bxy ; Y = cg. HerejXj = jy j = 1, so is normed. By the trnsition rules in Definition X genertes the trnsition grph in Figure 2 1. We hve tht L(Y )=fcgwheres L(X) =fb n c n j n 1g nd Tr(Y)=fcgbut Tr(X)=fb n jn1g[fb n c j j j n; n 1g. 2

28 28 Chpter 2. Bckground X b-xy b - XY 2 b?? c Y c? Y 2 c - ::: :::.. :::::: b c -XY n b? Y n c - ::: :::. :::::: Figure 2 1: Trnsition grph for X = + bxy ; Y = c (Exmple 1) Exmple The system of equtions = fx = X; Y = c + Xg is not normed, since there is no w such tht X w!. 2 Thus, becuse of the normedness restriction, normed BPA does not include ll regulr processes. Nevertheless, it is very rich fmily with processes tht cn hve infinitely mny sttes even fter quotienting by ny behviourl equivlence in the liner/brnching time hierrchy. For instnce, in Exmple bove, for ny two distinct sttes 1 nd 2 we hve Tr( 1 ) 6= Tr( 2 ), so no two distinct sttes re relted by ny equivlence in the liner/brnching time hierrchy, since they re not even trce equivlent. Thus, the trnsition grph is left unchnged with infinitely mny sttes fter quotienting by ny such equivlence Bisimultion equivlence on BPA processes In Chpter 4 we prove tht bisimultion equivlence [Pr81,Mil89] over normed BPA processes is decidble. Definition A reltion between processes is bisimultion if whenever pq then for ech 2 Act 1. p! p 0 )9q 0 :q!q 0 with p 0 q 0

29 2.2. Normed recursive BPA processes q! q 0 )9p 0 :p!p 0 with p 0 q 0 We ine by = f(p; q) j pq for some bisimultion g. Ifp q, p nd q re sid to be bisimultion equivlent or bisimilr. Proposition [BK88] is congruence reltion w.r.t. + nd Proposition For ny normed system, implies tht L() = L(), nd thus lso jj = jj. Exmple An exmple of bisimilr BPA process expressions is given by fx = Y X + b; Y = bx; A = C + b; C = baag. We hve tht X A, since the reltion f(x n ;A n )jn 0g[f(YX n+1 ;CA n )jn 0gis bisimultion (where V n here denotes n successive V s, V 2 Vr) Axiomtiztions of bisimultion equivlence In the usul presenttion of BPA (see e.g. [BK88]), much ttention is usully devoted to the so-clled BPA lws presented here in Tble 2 1. The BPA lws re esily shown to be sound w.r.t. bisimilrity 1, irrespective of ny restrictions on the processes involved. Proposition [BK88] For ny BPA expressions E 1 ;E 2 nd E 3 we hve tht E 1 + E 2 E 2 + E 1, (E 1 + E 2 )+E 3 E 1 +(E 2 +E 3 ), E 1 +E 1 E 1, (E 1 +E 2 )E 3 E 1 E 3 +E 2 E 3 nd (E 1 E 2 )E 3 E 1 (E 2 E 3 ). The BPA lws do not form complete xiomtiztion of BPA; some notion of fixedpoint induction must be dded in order to prove equtions involving recursively ined processes. In Section 4.3 we show how such n induction principle rises from the tbleu method used to decide strong bisimilrity nd use it together with n encoding of the BPA lws nd congruence lws of Proposition to give n equtionl theory for normed BPA processes in 3-GNF (see below). 1 In fct the BPA lws re sound for ll equivlences in the liner/brnching hierrchy [vg90].

30 30 Chpter 2. Bckground E 1 + E 2 = E 2 + E 1 A1 (E 1 + E 2 )+E 3 =E 1 +(E 2 +E 3 ) A2 E 1 + E 1 = E 1 A3 (E 1 + E 2 )E 3 = E 1 E 3 + E 2 E 3 A4 (E 1 E 2 )E 3 = E 1 (E 2 E 3 ) A5 Tble 2 1: The BPA lws Normed recursive BPA processes in Greibch Norml Form Any system of gurded BPA equtions hs unique solution up to bisimultion equivlence [BK84]. Moreover, in [BBK87] it is shown tht ny such system cn be effectively presented in wht we here cll Greibch Norml Form. In this thesis we restrict our ttention to normed BPA processes given in 3-GNF. Definition A system of BPA equtions is sid to be in Greibch Norml Form (GNF) if ll equtions re of the form fx i = Xn i j=1 ij ij j 1 i mg If for ech i; j the vrible sequence ij hs length( ij ) <k, is sid to be in k-gnf. The norml form is clled Greibch Norml Form by nlogy with context-free grmmrs (without the empty production) in Greibch Norml Form (see e.g. [HU79]). There is n obvious correspondence with grmmrs in GNF: process vribles correspond to non-terminls, the root is the strt symbol, ctions correspond to terminls, nd ech eqution X i = P n i j=1 ij ij cn be viewed s the fmily of productions fx i! ij ij j1

31 2.2. Normed recursive BPA processes 31 j n i g. The notion of normedness sys tht the grmmr must not hve useless productions. It is well-known tht ny context-free lnguge (without the empty string) is generted by grmmr in 3-GNF [HU79]. One should lso notice tht for systems in GNF, trnsition step in the opertionl semntics of Definition corresponds to leftmost derivtion step in the corresponding grmmr. Theorem [BBK87] If is gurded system of BPA equtions, we cn effectively find system 0 in 3-GNF such tht 0. Moreover, when is normed, so is 0. POOF: An effective procedure for rewriting into 3-GNF consists in first rewriting into GNF nd then rewriting the resulting system into 3-GNF. (We ssume tht the right-distributive lw A4 (Figure 2 1) hs been pplied (from left to right) s fr s possible.) For the rewriting into GNF, we first replce ll internl occurrences of tomic ctions by equtions. Thus, for ech tomic ction occurring in the initions introduce new vrible X, replce s mny occurrences of s possible while keeping the resulting system gurded nd dd the eqution X = to. We then remove ll unresolved sums from the outside in. An unresolved sum is sum F + G occurring in n expression of the form E(F + G). We now repet the following loop until there re no unresolved sums left: For ech outermost unresolved sum F + G introduce new vrible X F +G nd replce ll occurrences of F + G by X F +G. Add the eqution X F +G = F + G to. (This my mke the resulting system ungurded.) Cll the equtions dded in trversl i of the loop the ith strtum (letting the originl be strtum 0). After ll unresolved sums hve been removed, ll equtions in the resulting system re of the form X i = X j ij ij + X l il

32 32 Chpter 2. Bckground We then mke ll ungurded summnds gurded. Notice tht ll vribles introduced in strtum i hve initions using only vribles in strt < i. We replce ungurded vribles by their initions, using the following loop. It is esy to see tht when we rech strtum i ll equtions in strt <ire now gurded. For ech successive strtum do the following: For every eqution in the strtum, for ny ungurded summnd X 0 kl 0 kl replce X0 kl by its inition P ij ij ij nd use the right-distributive lw (A4) to obtin the new summnd P ij ij ij 0 kl. We now hve system of process equtions in k-gnf for some k. We cn then rewrite the system into 3-GNF in the following wy. We introduce new pir vrible U XY for every occurring vrible pir XY, dding the eqution U XY = XY. We then replce every occurrence of XY by U in ech eqution, going from left to right. For ech of the new ungurded equtions U XY = XY we use the sme trick s bove: X is replced by its inition nd A4 is pplied. This my hve introduced new instnces of the vrible pirs, which we then hve to replce by pproprite pir vribles. The resulting system is now in d k e-gnf. The whole procedure is repeted until we rech 2 3-GNF. Since ll steps used in the lgorithm described here either simply introduce new vribles tht renme expressions or use the BPA lws, we see tht bisimilrity nd thus normedness must be preserved, so clerly 0. 2 Exmple Let us rewrite the system : X Y Z = (Y + ZX)+Xb = Z(Y + bxxx)+z = in 3-GNF.

33 2.2. Normed recursive BPA processes 33 After removing internl occurrences of ctions, it becomes X Y Z X b = (Y + ZX)+XX b = Z(Y + X b XXX)+Z = = b We then remove unresolved sums, getting X Y Z X b X Y +ZX X Y +Xb XXX = X Y +ZX + XX b = ZX Y +Xb XXX + Z = = b = Y + ZX = Y + X b XXX After we hve got rid of ll ungurded sums, we hve X Y Z X b X Y +ZX X Y +Xb XXX = X Y +ZX + XX b = ZX Y +Xb XXX + Z = = b = ZX Y +Xb XXX + Z + X = ZX Y +Xb XXX + Z + bxxx This system is in 4-GNF. We introduce U XX = XX nd U XXb = X b X nd get X = X Y +ZX + XX b

34 34 Chpter 2. Bckground Y Z X b X Y +ZX X Y +Xb XXX U XX U XXb = ZX Y +Xb XXX + Z = = b = ZX Y +Xb XXX + Z + X = ZX Y +Xb XXX + Z + bxxx = XX = X b X which then becomes X Y Z X b X Y +ZX X Y +Xb XXX U XX U XXb = X Y +ZX + XX b = ZX Y +Xb XXX + Z = = b = ZX Y +Xb XXX + Z + X = ZX Y +Xb XXX + Z + bux = X + XX b X = bx finlly rriving t 0,whichis X Y Z X b X Y +ZX X Y +Xb XXX = X Y +ZX + XX b = ZX Y +Xb XXX + Z = = b = ZX Y +Xb XXX + Z + X = ZX Y +Xb XXX + Z + bux

35 2.2. Normed recursive BPA processes 35 U XX U XXb = X + U XXb X = bx 2 Becuse of the correspondence with context-free grmmrs, we immeditely see tht lnguge equivlence (or completed trce equivlence) is undecidble for normed BPA processes. This follows directly from the result for context-free grmmrs (see e.g. [HU79]). In our terminology this result reds s follows: Theorem For ny normed system of BPA process equtions in GNF it is undecidble whether L() =L()for ; 2 Vr. An esily estblished consequence is tht trce equivlence is undecidble. Theorem For ny normed system of BPA process equtions in GNF it is undecidble whether Tr()=Tr()for ; 2 Vr. POOF: We cn reduce lnguge equivlence to trce equivlence, since we hve L() = L()iff Tr( p ) = Tr( p ) where p is new ction (this is n observtion due to Lmbert Meertens). 2 An importnt dvntge of using GNF is tht the sttes in the trnsition grph for process given in this wy re elements of Vr. Moreover, the restriction to vrible sequences of length t most 2 gurntees limited growth of these sequences under single trnsitions. When pplying ining eqution to the leftmost vrible in string the length of the derivtive increses by t most 1: Proposition Suppose is in 3-GNF. Then, for ny 2 Vr, whenever! 0 we hve length( 0 ) length() +1.

36 36 Chpter 2. Bckground POOF: Suppose = X i 000. Then! 0 must be due to X i! 00. This in turn is due to the ining eqution X i = P n i j=1 ij ij hving summnd 00 with length( 00 ) 2. Since 0 = , the result follows. 2 Finlly, the following simple reltionship between lengths nd norms for vrible sequences becomes prticulrly useful in Chpter 3. Proposition For 2 Vr length() jjnd jjm length() Self-bisimultions For finite-stte processes nive decision procedure for the bisimultion problem p q consists in enumerting ll binry reltions over the stte spce nd determining if there is reltion mong them which is bisimultion contining (p; q). But since in generl bisimultions over normed BPA processes my be infinite - for instnce, the lest nonempty bisimultion for the trnsition grph in Exmple is the identity - decision procedure for the bisimultion problem for normed BPA cnnot rely on this. However, whenever, our tbleu system in Chpter 4 will construct self-bisimultion, finite reltion Vr Vr whose closure under congruence w.r.t. sequentil composition is bisimultion contining (; ). The notion of self-bisimultion ws introduced by Didier Cucl in [Cu90] (originlly published s [Cu88]). Here the notion of lest congruence is essentil. Definition For ny binry reltion on Vr,! is the lest precongruence w.r.t. sequentil composition tht contins,! the symmetric closure of! nd! the reflexive nd trnsitive closure of! nd thus the lest congruence w.r.t. sequentil composition contining. A self-bisimultion is then simply bisimultion up to congruence w.r.t. sequentil composition.