LATVIJAS UNIVERSITATE-UNIVERSITY OF LATVIA ALGORITHMIC PROBLEMS IN ANALYSIS OF REAL TIME SYSTEM SPECIFICATIONS Karlis Cerans Riga, 1992 LATVIJAS UNIVERSITATE.UNIVERSITY OF LATVIA Algorithmic Problems in Analysis of Real Time System Specifications Karhs Cerans A Thesis for the Dr.sc.cornp. Degree at University of Latvia Institute of Mathematics and Computer Science University of Latvia Riga, Rainis blvd. 29, Latvia. 226250 Copyright @1992 Karlis Cerans University of Latvia Riga, 1992 Abstr-act The thesis is devoted to the study of analysis automation possibilities (decidability and undecidability of reachability, infinite behaviour possibility and bisimulation equiva- lence problems) for various kinds of real time system specification formalisms. The investigated formalisms are based on finite state model control structure enriched in various ways to reflect data and time dependencies of the modelled system behaviour. Decidable proved are, first, the vertex rea.chability and infinite behaviour possi- bility (infinite fea.siblepath existence) problems for progra.ms in a simple theoretical programming language, ca.lled LTIBA, which is an enrichment of the Finite State Machine model with variables, suitable for modelling real time system behaviour de- pendencies both on quantitative time constraints (the LTIM system of commands) and external data (the LBASE system of commands). An effective symbolic charac- teristic of the sets of all fea.sibleprogram paths in the terms of path set projectivity is also given for LBASE, LTIM and LTIBA progra.ms whenever possible. The undecid- ability of the vertex rea.chability problem is proved for programs in a language LTIM' which is a slight variation of the considered time constraint specification language LTIM. In the thesis also the strong and weak (abstra.cted from system internal actions) hisimula.tionequivalence problems are proved decidable for the formalism of Parallel Timer Processes (PTP), which are provided with th~ real time labelled transition sys- tem semantics and allow to express in a direct way the quantitative time constraints on the behaviour of concurrent real time systems. Various enrichments of the basic PTP model by additional features (including the processes with the dependencies on external data) are also considered and investigated w.r.t. decidability of the vertex reachability, infinite path fea.sibilityand bisimulation equivalence problems. An unde- cidability result regarding the considered algorithmic problems is obtained for a class of timed processes supplied with memory cells for moving the timer value information along the time axis. Finally, in the appendix an example of the rea.chability and path fea.sibilityanalysis of a simple real time system, specified as a process in the C.C.LT.T. telecommunica- tion system specification language SDL, is presented. 11 .Anot.acija Disertacija veltita dazadu reala laika sist.ernu apr akst a forrnalisrnu arializes auto- matizacijas iespeju izpetei (petita atrisinamiba sasniedz arnibas. bezgaligas dar bibas iespejas un bisimulacijas ekvivalences algoritrniskajarn rnasu problernarn) Aplukoro apraksta formiilismu pamata Qemts gallga automata modelis, ka.s bagatinats ar Iidzek- [iem, ka.s [auj aprakstit modelejarno sisternu darbibas at.karibu no laika un datiern. Algoritrniska atrisinarniba pieradHa, vispirrns, programmas virsot.nes sasniedza- mibas un bezgallga realizejarna cela eksistences (bezgaligas program mas darbibas iespejas) problem am programrnarn vienkarsa teoretiska valoda LTIBA. Katra prog- ramma valoda LTIBA lidz ar galIgu vadibas grafu satur mainigos , kas at.lau] apr akstit rnodelejamas reala laika sistemas atkaribu gao no ternporali kvant.it ativiem nosa- cijumiem (LTIM komandu sistema), gan arf no arejiern datiem (LBASE komandu sistema). Situa.cija..s, kuras tas iespejams, apliikojarno va.lodu LBASE, LTIM un LTIBA prograrnmu realizejamo ce]u kopam dots efektivs simbolisks raksturojums projektivitates terminos. Virsotnes sasniedzamibas problemas algorit.rnisk a neatr isi- namiba pieradita prograrnrnarn valoda LTIM', ka.s tikai nedaudz atskiras no petitas temporalo atkaribu specifikaciju va.lodas LTIM. Darba pieradita ari' stingras un vaja..s (no sistemas darbibas ieksejiern noti kumiem abstrahetas] bisimulacijas ekvivalences problernu algoritrniska atrisinamiba paralelo tairneru procesu (PTP) modelim, kurarn defineta reala laika iez'imeto pareju sisternu sernantika, un kas atlauj dabiska veida aprakst.it paralelu sisternu atkaribu no tem- porali kvantitativajiern nosacijurniem. Virsotnes sesniedzamibas, bezgalIgas darbibas iespejes un bisimulacijas ekvivalences problemu algoritrniska atrisinarniba petit.a aTI daZa.d.iern PTP modela paplasinajurniern a.r papildus sisternu apr akst a lidzekliern (t.sk. procesiem ar arejo datu atkaribas atteJcianas iespejam). BisimuJii.cijas ekvivalences un virsotnes sasniedzamibes problemu neatrisinarnlbas rezultets ieguts ar at.minas silnam paplasinatu tairneru procesu klasei. Darba pielikuma apliikots pierners vienkarsas reala laika sistemas, kas uzdota ka process C.C.I.T.T. telekornunikaciju specifikaciju valoda SDL, sesniedzamibas un ce]u realizejamibas analizei. III AHHOT&Il:H.lI ~ pa60Ta ~ MCCJJell(lIl3HH ~ ~ aHaJlI.I3& pa3JlWlHbIXM¥l.eneHorreMptaJIHCrOBpet.eiH (~I Ha pa3pelllHM)CTh MllCCOIlbJe npo& .neMbI ~ JllBM:))JGl[lC' 6ecJ<oHe<nDI pa6am H 6~ 3KIlMll3JJeHI'- HOCTH~ PIlCOoCI'peJlHLJe ~ 6a3HpyJorcJI Ha I'IOIUrne ICOHe'HlrO llB'l'OolaTa, paClIlHPeHHOI"O 3,lIj!CI,pa31IH'IHblMH CIlOC06aMH!WI o6ecrJe1eoo:I ~ OmtC8HHll3<lBl£IDdX:'Df pa&m,I MO/l.~ CHCTeM OT IlpeMeHHIoIX ycII:lIlHHH H OT lUlHHbDC. ~ ~ n<lJ(33BHa CHIL'mlIa!WI MIIaX.I!bIX npo6neMLJ.()C'mlIGIM:l 'I'M BeJXIlllHhI H IlO3II.«llI<Kl ~ pa6am !WI nporpaMM B HeClJl)J(H)M ~ KOM Jl3hJKe LTIBA, ~ pac:JDHpeHHeM Ja:He'H>-1lIm:MlTI«lH MC¥lenH nyreM BIle,lJ£HIDI Ilepel-EHHblX H ~ IlI>lpIl3Ifil> ~ ~ CH:TeMI.I, KaK OT TeMr/()- pa11LHO KOIIH'IeC'l'BeHIX ~ (CK:TeMa ~ LTIM), TalC. OT BHeUIHHX ~IX (CHC'reMIl. J<OMM:flIlnLBASE).B CHI')'BUKlIX, ~~ IICfIM:DKRCl,~ CHMIlOI1H'If.CKO OI1HCllffile MHl>IlfCl'B BCeX peaJIJl3)'PMoIX nyreH B nptI1lIIMlolllX ~IX Jl3h1KDll LBASE, LTIM, LTIBA JJ,lIHO B Tep&IHllJC npoeI<TIIIHlCTH ~ Hepa3pellAlMlCTh npo& JJeMbI Lf.l)C'nrJIGlM: BeJIlHlbI .!I,CIal3llHll !WI IlJllX1BMM B Jl3hJKe LTIM', xoropoe 'I'OIlhW I1OHeMHOI"y ~ OT ~ BLIIIJe Jl3b1K& ornIl:8HHll' IlpeMeHHIoIX ~ LTIM. B pa6cYre JJ,OIal3llHa TtlIOfIe pa3pe1II1D.«lCn. C'I'JlOfOH H CJlll60H (~ OT BHY- TpeHHHX: co6LmIiiJ CHCTeMbI) ~ 3KBHIlaIIeHmOC' !WI ~ 11pODt.COl8 C napaJ1eln.IlblMH '1'aHMepaMH (PTP), ~ ~ ~ IlpeI>tiIH Onpe.lte1EHHYIO llOC'pe,I:II:'l CH:TeM IICNe'IeHHbIX ~ H ~ rIpllMbIM o6pa3oM OIIHCaTh 3&- ~ napane1lhHb1X CH:TeM pe1IJIYlOfO ape.eHH OT BpeMeHH> KOIIH4eC'llleHH ycJ1l:lIlMH. P83peIIlHt.DCTh npo6neM ~ BeJI11HHbI, BaN:l>KIlOC'l\I ~ pa6cym H 61t- c~ ~KmIl!ll1IeHll« HCC1IP,DPIlaHa TllIOIle!WI HeI<OrOJ:LIX JlIlCIIlHIleHHM 0CIDlH0fI Mene1IH PTP (~TaIOIQ!~!WI 0IlI«:aHHlI ~ ~IX CW;- TeMOT lJ,llHHhlX~ Pe1YJIbTaT ~ ~IX ~ npo6neM IIOlIy'leH!WI1UIaCCa TetoRlpIIJlLHLlX ~ HMeKJIlJfX JI'IclOOI nawmJ !WI nepeIlJIH)I<eHH ~IX 3Hll'eHMft no Bpet.rHHDH OCM. Ii.aKoftiJ, B ~ ~ I1JMM!P llH&1IH3a Jl.(lC'l"H)IOO H ~ nyreii B ~ HeClD>I<HCM CHC'I'eM1>I peam,mro BpeMejlH, 3a.IlllHHOO IlJlOl.'lCCOM Ha Jl3hJKe SDL, pa3jlIIliorlIHH M10CIT !WI omaHWI ~IX CHCTeM. iv Acknow ledgements First of all, I want to thank my thesis advisor Prof. Janis Barzdins for his sugges- tion to look at the time constrained real time system analysis, for his really noteworthy advices and continuous encouragement in my work on the thesis over the years. I am very thankful to all my colleagues at Department of Computer Science, In- stitute of Mathematics and Computer Science for the provided possibility to work on my thesis, for their understanding, support and encouragement. Among my col- leagues I want to thank especially Prof. Audris Kalnins for a number of interesting discussions we have had on the topic of the real time system analysis and automated test case generation and Lolita Zeltkalne for her help to improve the English in the most important parts of the thesis. I want to express also my thanks to my colleagues and friends abroad - Uno Holmer, Wang Vi, K.V.S. Prasad, Alan Jeffrey and Prof. Bengt Nordstrom in Gate- borg, as well as Kim G. Larsen and Jens Chr. Godskesen in Aalborg both for numerous interesting discussions on the real time process calculi, and for the provided possibility to have a look on a wide spectrum of the ongoing research in the general area of the program correctness in the world. My deepest thanks are also to Dr. Agnis Andzans and Prof. Rusins Freivalds who, though not being in a direct relation with my thesis work, have done very much for my mathematical and scientific ed uca.tion. Perhaps the most important help I recieved during the work on the thesis was that by my parents Silvija and Henrihs and my sister Kristine, let me say: Paldies Jums, rnilie vecaki un rnasa' Riga, Latvija June 17, 1992 Karlis Cerans. v Contents 1 Introduction 4 1.1 "Real Time Systems" . 4 1.2 Symbolic Models of Real Time Systems . 5 1.3 Automation of Model Analysis . 7 1.4 Languages for Specifying Data.and Time Dependencies 9 1.4.1 Dependencieson Integer Valued Data ... 9 1.4.2 Time Constraint Specification Languages 10 1.4.3 Results . 12 1.5 Parallel Timer Processes . 13 1.5.1 Modelling and Reachability ..... 15 1.5.2 Bisimula.tionEquivalencesfor PTPs . 16 1.6 Comparison with Related Work 18 1.7 Organization of the Material . 21 I Languages for Data and Time Dependencies 22 2 Mathematical Preliminaries 23 2.1 Labelled Gra.phs. 23 2.2 Projective Path Sets 25 3 Language Definitions 31 3.1 The Language LBASE 31 3.2 Languages LTIM and LTlBA 34 3.3 Feasibility end Reachability 37 4 Finite Path Feasibility 40 4.1 Variable Vector Value Set Partitionings . 41 4.2 Path Feasibility Graphs . 44 4.3 LBASE: Perfectness of BG(P) . 46 4.4 Basic Graphs for LBASQ programs 49 CONTENTS 2 4.5 LTIM: Perfectness of BG(P) .50 5 Path Inequality Systems 53 5.1 Path Inequality Systems for LBASE . 53 5.2 Path Inequality Systems: LTIM 57 5.3 Point Classes .... 61 6 LBASE: Infinite Feasible Paths 66 6.1 Accomplished Loops ..