Reachability Analysis of Hybrid Systems via Predicate Abstraction

Predicate abstraction has emerged to be a powerful technique for extracting finite-state models from infinitestate discrete programs. This paper presents algorithms and tools for reachability analysis of hybrid systems by combining the notion of predicate abstraction with recent techniques for approximating the set of reachable states of linear systems using polyhedra. Given a hybrid system and a set of user-defined boolean predicates, we consider the finite discrete quotient whose states correspond to all possible truth assignments to the input predicates. The tool performs an on-the-fly exploration of the abstract system. We demonstrate the feasibility of the proposed technique by analyzing a parametric timing-based mutual exclusion protocol and safety of a simple controller for vehicle coordination. Comments From the 6th International Workshop, HSCC 2003 Prague, Czech Republic, April 3–5, 2003. This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/cis_papers/202 ! #"$ % &' (*)+ , ./ 0% 12&3 (4 5 6 7 8 , -&3 9 3 : ;=A@ @ B C DFEHG IHJLKMANQZ[B?<OS,\ ] ^ _ ] `La bdc e[fAg a h?f3i jlk-i g bnmTfAaoe c hnp:qrhTjsi eAg c fAtuivhQwTx tua hyx a z9hyt|{vaoeA}[t~fr!i j€ aohnhn}‚Tƒu{ c hnt|c „ f[fAbM… †v† ‡3‡3‡5ˆ }[a c } ˆ mnb‰a hyhMˆ a pTm‰† „ TŠyeAt|pn† ‹ ŒŽ   ‘d’ v“ €ŽeAa pytux c fAa+c Šy}‚f[e”c xofAtuivh „ c }!a g aoeA• a p–fAi Š‰a+c%b‰i ‡-aoe[jsmyƒ—fAa x „y˜ hytF™?myaRjsi e=aošOf[e”c xofAtuhy•!›nhnt~fAa ˜ }‚f”c fAaRg iOpya ƒu}9jFeAivgœtuhy›nhnt~fAa ˜ }‚f”c fAa6pytu}[xoeAaofAa6byeAi ˜ • e”c g } ˆV „ t|}—bdc b‰aoeLbTeAa }[a h?fA}9c ƒu•vi eAt~f „ g }Lc hnpQfAi?ivƒu}—jži eLeAa c x „ c Šyt|ƒut~frŸc hdc ƒ~ ˜ }[tu}=i j „ TŠyeAt|pŸ}‚T}‚fAa g }5Š?+x ivgRŠntuhntuhy•!f „ a6hni fAtuivh i j byeAa pytux c fAa c Šn}‚f[e”c xofAtuivh ‡'t~f „ eAaox a h?f5fAa x „ hnt|™?mnao}9jsi e6c bnbTeAi šTtug c fAtuhy•!f „ a6}[aof=i j eAa c x „ c Šnƒua }‚f”c fAa }=i j ƒutuhna c e3}‚T}‚fAa g }mn}[tuhn•=b‰i ƒu „ a pTe”cOˆT¡9tu{¢a h c „ yŠTeAt|p6}‚T}‚fAa g£c hdp c5}[aof-i jHmn}[aoe ˜ pTao›dhya p Š‰i?i ƒua c h bTeAa pytux c fAa } ¤ ‡ a x i hn}[t|pya¥eŽf „ a-›nhnt~fAa pTt|}[x¥eAaofAa'™?mni fAtua h?f ‡ „ iv}[a }‚f”c fAa }Rxoi e[eAa }[b‰i hdpYfAi+c ƒuƒlb‰iv}[}[tuŠnƒua f[eAmyf „ c }[}[tu•vhng aohOfA}5fAiQf „ a t|hybnmTf=byeAa pTt ˜ x c fAa } ˆ  „ a-fAi?i ƒOb‰aoe[jsi eAg }Žc h ivh ˜ f „ a ˜§¦ 9aošTbnƒui e”c fAt|i h6i jnf „ a-c Šy}‚f[e”c xofŽ}‚T}‚fAa gQˆ ̈Ya9pTa g ivhy}‚f[e”c fAa'f „ a'jsa c }[t|Šytuƒ|t~f© i jHf „ a byeAivb‰i }[a p fAa x „ hnt|™?mna'Š? c hdc ƒ~Ta tuhn•Rc bnc e”c g a¥f[eAt|x,fAtug t|hy• ˜ Šdc }[a pLg6mTfAmdc ƒvaošTx ƒumn}[tui hRbTeAi fAi?x i ƒyc hdp5}Ac jžaofr=i jdc'}[tug bnƒua x i h?f[eAivƒuƒuaoe—jži e—{¢a „ t|xoƒ|a9x i?i e”pTt|hnc fAtuivhHˆ « ¬n­ ®‰ ̄V°5±R2 3 ́®VμA° ­ Z[SH¶”· ^ Go@ W1 ̧do1⁄4»¥KV@ ¶¥E ] ] @ ¶o¶ NT1⁄253⁄4¿N‰W‰@ D ] KV@ ] Àd^ SVU ^ SÁKH<TGoW‰ÂL<OGo@ŸBT@ G ^FÃM] <?» ^ NTS1<OSMW1·VG¥NT»¥NTÄ ] NTDl<OSH<TDFo‰¶ ^ ¶ Å ÆVI Ç¢ÈvɂI‰»oKV@ Go@ KH<y¶5 ̧M@ @ S ^ S ] Go@ <y¶ ^ SVUQGo@ ¶¥@ <OG ] KYNTS4W‰@ BT@ DžNT· ^ SVU¿»o@ ] KVS ^sÊ EV@ ¶ 1⁄2§NTGR<TE‰»¥Ny3⁄4Ÿ<?»¥@vWYBT@ G ^FÃM] <O» ^ NTS NO1⁄23Kdod ̧VG ^ W1˧3⁄4 ^FÌ @vW4W ^ ¶ ] G¥@ »¥@ Ä ] NTSn» ^ SdEVNTEH¶ ͗3⁄4QNdWV@ Ds¶=NO1⁄23@ 3⁄4:Ä ̧M@vWVW‰@ W ] NTSn»oG¥NyDFDž@ G ¶ Å|ÇyI¥ÎVI ÇvϢɂÐMJLKH@!¶A» <?»o@ ĂNO1⁄2ÑÄr»¥KV@ Ă<TG”» ] Ny3⁄4Q·VE‰»o<O» ^ NTSH<TDŽ»¥NdNyDž¶L1⁄2§NyG 3⁄4QN‰W‰@ D ] KV@ ] Àd^ SVU NO1⁄2LKnod ̧VG ^ WÒ¶¥o‰¶A»o@ 3⁄4Ÿ¶ <TG¥@QNO1⁄29»AÂ9N Àd^ SHWV¶ ÐlJlNdNTDs¶ ¶”E ] KÁ<y¶ ÓŸÔ ́ÕlÖ,Õ-× Å|ÇyÇ É‚IlØ¿ÙdÚ Ù¢ÛÛ,ÜÒÅ ÝTÞ?ɂIŽ<OSHWÒß àlá âMãMäåÅ|Çvæ?É D ^ 3⁄4 ^ » »oKV@ ] NTSn» ^ SdEVNTEM¶RW‰odSH<T3⁄4 ^ž] ¶ »oN%¶ ^ 3⁄4Q·VDž@Ÿ<O ̧H¶”»¥G < ] Ä » ^ NTSH¶+¶¥E ] Kç<T¶QGo@ ] »o<TSVUTEVDs<OG ^ S ] DFEH¶ ^ NTSH¶ Ëè@TÐ UHÐ é êìë ÅFÇTíoÝ?ÉÑÍ I <OSMW ] NT3⁄4Q·VE‰»o@ »¥KV@1⁄4¶”@ »+NO1⁄2 G¥@v< ] KH<T ̧VDF@Q¶”»o<?»o@ ¶ @ Ì < ] »oDFoÒ<OSHW1⁄4@ î @ ] » ^ BT@ DFo1⁄4 ̧noÒ¶”od3⁄4: ̧MNyD ^s] 3⁄4Ÿ<OS ^ ·VEVDs<?» ^ NySÒNT1⁄29D ^ SV@ <TG ^ S‰Ä @ Ê EH<OD ^ » ^ @ ¶ ЉïRSŸ»oKV@ NT»¥KV@ G—KM<OSHWŽIn@ 3⁄4Q@ G¥U ^ SVU »¥NdNTDs¶—¶¥E ] K <y¶5ðRäâMãMñ-ò=ۉóŽâ1⁄4ŠȢÉrIdôVõyôyöYÅ Ï?ÉrI <OSHW Dž@ By@ DFö¥@ » ¶53⁄4Q@ »¥KHNdW÷Å|Ç øHI¥ÝVÇ É‚IH<O·V·HG¥N ̉^ 3⁄4Q<O»¥@ »oKV@!¶¥@ » NT1⁄2 Go@ < ] KH<O ̧HDF@ ¶”»o<O»¥@ ¶ ̧do · ́NTDžoyÄ KV@ W‰G <RNTG3@ DžD ^ ·H¶¥N ^ WV¶LÅFÇ ù?ÉV ̧do Ny·‰» ^ 3⁄4 ^Fú <?» ^ NTS:»¥@ ] KVS ^sÊ EV@ ¶ ÐOû'By@ S »oKVNTEHUTK:»¥KV@v¶”@L»oNnNyDž¶-KH<¢By@