An E ective D ecision Procedure forLinear A rithm etic with Integerand R ealVariables ? BERNARD BOIGELOT,SEBASTIEN JODOGNE y ,and PIERRE W OLPER Universite de Liege Institut M onte ore,B28 4000 Liege,Belgium T hispaper considers nite-autom ata based algorithm sforhandling lineararithm etic w ith both real and integer variables. Previouswork hasshow n that thistheory can be dealtw ith by using nite autom ata on in nite words,butthisinvolves som e di cultand delicate to im plem entalgorithm s. T he contribution ofthis paper is to show ,using topologicalargum ents,that only a restricted class of autom ata on in nite w ords are necessary for handling realand integer linear arithm etic. T his allow s the use ofsubstantially sim pler algorithm s,w hich have been successfully im plem ented. C ategories and SubjectD escriptors: D .2.4 [ S oftw are E n gin eerin g]: Softw are/P rogram Veri ca- tion| Form al m ethods ;F.1.1 [ C om p u tation by ab stract d evices ]: M odelsofcom putation| Autom ata;F.4.1 [ M athem atical L ogic and form al languages ]: M athem atical Logic| C om - putational logic ;F.4.3 [ M athem atical L ogic and form al languages ]: Form allanguages| Classesde ned by gram m arsorautom ata. G eneralT erm s: A lgorithm s,T heory. A dditionalK ey W ords and P hrases: D ecision procedure,F inite-state representations,Integer and realarithm etic,W eak ! autom ata. 1. IN TRO D UCTIO N Am ong the techniques used to develop algorithm sfor deciding orchecking logical form ulas, niteautom ata haveplayed an im portantrolein a variety ofcases.Clas- sicalexam ples are the use ofin nite-w ord niteautomata by B uchi[B uchi1962] for obtaining decision procedures for the rst and second-order m onadic theories ofone successor,aswellasthe use oftree autom ata by Rabin [Rabin 1969]for deciding the second-order m onadic theory of n successors. M ore recent exam ples A uthors’e-m ail: fboigelot,jodogne,pwg@montefiore.ulg.ac.be A uthors’w ebsite : http://www.montefiore.ulg.ac.be/ fboigelot,jodogne,pwg/ ? T hiswork waspartially funded by a grantofthe \C om m unaute francaisedeB elgique -D irection de la recherche scienti que - A ctions de recherche concertees" and by the E uropean IST -F E T project A dvance (IST -1999-29082). A prelim inary version ofthis paper appeared as [Boigelot et al.2001]. y R esearch Fellow (\A spirant") for the N ationalFund for Scienti c R esearch (B elgium ). Permission to make digital/hard copy ofallor part ofthis materialwithout fee for personal or classroom use provided that the copies are notm ade ordistributed forpro torcom m ercial advantage,the A C M copyright/server notice,the title ofthe publication,and its date appear,and notice is given that copying is by perm ission ofthe ACM ,Inc. To copy otherwise,to republish, to post on servers,or to redistribute to lists requires prior speci c perm ission and/or a fee. c20YY ACM 1529-3785/20YY/0700-0001 $5.00 ACM T ransactions on C om putational L ogic, V ol. V , N o. N , M onth 20Y Y , P ages 1{0??.