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BDD-Driven First-Order Satisfiability Procedures (Extended Version)

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(1)BDD-Driven First-Order Satisfiability Procedures (Extended Version) David Déharbe, Silvio Ranise. To cite this version: David Déharbe, Silvio Ranise. BDD-Driven First-Order Satisfiability Procedures (Extended Version). [Research Report] RR-4630, INRIA. 2002, pp.24. �inria-00071955�. HAL Id: inria-00071955 https://hal.inria.fr/inria-00071955 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. BDD-Driven First-Order Satisfiability Procedures (Extended Version) David Déharbe — Silvio Ranise. N° 4630 Novembre 2002. ISSN 0249-6399. ISRN INRIA/RR--4630--FR+ENG. THÈME 2. apport de recherche.

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LNMEOQPSRGREOUTWVHXZY ½¾k[8 Ï ‰Š‰v’6d­> µ6w"z{kl°£¨p6‚™`. `ªbu™^{pAŒpsnqpsrs} u™p z{lmnyŒª|™wxu™ly|†¬+wslqnmlmu†}f‹‚„ptbLŒz‚„ba| £¨ps‚Nba€Hz{w—u„lmp6k{wxnAu„^b(p6‚™lqba| +ws|™baŒ"psk u™^b1‚™b ¼ ‚™lmu™lqkrvws‹‹‚„p6wst)^"ly|%‹{‚™p6‹+pH|~bLŒ³ Ï |™wxu™ly|†¬+wslqnmlmu†}¤‹{‚™ptbLŒz‚„bƒly|#wsk wxnqrsps‚„lmu™^` ¼ ^lyt)^ ly|jt(wx‹+wxnqbfps£Nt)^bat)¸Alqkr ¼ ^b(u™^ba‚jw3tp6k— †zk{tu™lqpsk¥Twxny|~p ¼ ‚™lmu~u„b(k“w6|¤w3|~b(u)«\8px£1rs‚„pszk{Œ?nqlmu™b(‚)wxny|¤lq| |„w—u„lq|~¬+wsnmb ¼ ³ ‚a³ ua³w?tba‚~u)wxlqk¥÷ba€Hz{w—u„lmp6k{wxn§«vu™^baps‚„}^] ¥ ¼ bªwsnq|™p?|™wº}?u„^{w—u_\ lq|`]o©)¥÷zk+«†|„w—u„lq|~¬+wsnmbº« ³ ]^{b3wx‹{‹‚™pHwst)^Wps£ 8 Ï ‰Š‰Š’6"d >w6|™|™z`ªba|fu™^+w—ucu™^b3u™^baps‚„a} ] t(wsk +b?w—ōlqps`!w—u„lmÃabaŒWA}Ww¯¬+kl°u„b|™bu bEced ]_f"ps£"¥÷ba€Hz{w—u„lmp6k{wxn§«jtnywxz{|™ba|ag ³ :\wskH}“zly€Hzl°u„psz{|ou™^baps‚„lmbL|o|™wxu™ly|†£¨}<u™^ly|o‚„ba€Hzlq‚™ba`cbakHuo|™z{t)^Wws| u„^bu™^baps‚„}Wpx£jnmly|†u)|(µˆwx‚„‚„wº}|aµƒwxk{Œ¿u™^{b(lq‚ªt(ps`˜lmk+w—u™lqpsk%³¿½¾ku™^b¯‚„b ¼ ‚„l°u„lmk{r“wx‹{‹‚™pHwst)^µ1w“|„w—u„lq|~¬+w—© {lmnql°u†}‹‚„pAt(baŒz{‚™b£¨psg‚ ] ws`ªpszkHu„|cu„p®u™^b¯b(ō^{wxz{|~u™lq–sb¯ws‹‹nqlqtaw—u™lqpsk8ps£¤u„^b¯lmk£¨b(‚„b(k{t(b¯‚™znqba|ªps£"w |™z‹Jb(‚„‹+pH|~lmu™lqpsk»t(wsnqt(znmz+|c¥T|~babªbs³ r{I³ 8 ”¤‰ŠU’ h>§«¤u„[p \ji bkc7d ]-f ³ Ï |"w?tpsk+|~bL€6z{b(k{t(bsµ²u™plq`c‹{nmba`cbakHuow |„w—u„lq|~¬+wslmnqlmu†}o‹‚™pt(baŒz‚„bsµ—lmu lq||™z¶·tlqb(kHuu™p"z{|™bw¤|~u„wxu™b©­px£§©Du™^{b©¾wx‚™u#‹‚„p—–sba‚#`ªbat)^+wxklqÃ(lqkrŠu„^bŠ|™z‹+ba‚~© ‹Jp6|™lmu™lqpsk¯t(wsnqt(znmz+|(³ˆ]^lq|¤wxny|~pªps±Jba‚„|Šwsk?b¶·tlqb(kHu¤wsn°u„b(‚„k{w—u„lm–6b¤u„p·|~‹Jbat(lqwsnmlqÃ(bLŒ?Œbat(lq|™lmp6k?‹‚™pt(baŒz‚„ba| ¥T|~baKb 8 Ï ÁˆE‰ lƒ’61d >£¨p6‚bō‹Jb(‚„lm`ªb(kHu)wxnb(–AlyŒb(k+tbopx£#u™^ly|£÷w6t u)«³ ]^bo|„w—u„lq|~¬+wx{lmnql°u†}!ps.£ "Z+ª+pApsnqbawsk3t(ps`f{lmk{wxu™lqpsJk m“ps£#rs‚„psz{k{Œ!nql°u„b(‚)wxny|ˆtawxk3+b"‚„baŒz{t(baŒ·u™pcu™^b |„w—u„lq|~¬+wslmnqlmu†}\px£w¯|~b(u px£|~b(u„|ops£ˆnqlmu™b(‚)wxny|"A}“t(pskA–sba‚~u„lmk[r m¦u™p\Œlq|÷ †zk{tu™lq–sb·kp6‚™`!wsn£¨ps‚„` ¥÷ÂjE” 6 « wsk{Œ®A}“|™‹nql°u™u™lqkr\psk Œlq|÷ †zk{tu™lqpsk{|an³ 6ps‚fbÅwx`ª‹nqbsµt)^bat)¸Alqkr ¼ ^b(u™^ba‚ bpojq ¥ ¼ ^ba‚™b b wxk{Œ q ws‚™bctp6k— †zk{tu™lqpsk{|¤px£Nr6‚™p6zk{Œ¯nqlmu™b(‚)wxny|)«ly| ]o©­|„w—u„lq|~¬+wx{nmb˜‚™bLŒz{tbL|Šu„pt)^{bat)¸Almk{r·u™^{-b ]"©¾|„w—u™ly|~¬+wxlqnql°u†} ê#ê. 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(49) # /" ") /F  / ®u™p¯Œbakpxu„b˜bal°u„^b(‚ wªt(psk{Œlmu™lqpsk+wxnJu™ba‚™`èp6‚Šw˜t(psk{Œl°u„lmp6k{wxn²£¨p6‚™`˜znqw{³ ]¼ ^bj¬{‚„|~uŠwx‚„rsz`ªbak6upx£wªtpsk+Œl°u„lmp6k{wxnb(ō‹‚™bL|™|™lqpsk ¥ ^p6|™bŠu„ps‹©­`ªp6|~u1|™}H`˜+p6n+ly| a«ƒly|ˆt(wxnqnqbaŒªu„^b (2 "·px£u„^bvb(ÅA‹{‚™bL|™|™lmp6k³ ēbjws|„|~z`ªbjtpsk+Œl°u„lmp6k{wxn b(¼Å‹‚™bL|™|™lqpsk{|1u™p!|„w—u„lq|~£¨}!u™^{b"£¨psnqnmp ¼ lqkrcu™^‚„b(b"‹‚„ps‹Jb(‚™u™lqba|a³ ŸÍ Œlq|~u™‚„lm{zu™bL|p—–sba‚ˆp6‹+ba‚„wxu™p6‚„|aµAlT³ bs³ d

(50) *,*,*+( d  ( & ( 2 f (*,*+* f‚„b ¼ ‚„l°u„ba|ˆu™p d  ( d

(51) *,*,*+( & ( *+*,* f ( d

(52) *,*,*+( 2 ('*+*,* ff µ ¼ ^{b(‚„b Xly|jb(lmu™^ba‚ow·JpHp6nmbLwxk<t(pskk{bat u„lm–6bsµJu™^{bcbL€Hz{wxnql°u†}¯|™}A`fJps!n 0˜µJp6‚jw |~}A`˜+p6nlm^k A¤µ Z&:( 2 ws‚™bot(psk{Œlmu™lqpsk+wxnbō‹‚„ba|„|™lmp6k{|(µwsk{Œ \lq|vwct(psk{Œl°u„lmp6k{wxn²£¨p6‚™`˜znqw{³ d  (  & (  2 f˜‚™b ‚™lmu™bL|˜u™p d  ŸÍ  &  - &f d   2  / &f ^ba‚™b  (  & µ wxk{Œ  2 wx‚„bXtp6k{Œlmu™lqpsk{ws¼ n £¨ps‚„`fz{nqwsbs³ ¥¨Ä»b3Œb(kpsu™bXA} D 1u„^bXb(ÅA‹{‚™bL|™|™lmp6k®¼ psu)wxlqkbaŒ®A} ‚™ba‹nywstlqkr!wxnqnpt(t(z‚™‚„b(k+tba|ˆps£ lmk ¼ lmu™^ D ³ò«ª]^ly|‹‚„ps‹Jb(‚™u†}!lq|vwxny|~pc¸Akp ¼ kw6| A^{wskkp6k  | Œbat(ps`ª‹Jp6|™l°u„lmp6k3‹‚„lqk{tlq‹nqb·¥÷|™b(bsµb6³ r+!³ 8 gjp6z ;x >§«³ d  ( ( fˆ|™lm`ª‹nqlm¬{ba|1u™p µ ¼ ^{b(‚„" ŸÍ b ly|1wctp6k{Œlmu™lqpsk{wsn{£¨p6‚™`˜znyw wxk{Œ lq|ˆwctpsk+Œl°u„lmp6k{wxn bō‹‚„ba|„|~lqpsk%³ ¹¤z{‚ rsp6wsnAlq|u„poŒba|™lmr6kcwskªb¶·tlqb(kHu u™bLt)^kly€Hzbˆu™p t)^bLt)¸ ^bu„^b(‚ƒw"t(psk{Œlmu™lqpsk+wxnA£¨ps‚„`fz{nqw !ly|ƒwjnqpsrx© lyt(wsnJt(psk{|™ba€Hzbak{tb¤px£ bEced ]KfNps‚ˆba€Hzlq–ºwsnmbakHu™nq}sµA|~lqk{tb ly|1¼ r6‚™p6zk{Œµ ¼ ^b(u™^ba‚ ?ly| ]"©­zk{|„w—u™ly|~¬+wxnqbs³ ƒws‚™lywxkHu)|#ps£+u„^ly|‹‚™p6nqb(` ^{wº–6b+bab(kªbÅAu™bak{|™lm–6b(nq} |†u„z{ŒlqbaŒµ6ba|™‹+bLtlywxnqnm} lqkcu™^bŠt(pskHu™b(ÅAups£+‹‚„psr6‚„ws`c© `ªlqkrnqwskrsz{wsrsbL| ¥÷|™b(bcbs³ r{I³ 8 :\tL‡ˆ6‘ >§«³o½­£ ]èly|¤u™^bc‹z‚„b˜u™^baps‚„}ps£ƒba€Hz{wsnmlmu†}sµJu™^{b(Qk 8 Ab(5u ' >ƒ|~^p | u„^{w—u˜u™bL|†u„lmkr<u™^b3ba€Hzlq–—wxnqb(k{t(bXpx£vt(psk{Œl°u„lmp6k{wxnNu™b(‚„`!| 3t(wskW+b?Œpsk{b·lmk¿‹+p6nm}Akp6`clywxn1u™lq`ªbXzk+Œ¼ b(‚ u„^b £¨p6nmnqp ¼ lqkrXtp6k{Œlmu™lqpsk{| o¥ ¾«jtpsk+Œl°u„lmp6k{wxn#bō‹‚„ba|„|~lqpsk+|Šwx‚„bf‚„b(‹‚„ba|™b(kHu„baŒ?ws|vu™‚„b(bL|vwxk+Œ ¥

(53) ¾«vu™^balm‚ r6z{wx‚)Œ|¤wx‚„b px£ u„^bf£¨p6‚™` $'& 0 $/2 µ ¼ ^{b(‚„b $'&N(-$/2 wx‚„b˜tp6k{|~u„wxkHu)|(³¤Ä“bLwx¸sbaklqkr·b(lmu™^{b(‚"tpsk+Œl°u„lmp6k¿¥ ¾« p6‚!¥

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(56) TO tp6k{Œlmu™lqpsk{wsn+b(ō‹‚™bL|™|™lqpsk'  Nd M(  (O 1f ly|NlqkQP 0/" "

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(64) 1f fµ ¼ ^ba‚™b 8 & 8 2 8 K 8

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(82)  n( YF A³v]^bakµ bf|™z`ªlmu i bEced ]-f ¥¨£¨ps‚vbLwst)^ j@ d  ( f~« u„p3w|™wxu™z‚)w—u„lmp6k¯‹‚„p—–sb(‚ ¼ ^lyt)^\u™‚„lqba|vu™p?Œb(‚„lm–6b u„¼ ^b˜ba`ª‹u†}\tny wxz{|™bs³j½­uolq|¤baw6|~}u„p|™^p  ¼ u™^+w—Cu »lq| nqpsr6lqtawxnqnm}!bL€Hzlm–—wsnmbak6uu„p       ³ Alqk{t(b1u„^bˆkHz{`fJb(‚px£{‹+w—u™^+|#lqk˜wjtp6k{Œlmu™lqpsk{wsn6k{ps‚„`ªwsns£¨p6‚™` tawxk Jbbō‹Jpskbak6u„lqwsn6lqk˜u™^bˆkAz`fJb(‚ ps£1w—u„ps`!| ¼ ^lqt)^“pt(tz{‚„|vlmk»l°uLµ ¼ bct(wsk<rsbakb(‚)w—u„b bō‹JpskbakHu™lywxnqnm}?`!wxkA}¯‹‚„pApx£ƒps{nmlqr6wxu™lqpsk{|Š£¨ps‚ju™^b ‹{‚™p—–6b(‚L³Áˆ}¿tp6k{|™lqŒba‚™lqkr»rsbakb(‚)wxnqlmÃabaŒ¿‹{w—u„^{|(µ1u™^{bkAz`˜+ba‚!px£"‹‚„pHps£¤p6nmlqr6wxu™lqpsk{|ct(wskJb¯`!wsŒb |™`!wxnqnmba‚<¥¨‹Jp6|„|™lmnq}8pskb ^b(k u„^b<k{pAŒb\lq|Xu™^{b<‚„pHpsuaµlD³ b6³Nu„^b“|™‹nql°u™u™lqkrŒba‹u™^ lq| 66«³ ¹¤k u™^b psu™^ba‚"^{wxk{Œµ%baw6t)^<p6kbªpx¼ £Nu„^ba|™bª‹‚„pHps£1p6nmlqr6wxu™lqpsk{|¤ly|"nmlq¸sbanm}¯u„p?+b!`ªps‚„bªtps`ª‹nqbÅ<u™^{wsk»w‹‚„pApx£ p6nqlmrHw—u™lqpsk“ws|„|~ptlyw—u„baŒ?u„pwX£÷wxny|™b˜p6‚vu„‚™zbc‚)wxk{t)^%³o½¾k\£÷wstuaµr6b(kba‚„wsnmlqÃ(bLŒ¯‹{wxu™^{|j‚„ba€Hzlq‚™bfu™p?t)^bat)¸ u„^9b ]"©­zk{|„w—u„lq|~¬+wslmnqlmu†}<px£Št(nqwsz{|~bL| ¼ l°u„^ `ªps‚„bªu™^{wskWpskb!nqlmu™b(‚)wxn ¼ ^b(‚„baw6|o‹{w—u„^{|op6knm}“‚™bL€Hzlm‚„bªu™p t)^{bat)¸ou„^Eb ]"©­zk{|„w—u„lq|~¬+wx{lmnql°u†}fpx£²nmlmu™ba‚„wsnq|a7³ 6 lqk{Œlqkr"u™^{bŠ‚™lqrs^Huu™‚)wsŒb©­px±XJbu b(bak˜u„^ba|™bu p"w6|~‹Jbatu„| ‚„ba€Hzlq‚„ba|%wvu™^{ps‚„pszr6^ bō‹Jb(‚„lm`ªb(kHu)w—u™lqpsk˜wxk{Œ u™^bŒb(–6b(nqps‹`ªbak6upx£+kb ¼ ^{b(z‚„¼ lq|~u™lyt(|a³%Ä»bˆ‚™ba¼ ‹+p6‚~u |~p6`ªb b(ō‹+ba‚™lq`ªb(kHu„|olqk AbLt u™lqpsk®+³ Ï ‚„b(`!ws‚™¸\ly|olqkWps‚)Œb(‚L³3]^{b3‡ˆE” 6 px£w£¨ps‚„`fznyw ¿psu)wxlqkbaŒ“£¨‚„ps` u„^bªtpsk+Œl°u„lmp6k{wxn kp6‚™`!wxn#£¨ps‚„`ÐpxI£ Wtp6k6u)wxlqk{| 1~nqb(`ª`!w—u„w 3)³ ]#p3|™b(b ¼ ^{wxu¤u„^lq|j`ªbawsk{|aµ²t(psk{|™lyŒb(‚  u„p¯+b·u™^b!£¨p6‚™`˜znyw 8  ¥ ¼ ^ba‚™b 8 wsk{Œ  wx‚„b!‹‚™p6‹+pH|~lmu™lqpsk+wxn nqbu~u„b(‚)|„«³]^{b 1†p6H–Alqpsz{| 3·‡ˆG” 6 ps£ ly| 8L8<8 AL( 8 ZA A ³”vp µ lm£ b!u™‚)wxk{|™nyw—u™b u„p<tp6k{Œlmu™lqpsk{wsnNkp6‚™`!wsnƒ£¨ps‚„` wxk{Œ®u„^b(k b·‚„baw6Œ®px± u„^bc‡ˆG” 61µ ¼ bop6u„wslmk ¼ 8L8<8 AL(¼ 8 8 ( ZA A µ ¼ ^ba‚™b 8<lq|u„^b nqb(`ª`!w³]^ba‚™boly|Šwxk¯p6A–Hlqpsz+|¼ ‚™bL|~p6nmzu„lmp6k |~u™ba‹¯+b(u bab(k?u„^bou p·tnywxz{|™ba|Š|™p·r6b(kba‚„wxu™baŒ³ Ï |vw!tp6k{|™ba€Hzb(k+tbsµ+l°u¤lq|vkpxu¤t(nmbLwx‚u„^{w—uvu™^bc‡ˆ”G6 p6u„wslmk{baŒ ¼ lqk˜u™^ly| ¼ wº¼ }"ly| 1†p6‹u™lq`!wxn 3N£¨ps‚ |„w—u„z‚„wxu™lqpskou„^b(p6‚™ba` ‹‚„p—–sb(‚)|a³½¾k˜r6b(kba‚„wsnTµºlmuƒt(wxk˜+b‹{‚™p—–6baŒ u„^{w—uu„^b(‚„b¤ly|wctnqp6|™b¤‚„b(nyw—u„lmp6k{|~^{lm‹3+b(u ¼ bab(k?ÁˆÂjÂj|wxk{Œ·u„wx{nmbLwxzÅ ¼ lmu™^nqb(`ª`!w—u„w¥÷|™b(bK8  3;=<>²£¨ps‚ Œbu„wslmny|)« ³ . . . F. F. . 6. 6. 6. . . .  . . `. < =)? B. ê#ê. "rUs  t. . . 7. D. =I<. %. 4. 6. M. 6. . M.

(83) hL’. !U '   G" $. ]^bŠ|™lmu™z{wxu™lqpskclq|ƒt(ps`ª‹nqlqtaw—u„baŒ˜H} u™^b£÷w6t u u™^+w—uƒ|~p6`ªb1u„^b(p6‚™}_]E`!wº} ‚„ba€Hzlq‚„bˆw6ŒŒlmu™lqpsk{wsnt(ws|™b© |™‹nql°u™u™lqkrcu™pcba|~u„wsnmly|™^]"©­zk{|„w—u„lq|~¬+wslmnqlmu†}s³Az{t)^wfu„^b(p6‚™}C] lq|tawxnqnmbLŒ  /"  0/  " 6³H6p6‚™`!wxnqnq}sµ ¼ b |„wº}u™^{wxu"w·£¨ps‚„`fznyw·ly-| F/   0/"Z  »l°£1l°u"b(kHu„wslmny|"wXŒly|÷ †zk{t u„lmp6k“ps£Nba€Hz{wsnmlmu™lqba|¤Jbu ¼ b(b(k»tp6k{|†u)wxkHu„| ¼lm£vlmu™ba^{–spsb(‚„z}®uctb(pskHkxu„ws †zlmnqk{lmk{t r“u„lmp6wxk kA}®px£Špx£nqlmu™u„b(^‚)bXwxnybL|€H¥¨z{lqkWwxnql°u™u„^lmbLb?|˜|™wslmr6nmp6k{kw—b u„1pz‚™xbXu„^pxb(£‚ ¼ u™^{lq|™bXbsµu„^lmucb(p6ly‚™|[}«o0 / ly |c "t ps6kA³ –6ÏbŃpu„^xb(u„p6^‚™b(}®‚ ly|ly|~b60µ /"Zly|˜  lq|  /"Z 0 /  " 6³W]^b?‹z‚„b·u„^b(p6‚™}®ps£vba€Hz{wsnmlmu†}Wlq|ªt(pskA–sb(Å Nu™^b3u™^{b(ps‚„}®px£¤ws‚™‚)wº} | !µ1w—ō¼ lmp6`!w—u™lqÃ(bLŒ A} 8   d 

(84)  db ( ( f ( f 0 ( 01     d 

(85)  dbC( ( f (  f 0   dbC(  f A µ !ly|¤k{psk© t(pskA–sb(ų#½¾k?psz{‚Šwx‹‹‚„p6w6t)^µHu™^b |„w—u„z‚)w—u™lqpsk3‹‚„p—–sb(‚‹Jb(‚™£¨ps‚„`!|ˆu™^b Œp6`!wxlqk|™‹+bLtlm¬+tot(w6|~b(©­|™‹nql°u™u™lqkr{³ 6{ps‚3u™^b“u™^{b(ps‚„ } !µ b“tawxk u„^lqk¸ px£f‚„b(nqlqb(–Almk{rWu™^b»‹‚„p—–sba‚·px£ u„^ly|3{z‚„Œb(k A}8u™‚)wxk+|†£¨p6‚™`ªlqkr u„^bflqk‹zu¤£¨p6‚™`˜znqwX|~pX¼ u™^{wxu¤baw6t)^¯pt(tz{‚™‚„b(k{t(b px£    d 

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