Unhiding Redundancy in SAT

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Elizabeth Flowers
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1 Armin Bir Institut or Forml Mols n Vriition Jonns Kplr Univrsity Linz, Austri joint work wit Mrijn Hul n Mtti Järvislo Dution t Sl 2011 Rinr Cstl, Trns, Grmny Mony, Mr 7, 2011

2 Motivtion 1/21 SAT solvrs ppli to u ormuls million o vrils ststs solvrs us prprossin/inprossin n p n tiv inprossin tniqus or millions o vrils tis tlk: uniin runny in lr ormuls lmost linr rnomiz loritm usin t inry implition rp st nou to ppli to lrn luss ppr sumitt, vill on rqust

3 Binry Implition Grp BIG 2/21 (ā ) (ā ) ( ) ( ) ( ) ( ) (ḡ ) ( ) (ḡ ) (ā ē ) ( ) ( ) }{{} non inry luss

4 Trnsitiv Rution TRD 3/21 (ā ) (ā ) ( ) ( ) ( ) ( ) (ḡ ) ( ) (ḡ ) (ā ē ) ( ) ( ) TRD

5 Hin Tutoloy Elimintion HTE [HulJärvisloBir LPAR 2010] 4/21 (ā ) (ā ) ( ) ( ) ( ) ( ) (ḡ ) ( ) (ā ē ) ( ) ( ) HTE

6 Hin Tutoloy Elimintion HTE [HulJärvisloBir LPAR 2010] 5/21 (ā ) (ā ) ( ) ( ) ( ) ( ) (ḡ ) ( ) ( ) ( ) HTE

7 Sl-Susumin Rsolution SSR [EénBir-SAT 05] 6/21 C l D D l C D l l rsolvnt D susums son ntnt D l ssum ivn CNF ontins ot ntnts...( l)( l) i D is to CNF tn D l n rmov wi in ssn rmovs l rom D l...( l)( )... us in SATLit prprossor now ommon in mny SAT solvrs

8 HTE = Rvrs SSR + Tutoloy Elimintion in litrl ition (HLA) uss SSR in rvrs orr [HulJärvisloBir LPAR 2010] 7/21 C l D D l C D l l ssum ivn CNF ontins rsolvnt n irst ntnt...( l)( ) w n rpl D y D l...( l)( l)... wi in ssn s l to D, rpt HLA until ix-point kp rminin non-tutoloil luss tr rmovin litrls in HTE = ssum C l is inry luss mor nrl vrsions in t ppr rmov luss wit litrl impli y ntion o notr litrl in t lus HTE onlunt n BCP prsrvin moulo quivlnt vril rnmin

9 Hin Litrl Elimintion HLE ttr xplin on inry implition rp [HulJärvisloBir sumitt] 8/21 rmov litrl rom lus wi implis notr litrl in t lus...(ā )( )( )......(ā )( )( )... rlt work or ll uss BCP: symmtri rnin implmnt in MiniSAT ut swit o y ult istilltion [HnSomnzi DAV 07] viviition [PittHmiSis ECAI 08] in tniqu in CryptoMiniSAT HTE/HLE only uss t inry implition rp!

10 Hin Litrl Elimintion HLE [HulJärvisloBir sumitt] 9/21 (ā ) (ā ) ( ) ( ) ( ) ( ) (ḡ ) ( ) ( ) HLE ll ut imply lso implis

11 Hin Litrl Elimintion HLE [HulJärvisloBir sumitt] 10/21 (ā ) (ā ) ( ) ( ) ( ) ( ) (ḡ ) ( ) ( )

12 TRD + HTE + HLE [HulJärvisloBir sumitt] 11/21 (ā ) (ā ) ( ) ( ) ( ) ( ) (ḡ ) ( ) ( )

13 Fil Litrl Elimintion FL 12/21 tully quit ol tniqu... [Frmn PTsis 95] [LBrr 01]... ssum litrl l, BCP, i onlit, unit l rtr ostly to run until ompltion onjtur: t lst qurti on BCP is linr n lso in prti n quit xpnsiv n to o it or ll vrils n rstrt i nw inry lus nrt usul in prti: lit ommon impli litrls or ssumption l n ssumption l vn on BIG (FL2) onjtur to qurti [VnGlr 05]...(ā )( )( )( ā)... unit lus ā susum y runnin on HLA until ompltion

14 Equivlnt Litrl Sustitution ELS [AspvllPlssTrjn 79] [Li 00] [Vl 01] [Brmn 04] [VnGlr 05] 13/21 ompos BIG into stronly onnt omponnts (SCCs) i tr is n l wit l n l in t sm omponnt unstisil otrwis rpl ll litrls y rprsnttiv linr loritm n ppli routinly urin r olltion ut s wit il litrl prprossin my nrt nw inry luss...(ā )( )( )( )......( )...

15 Tim Stmpin [HulJärvisloBir sumitt] 14/21 DFS tr wit isovr n inis tims: [s(l), in(l)] [29,32] [11,16] [17,28] [30,31] [14,15] [12,13] [20,27] [18,19] [2,5] [1,6] [25,26] [21,24] [7,10] [3,4] [22,23] [8,9] tr s prntsis torm: l nstor in DFS tr o k i [s(k), in(k)] [s(l), in(l)] wll known nstor rltionsip ivs nssry onitions or (trnsitiv) implition: i [s(k),in(k)] [s(l),in(l)] tn l k i [s( l), in( l)] [s( k), in( k)] tn l k

16 Uniin: Applyin Tim Stmpin to TRD/HTE/HLE/FL2/... [HulJärvisloBir sumitt] 15/21 tim stmpin in prvious xmpl os not ovr [11,16] = [s(),in()] [s(),in()] = [3,4] [17,28] = [s( ),in( )] [s( ),in( )] = [8,9] in xmpl still ot HTE unin, HLE works too (sin ) ovr vily pns on DFS orr s solution w propos multipl rnomiz DFS rouns/pss so w pproximt qurti prolm (rility) rnomly y linr loritm i BIG is tr on tim stmpin ovrs vrytin

17 Uniin trou Tim Stmpin [HulJärvisloBir sumitt] 16/21 Uniin (ormul F) 1 stmp := 0 2 or litrl l in BIG(F) o 3 s(l) := 0; in(l) := 0 4 prt(l) := l; root(l) := l 5 or r RTS(F) o 6 stmp := Stmp(r, stmp) 7 or litrl l in BIG(F) o 8 i s(l) = 0 tn 9 stmp := Stmp(l, stmp) 10 rturn Simpliy(F) Stmp (litrl l, intr stmp) 1 stmp := stmp s(l) := stmp 3 or ( l l ) F 2 o 4 i s(l ) = 0 tn 5 prt(l ) := l 6 root(l ) := root(l) 7 stmp := Stmp(l,stmp) 8 stmp := stmp in(l) := stmp 10 rturn stmp Simpliy (ormul F) 1 or C F 2 F := F \ {C} 3 i UHTE(C) tn ontinu 4 F := F {UHLE(C)} 5 rturn F

18 Uniin HTE [HulJärvisloBir sumitt] 17/21 UHTE (lus C) 1 l pos := irst lmnt in S + (C) 2 l n := irst lmnt in S (C) 3 wil tru 4 i s(l n ) > s(l pos ) tn 5 i l pos is lst lmnt in S + (C) tn rturn ls 6 l pos := nxt lmnt in S + (C) 7 ls i in(l n ) < in(l pos ) or ( C = 2 n (l pos = l n or prt(l pos ) = l n )) tn 8 i l n is lst lmnt in S (C) tn rturn ls 9 l n := nxt lmnt in S (C) 10 ls rturn tru S + (C) S (C) squn o litrls in C orr y s() squn o ntions o litrls in C orr y s() O( C lo C )

19 Uniin HLE [HulJärvisloBir sumitt] 18/21 UHLE (lus C) 1 inis := inis tim o irst lmnt in S + rv(c) 2 or l S + rv(c) strtin t son lmnt 3 i in(l) > inis tn C := C \ {l} 4 ls inis := in(l) 5 inis := inis tim o irst lmnt in S (C) 6 or l S (C) strtin t son lmnt 7 i in( l) < inis tn C := C \ {l} 8 ls inis := in( l) 9 rturn C S + rv(c) rvrs o S + (C) O( C lo C )

20 Avn Tim Stmpin [HulJärvisloBir sumitt] 19/21 Stmp (litrl l, intr stmp) 1 BSC stmp := stmp BSC s(l) := stmp; os(l) := stmp 3 ELS l := tru // l rprsnts SCC 4 ELS S.pus(l) // pus l on SCC stk 5 BSC or ( l l ) F 2 6 TRD i s(l) < os(l ) tn F := F \ {( l l )}; ontinu 7 FLE i s(root(l)) os( l ) tn 8 FLE l il := l 9 FLE 10 FLE wil s(l il ) > os( l ) o l il := prt(l il ) F := F {( l il )} 11 FLE i s( l ) 0 n in( l ) = 0 tn ontinu 12 BSC i s(l ) = 0 tn 13 BSC prt(l ) := l 14 BSC root(l ) := root(l) 15 BSC stmp := Stmp(l,stmp) 16 ELS i in(l ) = 0 n s(l ) < s(l) tn 17 ELS s(l) := s(l ); l := ls // l is quivlnt to l 18 OBS os(l ) := stmp // st lst osrv tim ttriut 19 ELS i l = tru tn // i l rprsnts SCC 20 BSC stmp := stmp ELS o 22 ELS l := S.pop() // t quivlnt litrl 23 ELS s(l ) := s(l) // ssin qul isovr tim 24 BSC in(l ) := stmp // ssin qul inis tim 25 ELS wil l l 26 BSC rturn stmp

21 Implmnttion [HulJärvisloBir sumitt] 20/21 implmnt s on inprossin ps in our SAT solvr Linlin si vril limintion, istilltion, lok lus limintion, proin,... ursts o rnomiz DFS rouns n swpin ovr t wol ormul st nou to pplil to lr lrn luss s wll uniin is prtiullry tiv or lrn luss si UHTE n UHLE w lso v ypr inry rsolution UHBR not usul in prti

22 Linlin 571 on SAT 09 Comptition Applition Bnmrks [HulJärvisloBir sumitt] 21/21 oniurtion sol st uns un simp lim v.stmp (no ur) % 33.0% 16.1% v.stmp (w/ur) % 32.8% 15.8% si stmp (no ur) % 32.3% 15.8% si stmp (w/ur) % 32.8% 15.8% no uniin % 28.6% 17.6% oniurtion t stmp runnt l runnt units stmp v.stmp (no ur) 22 64% 59% % % v.stmp (w/ur) 26 67% 70% % % si stmp (no ur) 6 0% 52% % 273 0% si stmp (w/ur) 7 0% 66% % 308 0% no uniin 0 0% 0% 0 0.0% 0 0% similr rsults or rt n SAT 10 R instns

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