Nenofex: Expanding NNF for QBF Solving

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1 Institute for Forml Models nd Verifiction (FMV) Johnnes Kepler University, Linz, Austri

2 Nenofex Negtion Norml Form Expnsion Solver for Quntified Boolen Formule (QBF) propositionl formul + quntified vribles generlizes SAT Fetures tree-bsed NNF representtion NNF expnsion: less size increse for -expnsion thn on CNF tight, estimted expnsion costs for greedy scheduling NNF redundncy removl: techniques from circuit optimiztion Results on QBFEVAL 07 benchmrk set less frequently out-of-memory thn resolution-bsed Quntor [Biere-SAT04] importnt, but expensive redundncy removl on NNF strong performnce on instnces from dder fmiliy (QBFLIB, Ayri)

3 Introduction QBF PSPACE-complete decision problem exponentilly more succint thn SAT CNF + quntifier prefix (prenex norml form): S S 2... S n S n φ {z } {z} quntifier prefix CNF S i : linerly ordered scopes two notions: sets of quntified vribles nd quntifier scopes (s usul) quntifier scope of x S i in prefix rnges over whole formul φ Solving QBF by vrible elimintion: from S n to S expnsion, Q-resolution or skolemiztion Our focus: solve by expnsion Quntor: CNF-bsed, -expnsion for S n, Q-resolution for S n Nenofex similr to Quntor but NNF-bsed, expnsion only

4 Motivtion (/2): NNF-expnsion vs. CNF-resolution Given: CNF φ R X 0 X with only -vribles R X 0 X 3 3 sets X 0, X, R: cluses with negtive, positive or no literl of vrible x Resolve x: φ res R V c (X 0 resx ) c R X 0 res X generlly: dd X 0 X resolvents worst cse: qudrtic size increse 3 3 = 9 Expnd x: φ exp R ((X 0 X )[x/0] (X 0 X )[x/]) R X 0 X (( ) ( ) 3 3 dd copy of φ by, fctor out R nd ssign x worst cse: liner size increse

5 Motivtion (2/2) = [x/0] [x/] Generl -expnsion on NNF φ exp grows linerly in the size of the subformul of x NNF llows compct representtion for expnding -vribles size increse in -expnsions: NNF nd CNF equivlent

6 Core Algorithm INIT UNITS UNATES GF RR EXP SAT True/Flse Elimintion of unit nd pure literls (untes) Redundncy Removl on smll subformul only, cutoff criterion Expnsion: S... S n S n φ from S n to S, expnd chepest vrible in S n or S n by scores score(x): tight upper bound on size increse of NNF when expnding x prtil score recomputtion SAT solving only ( )-vribles left generte CNF by Tseitin trnsformtion PicoSAT bckend

7 Formul Representtion (/2) Negtion Norml Form: only nd, pplied to literls only NNF-trees internl nodes: opertors nd leves: literl occurrence nodes (no shring) level(node) := distnce to root b 2 b (c d) Structurl Restrictions: flt nd compct NNF-trees (prticulrly for CNFs) number of children n 2: opertors denote n-ry boolen functions n = fter deletion: merge nodes c d 2 c = c b

8 Formul Representtion (2/2) lternting types: type(prent) type(child) pply ssocitivity of nd prerequisite: n-ry opertors 2 c = c b b one-level simplifiction: for vr. x, {, }, simplify x x, x x remove trivil redundncy bottom-up recursive effects 2 c = c

9 Expnsion (/2) Locl Expnsion for NNF: copy only relevnt prts Def.: ers(x) := expnsion-relevnt subformul of vrible x smllest subformul which contins ll occurrences of x finding ers(x) by scope reduction [AyriBsin02] in prenex formule: Qx(φ ψ) Qx(φ) ψ x Vrs(ψ), Q {, }, {, } In NNF-trees: for ers(x), find expnsion-relevnt subtree to be copied correspondence: smllest subformule nd subtrees Expnsion-relevnt LCAs of Vribles: scope reduction in NNF-tree LCA: lest common ncestor of set of nodes bottom-up pproch for computing ers(x) strting from literls of x expnsion-relevnt LCA of x denotes expnsion-relevnt subtree

10 Expnsion (2/2) Expnsion-relevnt LCAs of Vribles Def.: expnsion-relevnt LCA of x := node lc(x) nd set LCA-children set LCA-children: (proper) subset of children of node lc(x) LCA-child c: subtree of c contins t lest one occurrence of vrible x Expnsion: S... S n S n φ, x S n, type(s n) {, } replce ers(x) by (ers(x)[x/0] ers(x)[x/]), {, } Expnsion: x S n, type(s n ) = duplicte depending -vribles D x from S n 2 3 b D (0) x := {y S n y hs literls in ers(x)} D (k+) x := {z S n z hs literls in ers(y ) for some y D k x }, k 0 x 4 5 D x := [ k D k x D x: extended from CNF [BubeckKBüning-SAT07] to NNF universl expnsion-relevnt subformul urs(x) contins ll literls of x nd of depending vribles in D x x y x z

11 NNF Redundncy Removl Globl Flow (GF): globl nlysis of logicl flow of vlues implictions: trnsform circuit to reduce size x = 0 y = 0 : y x y x = y = : y x y Redundncy Removl (RR) by Automtic Test Pttern Genertion (ATPG) ATPG: structurl testing of circuits (NP-complete) ssume fult f t single signl s in circuit C: stuck-t-{0,} fult model find input v = (pi 0,..., pi n ) such tht C(v) C f (v) uniquely cused by f no such v: f is not testble, does not ffect C, cn remove HW t s GF+RR implementtion: incomplete, polynomil-time = = 2 3 b c b c b c

12 Experimentl Results (/2): QBFEVAL 07 full benchmrk set (36 instnces) from QBFEVAL 07 Pentium IV 3.0 GHz, Ubuntu Linux, limits 900 seconds nd.5 GB Quntor s reference: CNF-bsed, similr strtegy three versions of Nenofex: GF, RR enbled/disbled Nenofex Quntor GF, RR no GF, RR no GF, no RR Solved OOT OOM MEM-.0e6.5e6.7e6.23e6 MEM Quntor only Both Nenofex only Sum Solved OOT OOM Results less spce-outs thn CNF-bsed Quntor node implementtion in Nenofex not optimized for memory redundncy removl expensive but crucil for performnce GF, RR cuse more time outs 9 uniquely solved instnces

13 Experimentl Results (2/2): Ayri s dder benchmrks equivlence checking of n-bit ripple-crry dders [AyriBsin02] structured QBF-encodings of mondic second order formule hrd instnces in previous QBF evlutions optimiztions enbled optimiztions disbled Nme SAT-Vrs SAT-Cluses Time (Exp.) Mem SAT-Vrs SAT-Cluses Time (Exp.) Mem dder-2-unst (0.07) < <0.0 < dder-4-unst (0.36) (0.04) < dder-6-unst (.22) (0.0) 4.2 dder-8-unst (2.60) (0.24) 6.6 dder-0-unst (4.67) (0.50) 0.2 dder-2-unst (7.47) (0.89) 5. dder-4-unst (.0) (.54) 2.3 dder-6-unst (5.35) (2.47) 29.2 dder-2-st (0.04) < 76 8 <0.0 < dder-4-st (0.38) (0.04) < dder-6-st (.42) (0.3) 3.3 dder-8-st (3.23) (0.39) 4.7 dder-0-st (5.86) (.24) 7.7 dder-2-st (0.24) (3.34) 4.6 dder-4-st (23.45) (9.2) 23.5 dder-6-st (42.94) (4.50) 65.6 Results SAT-solving time domintes expnsion time in lrge instnces no optimiztions: less expnsion time but lrger CNFs Quntor, skizzo, squolem, ebddres: comprble on dder-{2,4}-{st,unst}, skizzo slower on dder-{2,...,0}-st bort on dder-{2,4,6}-{st,unst}, dder-{6,..,6}-unst

14 Summry: Nenofex Expnsion-bsed QBF solver for NNF -expnsion: liner vs. qudrtic size increse on NNF nd CNF NNF-trees: flt formul representtion Locl expnsion: scope reduction by quntifier rules expnsion-relevnt subformule, subtrees nd LCAs Vribles scores for greedy scheduling tight upper bound on ctul size increse of NNF-tree Redundncy removl: tret NNF-tree s circuit GF: deriving implictions for circuit trnsformtions ATPG-bsed RR: untestble fults correspond to redundnt HW implementtion: incomplete, on smll subtree only Experiments less spce-outs thn CNF-bsed solver Quntor GF+RR crucil for performnce, lthough NNF more compct thn CNF dder-benchmrks Future work optimize for run time nd memory incrementl mintinnce of scores

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