Enrico giunchiglia, armando tacchella theory and

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Lecture Notes in Computer Science 2919 Edited by G Goos, J Hartmanis, and J van Leeuwen

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Enrico Giunchiglia Armando Tacchella (Eds.)

Theory and Applications

of Satisfiability Testing

6th International Conference, SAT 2003

Santa Margherita Ligure, Italy, May 5-8, 2003 Selected Revised Papers

Springer

eBook ISBN: 3-540-24605-3

Print ISBN: 3-540-20851-8

©200 5 Springer Science + Business Media, Inc.

Print ©2004 Springer-Verlag Berlin Heidelberg

All rights reserved

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

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and the Springer Global Website Online at: http://www.springeronline.com

Dordrecht

This book is devoted to the 6th International Conference on Theory and

Ap-plications of Satisfiability Testing (SAT 2003) held in Santa Margherita Ligure

(Genoa, Italy), during May 5–8, 2003 SAT 2003 followed the Workshops on isfiability held in Siena (1996), Paderborn (1998), and Renesse (2000), and theWorkshop on Theory and Applications of Satisfiability Testing held in Boston(2001) and in Cincinnati (2002) As in the last edition, the SAT event hosted aSAT solvers competition, and, starting from the 2003 edition, also a QuantifiedBoolean Formulas (QBFs) solvers comparative evaluation

Sat-There were 67 submissions of high quality, authored by researchers from allover the world All the submissions were thoroughly evaluated, and as a result

42 were selected for oral presentations, and 16 for a poster presentation Thepresentations covered the whole spectrum of research in propositional and QBFsatisfiability testing, including proof systems, search techniques, probabilisticanalysis of algorithms and their properties, problem encodings, industrial appli-cations, specific tools, case studies and empirical results Further, the program

was enriched by three invited talks, given by Riccardo Zecchina (on “Survey

Propagation: from Analytic Results on Random to a Message-Passing gorithm for Satisfiability”), Toby Walsh (on “Challenges in SAT (and QBF)”)

Al-and Wolfgang Kunz (on “ATPG Versus SAT: Comparing Two Paradigms for

Boolean Reasoning”) SAT 2003 thus provided a unique forum for the

presenta-tion and discussion of research related to the theory and applicapresenta-tions of sitional and QBF satisfiability testing

propo-The book includes 38 contributions propo-The first 33 are revised versions of some

of the articles that were presented at the conference The last 5 articles presentthe results of the SAT competition and of the QBF evaluation, solvers that wonthe SAT competition, and results on survey and belief propagation All 38 paperswere thoroughly reviewed

We would like to thank the many people who contributed to the SAT 2003organization (listed in the following pages), the SAT 2003 participants for thelively discussions, and the sponsors

Armando Tacchella

John Franco, University of Cincinnati

Enrico Giunchiglia, Università di Genova

Henry Kautz, University of Washington

Hans Kleine Büning, Universität Paderborn

Hans van Maaren, University of Delft

Bart Selman, Cornell University

Ewald Speckenmayer, Universität Köln

SAT Competition Organizers

Daniel Le Berre, CRIL, Université d’Artois

Laurent Simon, LRI Laboratory, Université Paris-Sud

QBF Comparative Evaluation Organizers

Daniel Le Berre, CRIL, Université d’Artois

Laurent Simon, LRI Laboratory, Université Paris-Sud

Armando Tacchella, DIST, Università di Genova

Local Organization

Roberta Ferrara, Università di Genova

Marco Maratea, DIST, Università di Genova

Massimo Narizzano, DIST, Università di Genova

Adriano Ronzitti, DIST, Università di Genova

Armando Tacchella, DIST, Università di Genova (Chair)

SAT 2003 Organization VII

Program Committee

Dimitris Achlioptas, Microsoft Research

Fadi Aloul, University of Michigan

Fahiem Bacchus, University of Toronto

Armin Biere, ETH Zurich

Nadia Creignou, Université de la Méditerranée, Marseille

Olivier Dubois, Université Paris 6

Uwe Egly, Technische Universität Wien

John Franco, University of Cincinnati

Ian Gent, St Andrews University

Enrico Giunchiglia, DIST, Università di Genova

Carla Gomez, Cornell University

Edward A Hirsch, Steklov Institute of Mathematics at St Petersburg

Holger Hoos, University of British Columbia

Henry Kautz, University of Washington

Hans Kleine Büning, Universität Paderborn

Oliver Kullmann, University of Wales, Swansea

Daniel Le Berre, CRIL, Université d’Artois

Joo Marques-Silva, Instituto Superior Técnico, Univ Técnica de LisboaHans van Maaren, University of Delft

Remi Monasson, Laboratoire de Physique Théorique de l’ENS

Daniele Pretolani, Università di Camerino

Paul W Purdom, Indiana University

Jussi Rintanen, Freiburg University

Bart Selman, Cornell University

Malik Sharad, Princeton University

Laurent Simon, LRI Laboratory, Université Paris-Sud

Ewald Speckenmeyer, Universität Köln

Armando Tacchella, DIST, Università di Genova

Allen Van Gelder, UC Santa Cruz

Miroslav N Velev, Carnegie Mellon University

Toby Walsh, University of York

VIII SAT 2003 Organization

Sponsoring Institutions

CoLogNet, Network of Excellence in Computational Logic

DIST, Università di Genova

IISI, Intelligent Information Systems Institute at Cornell UniversityMicrosoft Research

MIUR, Ministero dell’Istruzione, dell’Università e della Ricerca

Table of Contents

Michael R Dransfield, Victor W Marek,

An Algorithm for SAT Above the Threshold

Hubie Chen

14

Ian Gent, Enrico Giunchiglia, Massimo Narizzano, Andrew Rowley,

Armando Tacchella

How Good Can a Resolution Based SAT-solver Be?

Eugene Goldberg, Yakov Novikov

37

A Local Search SAT Solver Using an Effective Switching Strategy and

an Efficient Unit Propagation

Xiao Yu Li, Matthias F Stallmann, Franc Brglez

53

Youichi Hanatani, Takashi Horiyama, Kazuo Iwama

Edmund Clarke, Muralidhar Talupur, Helmut Veith, Dong Wang

On Boolean Models for Quantified Boolean Horn Formulas

Hans Kleine Büning, K Subramani, Xishun Zhao

93

Local Search on SAT-encoded Colouring Problems

Steven Prestwich

105

A Study of Pure Random Walk on Random Satisfiability Problems

with “Physical” Methods

Guilhem Semerjian, Rémi Monasson

120

Hidden Threshold Phenomena for Fixed-Density SAT-formulae

Hans van Maaren, Linda van Norden

135

Improving a Probabilistic 3-SAT Algorithm by Dynamic Search and

Independent Clause Pairs

Sven Baumer, Rainer Schuler

150

Width-Based Algorithms for SAT and CIRCUIT-SAT

Elizabeth Broering, Satyanarayana V Lokam

162

Linear Time Algorithms for Some Not-All-Equal Satisfiability Problems

Stefan Porschen, Bert Randerath, Ewald Speckenmeyer

172

Using Problem Structure for Efficient Clause Learning

Ashish Sabharwal, Paul Beame, Henry Kautz

242

Abstraction-Driven SAT-based Analysis of Security Protocols

Alessandro Armando, Luca Compagna

257

A Case for Efficient Solution Enumeration

Sarfraz Khurshid, Darko Marinov, Ilya Shlyakhter, Daniel Jackson

Local Consistencies in SAT

Christian Bessière, Emmanuel Hebrard, Toby Walsh

299

Guiding SAT Diagnosis with Tree Decompositions

Per Bjesse, James Kukula, Robert Damiano, Ted Stanion,

Effective Preprocessing with Hyper-Resolution and Equality Reduction

Fahiem Bacchus, Jonathan Winter

341

Read-Once Unit Resolution

Hans Kleine Büning, Xishun Zhao

356

The Interaction Between Inference and Branching Heuristics

Lyndon Drake, Alan Frisch

370

Hypergraph Reductions and Satisfiability Problems

Daniele Pretolani

383

Table of Contents XI

John Franco, Michal Kouril, John Schlipf, Jeffrey Ward, Sean Weaver,

Michael Dransfield, W Mark Vanfleet

Computing Vertex Eccentricity in Exponentially Large Graphs:

QBF Formulation and Solution

Maher Mneimneh, Karem Sakallah

411

The Combinatorics of Conflicts between Clauses

Oliver Kullmann

426

Conflict-Based Selection of Branching Rules

Marc Herbstritt, Bernd Becker

441

The Essentials of the SAT 2003 Competition

Daniel Le Berre, Laurent Simon

452

Challenges in the QBF Arena: the SAT’03 Evaluation of QBF Solvers

Daniel Le Berre, Laurent Simon, Armando Tacchella

468

kcnfs: an Efficient Solver for Random Formulae

Gilles Dequen, Olivier Dubois

486

An Extensible SAT-solver

Niklas Eén, Niklas Sörensson

502

Survey and Belief Propagation on Random K-SAT

Alfredo Braunstein, Riccardo Zecchina

519

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Satisfiability and Computing van der Waerden

Department of Computer Science, University of Kentucky, Lexington,

KY 40506-0046, USA

Abstract In this paper we bring together the areas of combinatorics

and propositional satisfiability Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds The area of Ramsey theory is especially rich in such results Using the problem of computing van der Waerden numbers as an exam- ple, we show that these problems can be represented by parameterized propositional theories in such a way that decisions concerning their satis- fiability determine the numbers (function) in question We show that by using general-purpose complete and local-search techniques for testing propositional satisfiability, this approach becomes effective — competi- tive with specialized approaches By following it, we were able to obtain several new results pertaining to the problem of computing van der Waer- den numbers We also note that due to their properties, especially their structural simplicity and computational hardness, propositional theories that arise in this research can be of use in development, testing and benchmarking of SAT solvers.

1 Introduction

In this paper we discuss how the areas of propositional satisfiability and natorics can help advance each other On one hand, we show that recent dramaticimprovements in the efficiency of SAT solvers and their extensions make it pos-sible to obtain new results in combinatorics simply by encoding problems aspropositional theories, and then computing their models (or deciding that noneexist) using off-the-shelf general-purpose SAT solvers On the other hand, weargue that combinatorics is a rich source of structured, parameterized families

combi-of hard propositional theories, and can provide useful sets combi-of benchmarks fordeveloping and testing new generations of SAT solvers

In our paper we focus on the problem of computing van der Waerden bers The celebrated van der Waerden theorem [20] asserts that for every pos-itive integers and there is a positive integer such that every partition

num-of into blocks (parts) has at least one block with an arithmeticprogression of length The problem is to find the least such number This

E Giunchiglia and A Tacchella (Eds.): SAT 2003, LNCS 2919, pp 1–13, 2004.

2 Michael R Dransfield, Victor W Marek, and

are known only for five pairs For other combinations of and there aresome general lower and upper bounds but they are very coarse and do not giveany good idea about the actual value of In the paper we show thatSAT solvers such as POSIT [6], and SATO [21], as well as recently developed

local-search solver walkaspps [13], designed to compute models for propositional

theories extended by cardinality atoms [4], can improve lower bounds for vander Waerden numbers for several combinations of parameters and

Theories that arise in these investigations are determined by the two rameters and Therefore, they show a substantial degree of structure andsimilarity Moreover, as and grow, these theories quickly become very hard.This hardness is only to some degree an effect of the growing size of the theories.For the most part, it is the result of the inherent difficulty of the combinatorialproblem in question All this suggests that theories resulting from hard combi-natorial problems defined in terms of tuples of integers may serve as benchmarktheories in experiments with SAT solvers

pa-There are other results similar in spirit to the van der Waerden theorem.The Schur theorem states that for every positive integer there is an integersuch that every partition of into blocks contains a block that

is not sum-free Similarly, the Ramsey theorem (which gave name to this wholearea in combinatorics) [16] concerns the existence of monochromatic cliques inedge-colored graphs, and the Hales-Jewett theorem [11] concerns the existence ofmonochromatic lines in colored cubes Each of these results gives rise to a partic-ular function defined on pairs or triples of integers and determining the values ofthese functions is a major challenge for combinatorialists In all cases, only fewexact values are known and lower and upper estimates are very far apart Many

of these results were obtained by means of specialized search algorithms highlydepending on the combinatorial properties of the problem Our paper shows thatgeneric SAT solvers are maturing to the point where they are competitive andsometimes more effective than existing advanced specialized approaches

2 Van der Waerden Numbers

In the paper we use the following terminology By we denote the set of

X is a collection of nonempty and mutually disjoint subsets of X such that

Elements of are commonly called blocks.

Informally, the van der Waerden theorem [20] states that if a sufficientlylong initial segment of positive integers is partitioned into a few blocks, thenone of these blocks has to contain an arithmetic progression of a desired length.Formally, the theorem is usually stated as follows

Theorem 1 (van der Waerden theorem) For every there is

such that for every partition of there is

such that block contains an arithmetic progression of length at least

Satisfiability and Computing van der Waerden Numbers 3

We define the van der Waerden number to be the least number for

which the assertion of Theorem 1 holds Theorem 1 states that van der Waerden

numbers are well defined

case when

Little is known about the numbers In particular, no closed formula

has been identified so far and only five exact values are known They are shown

in Table 1 [1,10]

Since we know few exact values for van der Waerden numbers, it is important

to establish good estimates One can show that the Hales-Jewett theorem entails

the van der Waerden theorem, and some upper bounds for the numbers

can be derived from the Shelah’s proof of the former [18] Recently, Gowers

[9] presented stronger upper bounds, which he derived from his proof of the

Szemerédi theorem [19] on arithmetic progressions

In our work, we focus on lower bounds Several general results are known For

instance, Erdös and Rado [5] provided a non-constructive proof for the inequality

For some special values of parameters and Berlekamp obtained better bounds

by using properties of finite fields [2] These bounds are still rather weak His

strongest result concerns the case when and is a prime number

Namely, he proved that when is a prime number,

In particular, W(2,6) > 160 and W(2,8) > 896.

Our goal in this paper is to employ propositional satisfiability solvers to find

lower bounds for several small van der Waerden numbers The bounds we find

significantly improve on the ones implied by the results of Erdös and Rado, and

Berlekamp

We proceed as follows For each triple of positive integers we define

least in principle) to determine the satisfiability of and, consequently,

4 Michael R Dransfield, Victor W Marek, and

attention to We also show that more concise encodings are possible,leading ultimately to better bounds, if we use an extension of propositional logic

by cardinality atoms and apply to them solvers capable of handling such atoms

directly

To describe we will use a standard first-order language, without

function symbols, but containing a predicate symbol in_block and constants

specify this theory as finite (and independent of data) collections of

proposi-tional schemata, that is, open clauses in the language of first-order logic without

function symbols Given a set of appropriate constants (to denote integers andblocks) such theory, after grounding, coincides with In fact, we havedefined an appropriate syntax that allows us to specify both data and schemata

and implemented a grounding program psgrnd [4] that generates their equivalent

ground (propositional) representation This grounder accepts arithmetic sions as well as simple regular expressions, and evaluates and eliminates themaccording to their standard interpretation Such approach significantly simplifiesthe task of developing propositional theories that encode problems, as well asthe use of SAT solvers [4]

expres-Propositional interpretations of the theory can be identified with

determines an interpretation in which

all atoms in M are true and all other atoms are false In the paper we always

assume that interpretations are represented as sets

It is easy to see that clauses (vdW1) ensure that if M is a model of

(that is, is an interpretation satisfying all clauses of then for every

M contains at most one atom of the form Clauses (vdW2)ensure that for every there is at least one such that

M In other words, clauses (vdW1) and (vdW2) together ensure that if M is a

The last group of constraints, clauses (vdW3), guarantee that elements fromforming an arithmetic progression of length do not all belong to the sameblock All these observations imply the following result

Satisfiability and Computing van der Waerden Numbers 5

Proposition 1 There is a one-to-one correspondence between models of the

formula and partitions of into blocks so that no block contains

an arithmetic progression of length Specifically, an interpretation M is a model

of into blocks such that no block contains an arithmetic progression of length

Proposition 1 has the following direct corollary

Corollary 1 For every positive integers and with and

if and only if the formula is satisfiable.

It is evident that if has the property that is unsatisfiable thenfor every is also unsatisfiable Thus, Corollary 1 suggests thefollowing algorithm that, given and computes the van der Waerden number

is satisfiable If so, we continue If not, we return and terminate thealgorithm By the van der Waerden theorem, this algorithm terminates

It is also clear that there are simple symmetries involved in the van der

Waerden problem If a set M of atoms of the form is a model ofthe theory and is a permutation of then the corresponding set

so is the set of atoms

Following the approach outlined above, adding clauses to break these metries, and applying POSIT [6] and SATO [21] as a SAT solvers we were able to

sym-establish that W(4,3) = 76 and compute a “library” of counterexamples

(parti-tions with no block containing arithmetic progressions of a specified length) for

We were also able to find several lower bounds on van der Waerdennumbers for larger values of and

However, a major limitation of our first approach is that the size of ries grows quickly and makes complete SAT solvers ineffective Let

theo-us estimate the size of the theory The total size of clauses (vdW1)(measured as the number of atom occurrences) is The size of clauses(vdW2) is Finally, the size of clauses (vdW3) is (indeed, thereare arithmetic progressions of length in 3 Thus, the total size of

To overcome this obstacle, we used a two-pronged approach First, as a eling language we used PS+ logic [4], which is an extension of propositionallogic by cardinality atoms Cardinality atoms support concise representations ofconstraints of the form “at least and at most elements in a set are true”and result in theories of smaller size Second, we used a local-search algorithm,

mod-walkaspps, for finding models of theories in logic PS+ that we have designed and

Goldstein [8] provided a precise formula When and

in

3

6 Michael R Dransfield, Victor W Marek, and

implemented recently [13] Using encodings as theories in logic PS+ and

walka-spps as a solver, we were able to obtain substantially stronger lower bounds for

van der Waerden numbers than those know to date

We will now describe this alternative approach For a detailed treatment

of the PS+ logic we refer the reader to [4] In this paper, we will only reviewmost basic ideas underlying the logic PS+ (in its propositional form) By a

propositional cardinality atom ( for short), we mean any expression of the

and are non-negative integers and are propositional atoms from

At The notion of a clause generalizes in an obvious way to the language with

cardinality atoms Namely, a is an expression of the form

where all and are (propositional) atoms or cardinality atoms

Let At be a set of atoms We say that M satisfies a cardinality atom

if

satisfies a c-clause C of the form (1) if M satisfies at least one atom or does not

the quantifier “There exists exactly one ” - commonly used in mathematicalstatements

It is now clear that all clauses (vdW1) and (vdW2) from can berepresented in a more concise way by the following collection of c-clauses:

for everyIndeed, c-clauses enforce that their models, for every containexactly one atom of the form — precisely the same effect as that

of clauses (vdW1) and (vdW2) Let be a PS+ theory consisting ofclauses and (vdW3) It follows that Proposition 1 and Corollary 1

Consequently, any algorithm for finding models of PS+ theories can be used tocompute van der Waerden numbers (or, at least, some bounds for them) in theway we described above

The adoption of cardinality atoms leads to a more concise representation ofthe problem While, as we discussed above, the size of all clauses (vdW1) and

In our experiments, for various lower bound results, we used the local-search

algorithm walkaspps [13] This algorithm is based on the same ideas as

walk-sat [17] A major difference is that due to the presence of c-atoms in c-clauses walkaspps uses different formulas to calculate the breakcount and proposes sev-

eral other heuristics designed specifically to handle c-atoms

Satisfiability and Computing van der Waerden Numbers 7

3 Results

Our goal is to establish lower bounds for small van der Waerden numbers byexploiting propositional satisfiability solvers Here is a summary of our results.Using complete SAT solvers POSIT and SATO and the encoding of theproblem as we found a “library” of all (up to obvious symmetries)counterexamples to the fact that W(4, 3) > 75 There are 30 of them We listtwo of them in the appendix A complete list can be found at http://www.cs.uky.edu/ai/vdw/ Since there are 48 symmetries, of the types discussedabove, the full library of counterexamples consists of 1440 partitions

We found that the formula is unsatisfiable Hence, we found that

a “generic” SAT solver is capable of finding that W(4,3) = 76.

We established several new lower bounds for the numbers They

are presented in Table 3 Partitions demonstrating that W(2,8) > 1295,

W(3,5) > 650, and W(4,4) > 408 are included in the appendix

Counterex-ample partitions for all other inequalities are available at http://www.cs.uky.edu/ai/vdw/ We note that our bounds for W(2,6) and W(2,8) aremuch stronger than those implied by the results of Berlekamp [2], which westated earlier

8 Michael R Dransfield, Victor W Marek, and

to van der Waerden numbers can be naturally cast as questions on the existence

of satisfying valuations for some propositional CNF-formulas

Computing combinatorial objects such as van der Waerden numbers is hard.They are structured but as we pointed out few values are known, and newresults are hard to obtain Thus, the computation of those numbers can serve

as a benchmark (‘can we find the configuration such that ’) for complete andlocal-search methods, and as a challenge (‘can we show that a configuration suchthat ’ does not exist) for complete SAT solvers Moreover, with powerful SATsolvers it is likely that the bounds obtained by computation of counterexamplesare “sharp” in the sense that when a configuration is not found then none exist

For instance it is likely that W(5, 3) is close to 126 (possibly, it is 126), because

125 was the last integer where we were able to find a counterexample despitesignificant computational effort This claim is further supported by the factthat in all examples where exact values are known, our local-search algorithmwas able to find counterexample partitions for the last possible value of Thelower-bounds results of this sort may constitute an important clue for researcherslooking for nonexistence arguments and, ultimately, for the closed form of vander Waerden numbers

A major impetus for the recent progress of SAT solvers comes from cations in computer engineering In fact, several leading SAT solvers such as

appli-zCHAFF [15] and berkmin [7] have been developed with the express goal of

aid-ing engineers in correctly designaid-ing and implementaid-ing digital circuits Yet, thefact that these solvers are able to deal with hard optimization problems in onearea (hardware design and verification) carries the promise that they will be ofuse in another area — combinatorial optimization Our results indicate that it

is likely to be the case

The current capabilities of SAT solvers has allowed us to handle large stances of these problems Better heuristics and other techniques for pruningthe search space will undoubtedly further expand the scope of applicability ofgeneric SAT solvers to problems that, until recently, could only be solved usingspecialized software

in-Satisfiability and Computing van der Waerden Numbers 9

Acknowledgments

The authors thank Lengning Liu for developing software facilitating our mental work This research has been supported by the Center for CommunicationResearch, La Jolla During the research reported in this paper the second andthird authors have been partially supported by an NSF grant IIS-0097278

experi-References

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satis-fiability, IEEE Transactions on Computers, 48:506–521, 1999.

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10 Michael R Dransfield, Victor W Marek, and

Appendix

Using a complete SAT solver we computed the library of all partitions (up to

isomorphism) of [75] showing that 75 < W(4,3) Two of these 30 partitions are

Satisfiability and Computing van der Waerden Numbers 11

Block 2:

Next, we exhibit a partition of [650] into three blocks demonstrating that W(3, 5)

>650

Block 1:

12 Michael R Dransfield, Victor W Marek, and

Satisfiability and Computing van der Waerden Numbers 13

Block 4:

Configurations showing the validity of other lower bounds listed in Table 3 areavailable at http://www.cs.uky.edu/ai/vdw/.

An Algorithm for SAT Above the Threshold

Hubie Chen

Department of Computer Science, Cornell University, Ithaca, NY 14853, USA

hubes@cs.Cornell.edu

Abstract We study algorithms for finding satisfying assignments of

randomly generated 3-SAT formula In particular, we consider butions of highly constrained formulas (that is, “above the threshold” formulas) restricted to satisfiable instances We obtain positive algorith- mic results, showing that such formulas can be solved in low exponential time.

distri-1 Introduction

Randomly generated instances of the boolean satisfiability problem have beenused to study typical-case problem complexity Consider the model for randomlygenerating a 3-SAT formula over a variable set of size which includes each ofthe clauses independently with probability For very “low” such a for-mula tends not to have many clauses, and is usually satisfiable; for very “high”such a formula tends to have many clauses, and is likely to be unsatisfiable A

fundamental conjecture, the satisfiability threshold conjecture, states that there

is a transition point at which such random formulas abruptly change from beingalmost always satisfiable to almost always unsatisfiable Precisely, this conjec-

ture posits the existence of a constant c such that a random formula is satisfiable

almost always if that is, the expected number of clauses is less

than cn; and is unsatisfiable almost always if (By almost always,

we mean with probability tending to one as approaches infinity.) There isample empirical evidence for this conjecture [23], and theoretical work has pro-vided rigorous lower and upper bounds on the value of the purported constant[16,6,15,1]

There are a number of positive algorithmic results for finding satisfying

as-signments of randomly generated formulas below the threshold – namely, results giving polynomial time algorithms which find satisfying assignments almost al-

ways on below-threshold formula distributions In contrast, satisfiability

algo-rithms for formulas above the threshold have seen relatively little attention.

The main result of this paper is the analysis of a novel random walk algorithmfor finding satisfying assignments of 3-SAT formulas from above the threshold

Of course, deciding satisfiability of such formulas is rather uninteresting on thedescribed distribution: the trivial algorithm which outputs “unsatisfiable” is cor-rect almost always Hence, we restrict the described distribution of formulas tosatisfiable formulas, and seek algorithms which output satisfying assignments

E Giunchiglia and A Tacchella (Eds.): SAT 2003, LNCS 2919, pp 14–24, 2004.

An Algorithm for SAT Above the Threshold 15

almost always In essence, our result shows that almost all satisfiable random

3-SAT formulas with enough clauses have a tractable structure that can be

ex-ploited by an algorithm to obtain a satisfying assignment This paper thus forms

a contribution to the long-term research program aiming to fully understand thetypical complexity of random satisfiability formulas

1.1 Three Formula Distributions

Before formally stating our results and comparing them to previous work, it will

be useful to identify three distinct 3-SAT formula distributions.

Standard distribution, Formulas over a variable set of size are generated

by including each of the clauses independently with probability Let

be a formula over variables with exactly clauses; clearly,

Satisfiable distribution, This is the standard distribution, but tioned on the property of satisfiability That is, unsatisfiable formulas haveprobability zero, and satisfiable formulas have probability proportional to theirprobability in the standard distribution Formally, we define, for all formulas

satisfiable

Planted distribution, Formulas over a variable set of size are ated by first selecting an assignment uniformly at random from the set of allassignments Then, each of the clauses under which is true are includedindependently with probability intuitively, is a “forced” or “planted” satis-fying assignment of the resulting formula Formally, we define, for all formulas

over all assignments to the variables of and is the event that is asatisfying assignment

The standard distribution is clearly different from each of the other two,since in the standard distribution, every formula has a non-zero probability ofoccurring, whereas in the other two distributions, unsatisfiable formulas havezero probability of occurring Moreover, the satisfiable and planted distributionsare different, as the planted distribution is biased towards formulas with manysatisfying assignments For instance, the empty formula containing no clauses has

a probability of occurring in the planted distribution, whereas theempty formula has a lower probability of occurring in the satisfiable distribution,

16 Hubie Chen

1.2 Our Results

Roughly speaking, we show that there are “low” exponential time algorithms

clause-to-variable ratios More precisely, our main theorem is that there is a monotonically

finds a satisfying assignment in time for almost all formulas according to the

number of clauses is Put differently, for any we demonstrate thatthere is a constant such that an algorithm running in time finds a satisfying

rephrasing follows from the initial description by defining to be for an

such that We also prove that this theorem holds for almost allformulas from

An intriguing corollary we obtain is that if is set so that the expectednumber of clauses is super-linear – formally, if is – then, for everythere is an algorithm finding a satisfying assignment for almost all formulasfrom (as well as from in time In other words, for such a setting

of we obtain a distribution of formulas which can be solved almost always

in time – for all The obvious question suggested by this corollary iswhether or not such distributions can be solved almost always in polynomialtime, and we conjecture this to be the case

1.3 Related Work

Our results are most directly comparable to the works of Koutsoupias and padimitriou [17], Gent [9], and Flaxman [7] Koutsoupias and Papadimitriouproved that the “greedy” algorithm almost always finds a satisfying assignmentwhen is a sufficiently large constant times – that is, the expected num-ber of clauses is a sufficiently large quadratic function of

Pa-Theorem 1 [17] There exists a constant such that when

the greedy algorithm finds a satisfying assignment in polynomial time, for almost all formulas drawn according to

Their theorem also holds for in place of

Using a similar analysis, Gent showed that a simple algorithm almost alwaysfinds a satisfying assignment when the expected number of clauses is larger than

a particular constant times However, his result is only shown to hold forthe planted distribution

Theorem 2 [9] There exists a constant and a polynomial time algorithm A such that when the algorithm A finds a satisfying assignment for almost all formulas drawn according to

Flaxman also studied the planted distribution, using spectral techniques todemonstrate polynomial time tractability at a sufficiently high linear clause den-sity

An Algorithm for SAT Above the Threshold 17

Theorem 3 [7] There exists a constant and a polynomial time algorithm A such that when the algorithm A finds a satisfying assignment for almost all formulas drawn according to

In fact, Flaxman [7] studies a more general model of “planted” random mulas which includes our definition of planted distribution as a particular pa-rameterization; we refer the reader to his paper for more details

for-To review, our result is that for every there exists a constant suchthat when an algorithm running in time finds a satisfying as-signment for almost all formulas drawn according to (as well as

Thus, we basically lower by a factor of the clause density required to give theresult of Koutsoupias and Papadimitriou, but give an algorithm which requiresexponential time Although our algorithms require more time than those of Gentand Flaxman, our results on the satisfiable distribution are not directly compa-rable to their results, which concern only the planted distribution We emphasizethat Flaxman’s Theorem 3 does not imply our results on the satisfiable distri-bution This theorem does imply the results we give on the planted distribution;however, we still consider our proof technique of analyzing a simple random walk

to be of independent interest

We briefly discuss three other groups of results related to our study

“Below the threshold” algorithms For randomly generated formulas from

be-low the threshold, polynomial time algorithms that work almost always havebeen given [2,8] It is worth noting that at the times these results were initiallypresented, they gave the best threshold lower bounds to date

Worst-case algorithms The problem of finding a satisfying assignment for a

3-SAT formula has the trivial exponential time upper bound of a sequence ofresults has improved this trivial upper bound [21,5,22] For instance, [22] gives

a probabilistic algorithm solving 3-SAT in time That is, they exhibit

an algorithm which solves any given 3-SAT formula in time with highprobability Hirsch [13] took the slightly different route of performing a worst-case analysis of particular local search algorithms that had been shown to workwell in practice

Our results do not by any means strictly improve these results, nor are theyimplied by these results These upper bounds establish specific values of

such that any formula from the entire class of all 3-SAT formulas can be solved

in time In contrast, we establish that for any there is a particular class of distributions that can be solved almost always in time

Proof complexity Formulas from above the threshold are almost always

unsat-isfiable A natural question concerning such formulas is how difficult or easy

it is to certify their unsatisfiability The lengths of unsatisfiability proofs fromabove-threshold formula distributions is taken up in [3,11,10,12], for example;

we refer the reader thither for further pointers into the current literature

18 Hubie Chen

2 Preliminaries

We first note a lemma; both it and its proof are from [17]

Lemma 1 Let M, be independent binomial random variables If

is monotonically increasing.

Proof For any the probability is bounded above by

bounded above by routine computations using the Chernoff bounds

and

We now present the notation and definitions used throughout the paper Let

denote a 3-SAT formula, and let A and denote true/false assignments tothe variable set of

Let S denote the event that a formula is satisfying, and let denotethe event that a formula is satisfied by A property X holds for almost all

satisfiable formulas if A property X holds for almost all

for all formulas

the set of variables such that (Notice that when is the all-trueassignment is the set of variables that are true in A, and is the

set of variables that are false in A.) Let denote the complement of assignment

A, so that assigns a variable to false if and only if A assigns the variable to

true Let denote the set of assignments within Hamming distance of

A.

The following definitions are relative to a formula but we do not explicitlyinclude in the notation, as the relevant formula will be clear from context.Let denote the set of clauses that are “lost” at assignment A if variable

is flipped, that is, the set of clauses in that are true under A but false

which have one of the lowest values of We assume that ties arebroken in an arbitrary but fixed manner, so that is always of size

would move the assignment A further away from the assignment if flipped.

Let us say that a formula is relative to if is a satisfying

Let us say that a formula is if it is relative to every satisfyingassignment Let us say that a formula is if it is not that

is, and there exists a satisfying assignment such that for all

The next two sections constitute the technical portion of the paper Roughlyspeaking, these sections will establish that the algorithm succeeds on any good

An Algorithm for SAT Above the Threshold 19

formula, and that almost all satisfiable formulas are good – hence, the algorithmcan handle almost all satisfiable formulas

3 Algorithm

In this section, we present our algorithm, and show that it finds a satisfyingassignment on all good formulas The algorithm, which is parameterized byand takes a 3-SAT formula as input, is as follows:

Pick a random assignment A

Randomly pick a variable from and flip it to obtain a new A

Output satisfying assignment

The key property of this algorithm is given by the following theorem Roughly,the theorem states that the property of is sufficient to direct the al-gorithm to a satisfying assignment in a polynomial number of loop iterations

Theorem 4 Suppose that is a formula Then, the expected number

of flips required by the algorithm (parameterized with to find a satisfying assignment for is

Proof Consider a Markov chain with state set such that when thechain is in state (for it is only possible to make transitions

to the two states If the probability of transitioning from state to is

the expected time to hit either state 0 or state is [19, pp 246-247].Since is there exists a satisfying assignment such that for all

We can consider the algorithm as taking a random walk on the Markov chaindescribed above, where its state is the Hamming distance to the assignment

Each time the loop test fails on an assignment A, we have

which implies that Thus,

and randomly picking a variable from will result in picking a variablefrom at least 1/2 of the time, and thus the Hamming distance of A to has a probability of at least 1/2 of decreasing If the Hamming distance of A to

is less than or greater than the loop test is true, and the satisfyingassignment is found By the initially stated fact on the type of Markov chainconsidered, this will occur in expected flips

4 Analysis: Almost All Satisfiable Formulas Are Good

We now prove a complement to the theorem of the last section: that almost all

that we give This will allow us to conclude that the algorithm “works” foralmost all satisfiable formulas

is the same function given by Lemma 1.

Proof We assume without loss of generality that is the all-true assignment.

respect to any formula) This in turn implies that there exist distinct variables

all

Thus, we have

which is bounded above by

the so-called union bound, we see that the previous expression is

where the summation is over the same set of sequences as the union in theprevious expression

Observe that the clause set is a subset of those clauses containingpositively and two other variables occurring positively if and only if they are

false in A Also, the clause set is a subset of those clauses containingnegatively and two other variables occurring positively if and only if they are

false in A, but without clauses containing three negative literals (as they are not

satisfied by the forced all-true assignment)

a clause with appearing positively, and all clauses of do; likewise, for

a variable only can contain a clause with appearing negatively,

random variables! It follows that the previous expression is

As just described, is the sum of Bernoulli trials, and

is the sum of Bernoulli trials, where is the number of variables

assigned to the value true in A – that is, Using Lemma 1 along with thedeveloped chain of inequalities, we can upper bound the expression of interest,

An Algorithm for SAT Above the Threshold 21

Lemma 3 There exists a monotonically decreasing function

all assignments on variables.

Proof Let denote the set of assignments A such that

We establish an upper bound on the probability of interest

The third inequality holds by Lemma 2

It suffices to choose to be a sufficiently large monotonically decreasing

The next lemma connects the distribution to the distribution,using the prior two lemmas to show the rareness of formulas in

Lemma 4 There exists a monotonically decreasing function

Proof Let denote the set containing all assignments on variables, andlet be as in Lemma 3 Fix and let D denote the event that a formula

is We have the following chain of inequalities

By Lemma 3, the last expression approaches zero as approaches infinity

Having established that almost all formulas are we are now in aposition to combine this result with the theorem which showed that the algorithm

Theorem 5 There exists a monotonically decreasing function

such that when the given randomized algorithm finds a satisfying assignment for almost all satisfiable formulas in expected time (where “expected time” is with respect to the random bits used by the algorithm).

Proof For all define where is sufficiently small so that

434].) By Lemma 4, when

22 Hubie Chen

Thus, for such almost all satisfiable formulas are and by Theorem 4,

the algorithm will find a satisfying assignment for almost all satisfiable formulas

in flips

to see if it is satisfying; there are such assignments, and checking

each takes time for some polynomial Thus, flips require time

which is

By a similar proof, we obtain the same theorem for planted formulas

Theorem 6 There exists a monotonically decreasing function

such that when the given randomized algorithm finds a

satisfying assignment for almost all planted formulas in expected time

(where “expected time” is with respect to the random bits used by the algorithm).

Proof Notice that Lemma 4 is true with in place of this is

immediate from the definition of (Notice that the extra factor of

Lemma 3 is not needed.) Using this modified version of Lemma 4, the proof of

the theorem is exactly the same as the proof of Theorem 5

Finally, we observe in the next two corollaries that when the clause density

is super-linear, then both the satisfiable and planted distributions are solvable

almost always in time – for all

Corollary 1 Suppose that is Then for every there is a

randomized algorithm finding a satisfying assignment for all satisfiable formulas

in expected time (where “expected time” is with respect to the random bits

used by the algorithm).

Proof Let be sufficiently small so that Observe that

for all but finitely many Modify the algorithm given by Theorem

output is hard-coded

Corollary 2 Suppose that is Then for every there is a

randomized algorithm finding a satisfying assignment for all planted formulas in

expected time (where “expected time” is with respect to the random bits

used by the algorithm).

The proof of Corollary 2 is similar to that of Corollary 1

ex-pected number of clauses is a small constant times Then there is

an algorithm finding satisfying assignments for almost all satisfiable formulas in

expected time

5 Discussion

The work in this paper points to the question of whether or not there is a

polynomial time algorithm that works almost always, for “above the threshold”

An Algorithm for SAT Above the Threshold 23

satisfiable formulas A number of SAT algorithms, for example, GSAT [24], can

be viewed as performing random walks on a Markov chain with state set equal

to the set of all assignments From this perspective, each SAT formula induces a

different Markov chain, and what we have really done here is shown that almost

all such induced Markov chains (for the distribution of formulas of interest)

have a desired convergence property Perhaps one can perform a more detailed

analysis of the Markov chains for some such random walk algorithm, without

collapsing assignments with the same Hamming distance from some “target”

assignment, together into one state – as is done here and in the analysis of a

2-SAT algorithm given in [20]

It may also be of interest to attempt to prove time lower bounds on restricted

classes of random-walk algorithms For instance, the algorithm in this paper

picks a variable to flip based only on the number of clauses that would be lost by

flipping each variable (that is, the quantity for variable v at assignment

A) and not based on, for instance, what the clauses of look like, and at

which variables they overlap Can it be shown that all such algorithms with this

property have an exponential expected time before converging on a satisfying

assignment, say, for the distribution of formulas studied in this paper?

A broader but admittedly more speculative question we would like to pose is

whether or not it is possible to develop a complexity theory for distributions of

problem instances which takes almost always polynomial time as its base notion

of tractability Along these lines, we are curious whether or not distributions that

are in almost always time for all – such as those identified by Corollaries

1 and 2 – are always in almost always polynomial time Can this hypothesis,

perhaps restricted to some class of “natural” or computable distributions, be

related to any better-known complexity-theoretic hypotheses?

Acknowledgements The author wishes to thank Amy Gale, Jon Kleinberg,

Ric-cardo Pucella, and Bart Selman for useful discussions and comments The author

also thanks the anonymous referees for their many helpful comments

References

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random k-CNF formulas In Proceedings of the 30th Annual ACM Symposium on

Theory of Computing, pages 561-571, Dallas, TX, May 1998.

P Beame, T Pitassi Propositional Proof Complexity: Past, Present and Future.

Electronic Colloquium on Computational Complexity (ECCC) 5(067): (1998).

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P Raghavan, U Schning A Deterministic Algorithm for k-SAT Based on Local Search Theoretical Computer Science, 289(2002).

24 Hubie Chen

O Dubois, Y Boufkhad, J Mandler Typical random 3 SAT formulae and the satisfiability threshold Proc 11th ACM SIAM Symp on Discrete Algorithms, San Franscisco, CA, January 2000, pp 124-126 and Electronic Colloquium on Computational Complexity, TR03-007 (2003).

A Flaxman A spectral technique for random satisfiable 3CNF formulas SODA 2003.

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I Gent On the Stupid Algorithm for Satisfiability APES Technical Report 1998

03-A Goerdt, T Jurdzinski Some Results On Random Unsatisfiable K-Sat Instances and Approximation Algorithms Applied To Random Structures Combinatorics, Probability & Computing, Volume 12, 2003.

A Goerdt, M Krivelevich Efficient recognition of random unsatisfiable k-SAT stances by spectral methods Proceedings of the 18th International Symposium on Theoretical Aspects of Computer Science (STACS 2001) Lecture Notes in Com- puter Science, 2010:294-304, 2001.

in-A Goerdt, in-A Lanka Recognizing more random unsatisfiable 3-SAT instances efficiently LICS’03, Workshop on Typical case complexity and phase transitions, June 21, 2003, Ottawa, Canada.

E Hirsch SAT Local Search Algorithms: Worst-Case Study Journal of Automated Reasoning 24(1/2):127-143, 2000.

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al-B Selman, H Levesque, D Mitchell Hard and Easy Distributions of SAT lems Proc of the 10th National Conference on Artificial Intelligence, 1992, pp 459-465.

Prob-B Selman, H Levesque, D Mitchell A New Method for Solving Hard Satisfiability Problems Proc of the 10th National Conference on Artificial Intelligence, 1992,

Watched Data Structures for QBF Solvers

Ian Gent1, Enrico Giunchiglia2, Massimo Narizzano2, Andrew Rowley1, and

Armando Tacchella2

1 Dept of Computer Science, University of St Andrews

North Haugh, St Andrews, Fife, KY16 9SS, Scotland

{ipg, agdr}@dcs.st–and.ac.uk 2

DIST - Università di Genova Viale Causa 13, 16145 Genova, Italy {enrico,mox,tac}@mrg.dist.unige.it

Abstract In the last few years, we have seen a tremendous boost in the

efficiency of SAT solvers, this boost being mostly due to C HAFF C HAFF

owes some of its efficiency to its “two-literal watching” data structure.

In this paper we present watched data structures for Quantified Boolean

Formula (QBF) satisfiability solvers In particular, we propose (i) two

C HAFF-like literal watching schemes for unit clause detection; and (ii)

two other watched data structures, one for detecting pure literals and the other for detecting void quantifiers We have conducted an experi- mental evaluation of the proposed data structures, using both randomly generated and real-world benchmarks Our results indicate that clause watching is very effective, while the 2 and 3 literal watching data struc- tures become more effective as the clause length increases The quantifier watching structure does not appear to be effective on the instances con- sidered.

1 Introduction

In the last few years, we have seen a tremendous boost in the efficiency of SATsolvers, this boost being mostly due to CHAFF CHAFF is based on DPLL pro-cedure [1,2], and owes part of its efficiency to its data structures designed forthe specific look-ahead it implements, i.e., unit-propagation The basic idea is

to detect unit clauses by watching two unassigned literals per clause As soon

as one of the watched literals is assigned, another unassigned literal is lookedfor in the clause: failure to find one implies that the clause is unit The mainadvantage of any such procedure is that, when a literal is given a truth value,only its watched occurrences are assigned This is to be contrasted to traditionalDPLL implementations where, when assigning a variable, all its occurrences areconsidered This simple idea can be realized in various ways, differing for thespecific operations done when assigning a watched literal or when backtrack-ing (see, e.g., [3,4,5]) In CHAFF, backtracking requires a constant number ofoperations See [4] for more details

In this paper we tackle the problem of designing, implementing and imenting with watching data structures for DPLL-based QBF solvers In par-

exper-ticular, we propose (i) two CHAFF-like literal watching schemes for unit clause

E Giunchiglia and A Tacchella (Eds.): SAT 2003, LNCS 2919, pp 25–36, 2004.

26 Ian Gent et al.

detection; and (ii) two other watched data structures, one for detecting pure

literals and the other for detecting void quantifiers We have implemented suchwatching structures, and we conducted an experimental evaluation, using bothrandomly generated and real-world benchmarks Our results indicate that clausewatching is very effective, while the 2 and 3 literal watching data structuresbecome more effective as the clause length increases The quantifier watchingstructure does not appear to be effective on the instances considered

The paper is structured as follows We first introduce some basic terminologyand notation (§2) In §3, we briefly present the standard data structures Thewatched literal data structures that we propose are comprehensively described

in §4 and the other watched data structures are described in §5 We end thepaper with the experimental analysis (§6)

2 Basic Definitions

We take for granted the definitions of variable, literal, clause Notationally, if

is a literal, we write as an abbreviation for if and for otherwise

A QBF is an expression of the form

is the matrix, and is the bounding quantifier of each variable in

The semantics of a QBF can be defined recursively as follows:

1

2

3

4

If the matrix of contains an empty clause then is FALSE

If the matrix of is the empty set of clauses then is TRUE.

If is a QBF and is a literal, is the QBF

1

2

whose matrix is obtained from the matrix of by deleting the clauses C

whose prefix is obtained from the prefix of by deleting the variables notoccurring in Void quantifiers (i.e., quantifiers not binding any variable)are also eliminated

As usual, we say that a QBF is satisfiable iff is TRUE.

On the basis of the semantics, a simple recursive procedure for determiningthe satisfiability of a QBF simplifies to and/or if is in the leftmostset of variables in the prefix, until either an empty clause or the empty set

of clauses is produced: On the basis of the satisfiability of and thesatisfiability of can be determined according to the semantics of QBFs

Watched Data Structures for QBF Solvers 27

Most of the current QBF solvers are based on such simple procedure ever, in order to prune the search tree, they introduce some improvements.The first improvement is that it is possible to directly conclude that a QBF

How-is unsatHow-isfiable if the matrix contains a contradictory clause, i.e., a clause with

no existential literals (Notice that the empty clause is also contradictory).Then, if a literal is unit or pure in a QBF then can be simplified to

We say that a literal is

Unit if the matrix contains a unit clause in i.e., a clause of the form

with (i) existential; and (ii) each literal

universally quantified inside the quantifier binding For example, bothand are unit in any QBF of the form:

Pure if either is existential and does not belong to any clause in or

is universal and does not belong to any clause in For example, in thefollowing QBF, the pure literals are and

In the above example, notice that after and are assigned, can

be assigned because it is pure, and then can be assigned because it isunit This simple example shows the importance of implementing pure literalfixing in QBFs: The assignment of a pure existential literal may cause thedetection of a pure universal literal, and the assignment of a pure universalliteral may cause the detection of unit literals

Finally, all QBF solvers implement some heuristic in order to decide the best(among those admissible) literal to be used for branching

3 Unwatched Data Structures

We need to compare our new, watched, data structures with an implementationwhich is identical in terms of propagation, heuristics, etc, but in which standarddata structures are used To do this, we provided an alternative implementation

of CSBJ identical except for the use of unwatched data structures Thus, allour results in terms of run time compare executions with identical number ofbacktracks For a fair comparison, we tried our best to implement ‘state-of-the-art’, but unwatched, data structures In the rest of this section we describe these.The main requirements of any data structure in a QBF solver is to detectkey events The key events that we want to detect are

1

2

3

The occurrence of unit or pure literals

The presence of contradictory clauses in the matrix

The presence of void quantifiers in the prefix: This allows the removal of thequantifier from the prefix