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## Quantified satisfiability (QSAT) Algorithms in addition to Problems in consideration of Quantified SAT Algorithms in addition to Problems in consideration of Quantified SAT

Emory & Henry College, US has reference to this Academic Journal, Algorithms in addition to Problems in consideration of Quantified SAT Toby Walsh Department of Computer Science University of York England Ian P. Gent School of Computer Science University of St Andrews Scotland Algorithms in addition to Problems in consideration of Quantified SAT 1. Quantified Satisfiability QSAT 2. The Evaluate Algorithm 3. A New Algorithm 4. Phase Transitions 5. Phase Transition in QSAT Quantified satisfiability (QSAT) Existential quantifiers as in propositional SAT Universal quantifiers “x.$ y. (x v y) & (-x v -y) x=true, then y=false satisfies x=false, then y=true satisfies also called quantified Boolean formulae (QBF) QSAT can be seen as an alternating game between existentials which want so that make formula true, in addition to universals which want so that make it false

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Complexity of QSAT PSPACE-complete no limit on number of alternating quantifiers problems needing polynomial space on a Turing machine k-QSAT k alternations, $ innermost Sk P-complete CNF formulae does not change complexity Algorithms in consideration of QSAT Recent Algorithms by. Cadoli, Giovanardi & Schaerf, AAAI 98, SAT 2000. simple so that avoid Both based on similar idea Work `outside in’ Set outermost quantifiers first, then next level . Both make extensive use of propagation rules I will present a simplified version of Cadoli’s Cadoli et al’s Evaluate algorithm Two mutually recursive functions U-evaluate in consideration of universal quantifiers E-evaluate in consideration of existential quantifier Many propagation rules in each e.g. `unit propagation’ If a clause exists alongside only a single existential literal . commit so that the E-variable having the relevant value I will not detail propagation rules Focus on branching nature of search . . in addition to an unusual preprocessing step

An unusual preprocessing step Given a QSAT formula F Form existential simplification E(F) Discard universal literals in F Solve E(F) as SAT problem (not as QSAT) Use classic Davis-Putnam If DP succeeds, F is soluble as QSAT Same values of e-variables works in consideration of all values of u-variables Function U-Evaluate(F) If clause set is empty, succeed If there is empty/all universal clause, fail Try Davis-Putnam on Existential Simplification If this solves all clauses, succeed Apply Propagation rules If the outermost quantifier is now Existential Return result of E-evaluate(F) Choose a U-variable u Fail if U-Evaluate(F u:=true) fails Else return result of U-Evaluate(F u:=false) Function E-Evaluate(F) If clause set is empty, succeed If there is empty/all universal clause, fail Try Davis-Putnam on Existential Simplification If this solves all clauses, succeed Apply Propagation rules If the outermost quantifier is now Universal Return result of U-evaluate(F) Choose a E-variable e Succeed if E-Evaluate(F e:=true) succeeds Else return result of E-Evaluate(F e:=false)

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Thoughts about theAlgorithms Both Rintanen/Cadoli algorithms successful Cadoli et al on Random Problems Rintanen on `Conditional Planning’ Problems Blind search successful especially notable in U-evaluate simple enumeration of possibilities SAT preprocessing successful yet work often discarded New algorithm should: avoid blind search/avoid discarding SAT work Failure Driven Search Any partial solution (e.g. from DP) is helpful . . as long as it solves all clauses without universals The existential assignment solves many clauses applies so that all universal assignments . except those invalidating universals in unsat clauses Future tests need only solve remaining cases Failure-driven rather than blind search Exploits work done by failed calls so that DP No-commitment Search We can try hard so that avoid guessing universals Set them as unknown initially Treat all literals as false if value = unknown Attempt so that solve without choosing value Only forced so that choose a value on backtracking in consideration of universal values not solved by existentials

Hard in addition to Soft Clauses Soft Clauses clauses that can be left unsolved on this attempt have some free universal variables e.g. U or V or W or E if we set E = false, have solution when U/V/W true next test we set U=V=W=false in addition to force new solution Hard Clauses clauses that must be solved in consideration of problem so that be soluble i.e. No universal variables, e.g. E Revised E-evaluate Takes hard in addition to soft clauses as input Fails if hard clauses alone are insoluble Succeeds if hard clauses are soluble solves as many soft clauses as possible can use heuristic method on soft clauses returns list of unsatisfied soft clauses Calls U-evaluate recursively in consideration of hard clauses Revised U-evaluate([F]) Uses ToDo list of subproblems Initially ToDo = [ F ] Succeed when ToDo empty Removes element G of ToDo list Call E-evaluate(G) fail if E-evaluate fails if E-evaluate succeeds . in consideration of each unsat soft clause returned assign universals so as so that unsatisfy the soft clause add the resulting simplified problem so that ToDo Return result of U-evaluate on revised ToDo list

Comments Where did DP go? DP call evaporates! First step in U-evaluate is so that call E-evaluate We are always reasoning on existentials first DP could be used as heuristic on soft clauses As could WalkSAT . Where is unknown value set? Unknown value is set implicitly when U-evaluate calls E-evaluate without setting universal Conclusions (Algorithms) Recent algorithms making QSAT feasible Existing algorithms have possible drawbacks blind search wasted work Can remove these drawbacks Exercises in consideration of the interested author. Correctness proof (except 2-QSAT) Implementation Empirical Evaluation Phase transition behaviour Seen in many NP-hard problems SAT, CSP, number partitioning, TSP . seminal IJCAI-91 paper by Cheeseman, Kanefsky & Taylor rapid solubility transition around ¯ 1

Phase transition behaviour Complexity peak associated alongside solubility transition easy-hard-easy problems on “knife-edge” a theory of constrainedness models such behaviour k = 1-log()/n AAAI-96, 97, 98 Other complexity classes Polynomial problems enforcing arc-consistency in CSPs [CP-97] worst-case complexity seen at phase boundary k useful so that predict location of transition What about higher complexity classes? QSAT phase transition Fixed clause length model k alternating quantifiers each alongside n variables l clauses each alongside h literals Cadoli et al (AI*IA97, AAAI98) not always an easy-hard-easy pattern solubility transition at l æn

Flawed problem generation! Propositional satisfiability may generate unit clauses, x in addition to -x just as 2 people here are likely so that have same birthday Quantified satisfiability may generate clause alongside single existential, x in addition to another alongside -x no satisfying assignment simple argument gives l æn Two fixes: Model A, discard clauses alongside one or fewer existentials Model B, fix number of existentials in each clause Easy fix: discard unit clauses QSAT phase transition QSAT phase transition Easy-hard-easy pattern now clearly visible phase transition around fixed l/n 2-QSAT, 3-cnf

QSAT phase transition Clear complexity “ridge” varying l/n, in addition to proportion of universals larger gap between higher percentiles than in NP? ?J. Rintanen Conclusions (Transitions) Be careful of “flaws” SAT, CSP, QSAT . simple so that avoid QSAT phase transition similar so that that seen in NP constrainedness, k useful but less accuate We predict similar results in other PSPACE problems game playing in addition to planning

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This Particular Journal got reviewed and rated by QSAT phase transition Clear complexity “ridge” varying l/n, in addition to proportion of universals larger gap between higher percentiles than in NP? ?J. Rintanen Conclusions (Transitions) Be careful of “flaws” SAT, CSP, QSAT . simple so that avoid QSAT phase transition similar so that that seen in NP constrainedness, k useful but less accuate We predict similar results in other PSPACE problems game playing in addition to planning and short form of this particular Institution is US and gave this Journal an Excellent Rating.