Much recent work has gone into adapting techniques that were originally developed for SAT solving to QBF solving. In particular, QBF solvers are often based on SAT solvers. Most competitive QBF solvers are search-based. In this work we explore an alternative approach to QBF solving, based on symbolic quantifier elimination. We extend some recent symbolic approaches for SAT solving to symbolic QBF solving, using various decision-diagram formalisms such as OBDDs and ZDDs. In both approaches, QBF formulas are solved by eliminating all their quantifiers. Our first solver, QMRES, maintains a set of clauses represented by a ZDD and eliminates quantifiers via multi-resolution. Our second solver, QBDD, maintains a set of OBDDs, and eliminate quantifier by applying them to the underlying OBDDs. We compare our symbolic solvers to several competitive search-based solvers. We show that QBDD is not competitive, but QMRES compares favorably with search-based solvers on various benchmarks consisting of non-random formulas.

Abstract. Recent work has shown how to use OBDDs for satisfiability solving. The idea of this approach, which we call symbolic quantifier elimination, is to view an instance of propositional satisfiability as an existentially quantified propositional formula. Satisfiability solving then amounts to quantifier elimination; once all quantifiers have been eliminated we are left with either 1 or 0. Our goal in this work is to study the effectiveness of symbolic quantifier elimination as an approach to satisfiability solving. To that end, we conduct a direct comparison with the DPLL-based ZChaff, as well as evaluate a variety of optimization techniques for the symbolic approach. In comparing the symbolic approach to ZChaff, we evaluate scalability across a variety of classes of formulas. We find that no approach dominates across all classes. While ZChaff dominates for many classes of formulas, the symbolic approach is superior for other classes of formulas. Once we have demonstrated the viability of the symbolic approach, we focus on optimization techniques for this approach. We study techniques from constraint satisfaction for finding a good plan for performing the symbolic operations of conjunction and of existential quantification. We also study various variable-ordering heuristics, finding that while no heuristic seems to dominate across all classes of formulas, the maximum-cardinality search heuristic seems to offer the best overall performance. 1

Bisimulation is a seemingly attractive state-space minimization technique because it can be computed automatically and yields the smallest model preserving all ¯-calculus formulas. It is considered impractical for symbolic model checking, however, because the required BDDs are prohibitively large for most designs. We revisit bisimulation minimization, this time in an automata-theoretic framework. Bisimulation has potential in this framework because after intersecting the design with the negation of the property, minimization can ignore most of the atomic propositions. We compute bisimulation using an algorithm due to Lee and Yannakakis that represents bisimulation relations by their equivalence classes and only explores reachable classes. This greatly improves on the time and memory usage of naive algorithms. We demonstrate that bisimulation is practical for many designs within the automatatheoretic framework. In most cases, however, the cost of performing this reduction still outweigh...

This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps read-once branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...

We present a compositional proof of correctness for a binary decision diagram (BDD) manager used in the context of a propositional satisfiability checker implemented using SingleThreaded Objects (stobjs) in ACL2. The use of stobjs affords the definition of an efficient BDD manager which ensures unique construction, allows constant-time comparison, and caches previously computed results. The use of ACL2 means we can prove that the BDD manager implements the prescribed task of building a normal-form representation of a boolean formula. We divide the proof requirements into (1) showing that a simpler set of BDD functions is correct, and (2) showing that the stobj-based BDD functions return values consistent with these simpler functions. We conclude the paper with a discussion of future extensions and refinements to the BDD manager presented.

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this article, there is considerable activity in MVL deduction which is why we felt justified in writing this survey. Needless to say, we cannot give a general introduction to MVL in the present context. For this, we have to refer to general treatments such as [153, 53, 93]. 2 A classification of many-valued logics according to their intended application

This paper introduces a new normal form for Exclusive Sums-of-Products (ESOPs) of completely specified Boolean functions. We study the properties of the SNF and show its special place among canonical Reed-Muller representations. We propose to use the SNF in a number of applications related to the exact ESOP minimization. We describe an efficient way to compute the SNF with the complexity proportional to the number of nodes in the BDD of the given function. Experimental results speak for the potential usefulness of the SNF. 1

Abstract. Binary Decision Diagrams (BDDs) have recently become widely accepted as a space-efficient method of representing relations in points-to analyses. When BDDs are used to represent relations, each element of a domain is assigned a bit pattern to represent it, but not every bit pattern represents an element. The circuit design, model checking, and verification communities have achieved significant reductions in BDD sizes using Zero-Suppressed BDDs (ZBDDs) to avoid the overhead of these don’t-care bit patterns. We adapt BDD-based program analyses to use ZBDDs instead of BDDs. Our experimental evaluation studies the space requirements of ZBDDs for both context-insensitive and contextsensitive program analyses and shows that ZBDDs can greatly reduce the space requirements for expensive context-sensitive points-to analysis. Using ZBDDs to reduce the size of the relations allows a compiler or other software analysis tools to analyze larger programs with greater precision. We also provide a metric that can be used to estimate whether ZBDDs will be more compact than BDDs for a given analysis. 1

Several variants of Bryant's ordered binary decision diagrams have been suggested in the literature to reason about discrete functions. In this paper, we introduce a generic notion of weighted decision diagrams that captures many of them and present criteria for canonicity. As a special instance of such weighted diagrams, we introduce a new BDD-variant for real-valued functions, called normalized algebraic decision diagrams. Regarding the number of nodes and arithmetic operations like addition and multiplication, these normalized diagrams are as efficient as factored edge-valued binary decision diagrams, while several other operators, like the calculation of extrema, minimum or maximum of two functions or the switch from real-valued functions to boolean functions through a given threshold, are more efficient for normalized diagrams than for their factored counterpart.