Showing papers on "Conjunctive normal form published in 2009"

Satisfiability Modulo Theories: An Appetizer

Abstract: Satisfiability Modulo Theories (SMT) is about checking the satisfiability of logical formulas over one or more theories. The problem draws on a combination of some of the most fundamental areas in computer science. It combines the problem of Boolean satisfiability with domains, such as, those studied in convex optimization and term-manipulating symbolic systems. It also draws on the most prolific problems in the past century of symbolic logic: the decision problem, completeness and incompleteness of logical theories, and finally complexity theory. The problem of modularly combining special purpose algorithms for each domain is as deep and intriguing as finding new algorithms that work particularly well in the context of a combination. SMT also enjoys a very useful role in software engineering. Modern software, hardware analysis and model-based tools are increasingly complex and multi-faceted software systems. However, at their core is invariably a component using symbolic logic for describing states and transformations between them. A well tuned SMT solver that takes into account the state-of-the-art breakthroughs usually scales orders of magnitude beyond custom ad-hoc solvers.

veriT: An Open, Trustable and Efficient SMT-Solver

Abstract: This article describes the first public version of the satisfiability modulo theory (SMT) solver veriT. It is open-source, proof-producing, and complete for quantifier-free formulas with uninterpreted functions and difference logic on real numbers and integers.

Polylogarithmic Independence Can Fool DNF Formulas

Abstract: We show that any $k$-wise independent probability distribution on $\{0,1\}^n$ $O(m^{2.2}$ $2^{-\sqrt{k}/10})$-fools any boolean function computable by an $m$-clause disjunctive normal form (DNF) (or conjunctive normal form (CNF)) formula on $n$ variables. Thus, for each constant $e>0$, there is a constant $c>0$ such that any boolean function computable by an $m$-clause DNF (or CNF) formula is $m^{-e}$-fooled by any $c\log^2m$-wise probability distribution. This resolves up to an $O(\log m)$ factor the depth-2 circuit case of a conjecture due to Linial and Nisan [Combinatorica, 10 (1990), pp. 349-365]. The result is equivalent to a new characterization of DNF (or CNF) formulas by low degree polynomials. It implies a similar statement for probability distributions with the small bias property. Using known explicit constructions of small probability spaces having the limited independence property or the small bias property, we directly obtain a large class of explicit pseudorandom generators of $O(\log^2m\log n)$-seed length for $m$-clause DNF (or CNF) formulas on $n$ variables, improving previously known seed lengths.

Indexing Boolean expressions

Abstract: We consider the problem of efficiently indexing Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF) Boolean expressions over a high-dimensional multi-valued attribute space. The goal is to rapidly find the set of Boolean expressions that evaluate to true for a given assignment of values to attributes. A solution to this problem has applications in online advertising (where a Boolean expression represents an advertiser's user targeting requirements, and an assignment of values to attributes represents the characteristics of a user visiting an online page) and in general any publish/subscribe system (where a Boolean expression represents a subscription, and an assignment of values to attributes represents an event). All existing solutions that we are aware of can only index a specialized sub-set of conjunctive and/or disjunctive expressions, and cannot efficiently handle general DNF and CNF expressions (including NOTs) over multi-valued attributes.In this paper, we present a novel solution based on the inverted list data structure that enables us to index arbitrarily complex DNF and CNF Boolean expressions over multi-valued attributes. An interesting aspect of our solution is that, by virtue of leveraging inverted lists traditionally used for ranked information retrieval, we can efficiently return the top-N matching Boolean expressions. This capability enables emerging applications such as ranked publish/subscribe systems [16], where only the top subscriptions that match an event are desired. For example, in online advertising there is a limit on the number of advertisements that can be shown on a given page and only the "best" advertisements can be displayed. We have evaluated our proposed technique based on data from an online advertising application, and the results show a dramatic performance improvement over prior techniques.

Circuit complexity and decompositions of global constraints

Abstract: We show that tools from circuit complexity can be used to study decompositions of global constraints. In particular, we study decompositions of global constraints into conjunctive normal form with the property that unit propagation on the decomposition enforces the same level of consistency as a specialized propagation algorithm. We prove that a constraint propagator has a a polynomial size decomposition if and only if it can be computed by a polynomial size monotone Boolean circuit. Lower bounds on the size of monotone Boolean circuits thus translate to lower bounds on the size of decompositions of global constraints. For instance, we prove that there is no polynomial sized decomposition of the domain consistency propagator for the ALLDIFFERENT constraint.

Narrow Proofs May Be Spacious:Separating Space and Width in Resolution

Abstract: The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously been studied and related to the resolution refutation size of unsatisfiable conjunctive normal form (CNF) formulas. Also, the minimum refutation space of a formula has been proven to be at least as large as the minimum refutation width, but it has been open whether space can be separated from width or the two measures coincide asymptotically. We prove that there is a family of $k$-CNF formulas for which the refutation width in resolution is constant but the refutation space is nonconstant, thus solving a problem mentioned in several previous papers.

Circuit Complexity and Decompositions of Global Constraints

Abstract: We show that tools from circuit complexity can be used to study decompositions of global constraints. In particular, we study decompositions of global constraints into conjunctive normal form with the property that unit propagation on the decomposition enforces the same level of consistency as a specialized propagation algorithm. We prove that a constraint propagator has a a polynomial size decomposition if and only if it can be computed by a polynomial size monotone Boolean circuit. Lower bounds on the size of monotone Boolean circuits thus translate to lower bounds on the size of decompositions of global constraints. For instance, we prove that there is no polynomial sized decomposition of the domain consistency propagator for the ALLDIFFERENT constraint.

Quantifier Elimination via Functional Composition

Abstract: This paper poses the following basic question: Given a quantified Boolean formula *** x . φ , what should a function/formula f be such that substituting f for x in φ yields a logically equivalent quantifier-free formula? Its answer leads to a solution to quantifier elimination in the Boolean domain, alternative to the conventional approach based on formula expansion. Such a composite function can be effectively derived using symbolic techniques and further simplified for practical applications. In particular, we explore Craig interpolation for scalable computation. This compositional approach to quantifier elimination is analyzably superior to the conventional one under certain practical assumptions. Experiments demonstrate the scalability of the approach. Several large problem instances unsolvable before can now be resolved effectively. A generalization to first-order logic characterizes a composite function's complete flexibility, which awaits further exploitation to simplify quantifier elimination beyond the propositional case.

Beyond CNF: A Circuit-Based QBF Solver

Abstract: State-of-the-art solvers for Quantified Boolean Formulas (QBF) have employed many techniques from the field of Boolean Satisfiability (SAT) including the use of Conjunctive Normal Form (CNF) in representing the QBF formula. Although CNF has worked well for SAT solvers, recent work has pointed out some inherent problems with using CNF in QBF solvers. In this paper, we describe a QBF solver, called CirQit (Cir-Q-it) that utilizes a circuit representation rather than CNF. The solver can exploit its circuit representation to avoid many of the problems of CNF. For example, we show how this approach generalizes some previously proposed techniques for overcoming the disadvantages of CNF for QBF solvers. We also show how important techniques like clause and cube learning can be made to work with a circuit representation. Finally, we empirically compare the resulting solver against other state-of-the-art QBF solvers, demonstrating that our approach can often outperform these solvers.

A solver for QBFs in negation normal form

Abstract: Various problems in artificial intelligence can be solved by translating them into a quantified boolean formula (QBF) and evaluating the resulting encoding. In this approach, a QBF solver is used as a black box in a rapid implementation of a more general reasoning system. Most of the current solvers for QBFs require formulas in prenex conjunctive normal form as input, which makes a further translation necessary, since the encodings are usually not in a specific normal form. This additional step increases the number of variables in the formula or disrupts the formula's structure. Moreover, the most important part of this transformation, prenexing, is not deterministic. In this paper, we focus on an alternative way to process QBFs without these drawbacks and describe a solver, $\ensuremath{\sf qpro}$ , which is able to handle arbitrary formulas. To this end, we extend algorithms for QBFs to the non-normal form case and compare $\ensuremath{\sf qpro}$ with the leading normal form provers on several problems from the area of artificial intelligence. We prove properties of the algorithms generalized to non-clausal form by using a novel approach based on a sequent-style formulation of the calculus.

On-the-Fly Clause Improvement

Abstract: Most current propositional SAT solvers apply resolution at various stages to derive new clauses or simplify existing ones. The former happens during conflict analysis, while the latter is usually done during preprocessing. We show how subsumption of the operands by the resolvent can be inexpensively detected during resolution; we then show how this detection is used to improve three stages of the SAT solver: variable elimination, clause distillation, and conflict analysis. The "on-the-fly" subsumption check is easily integrated in a SAT solver. In particular, it is compatible with the strong conflict analysis and the generation of unsatisfiability proofs. Experiments show the effectiveness of this technique and illustrate an interesting synergy between preprocessing and the DPLL procedure.

The Lovász Local Lemma and Satisfiability

Abstract: We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovasz Local Lemma, a probabilistic lemma with many applications in combinatorics. In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NP-completeness. Several open problems remain, some of which we mention in the concluding section.

Towards a notion of unsatisfiable cores for LTL

Abstract: Unsatisfiable cores, i.e., parts of an unsatisfiable formula that are themselves unsatisfiable, have important uses in debugging specifications, speeding up search in model checking or SMT, and generating certificates of unsatisfiability. While unsatisfiable cores have been well investigated for Boolean SAT and constraint programming, the notion of unsatisfiable cores for temporal logics such as LTL has not received much attention. In this paper we investigate notions of unsatisfiable cores for LTL that arise from the syntax tree of an LTL formula, from converting it into a conjunctive normal form, and from proofs of its unsatisfiability. The resulting notions are more fine-granular than existing ones.

Faster SAT solving with better CNF generation

Abstract: Boolean satisfiability (SAT) solving has become an enabling technology with wide-ranging applications in numerous disciplines. These applications tend to be most naturally encoded using arbitrary Boolean expressions, but to use modern SAT solvers, one has to generate expressions in Conjunctive Normal Form (CNF). This process can significantly affect SAT solving times. In this paper, we introduce a new linear-time CNF generation algorithm. We have implemented our algorithm and have conducted extensive experiments, which show that our algorithm leads to faster SAT solving times and smaller CNF than existing approaches.

Optimal Length Resolution Refutations of Difference Constraint Systems

Abstract: This paper is concerned with determining the optimal length resolution refutation (OLRR) of a system of difference constraints over an integral domain The problem of finding short explanations for unsatisfiable difference constraint systems (DCS) finds applications in a number of design domains including program verification, proof theory, real-time scheduling and operations research It is well-known that resolution refutation is a sound and complete procedure to establish the unsatisfiability of boolean formulas in clausal form The literature has also established that a variant of the resolution procedure called Fourier-Motzkin elimination is a sound and complete procedure for establishing the unsatisfiability of linear constraint systems (LCS) Our work in this paper first establishes that every DCS has a short (polynomial in the size of the input) resolution refutation and then shows that there exists a polynomial time algorithm to compute the optimal size refutation One of the consequences of our work is that the Minimum Unsatisfiable Subset (MUS) of a DCS can be computed in polynomial time; computing the MUS of an unsatisfiable constraint set is an extremely important aspect of certifying algorithms

Efficient SAT solving for non-clausal formulas using DPLL, graphs, and watched cuts

Abstract: Boolean satisfiability (SAT) solvers are used heavily in hardware and software verification tools for checking satisfiability of Boolean formulas. Most state-of-the-art SAT solvers are based on the Davis-Putnam-Logemann-Loveland (DPLL) algorithm and require the input formula to be in conjunctive normal form (CNF). We present a new SAT solver that operates on the negation normal form (NNF) of the given Boolean formulas/circuits. The NNF of a formula is usually more succinct than the CNF of the formula in terms of the number of variables. Our algorithm applies the DPLL algorithm to the graph-based representations of NNF formulas. We adapt the idea of the two-watched-literal scheme from CNF SAT solvers in order to efficiently carry out Boolean Constraint Propagation (BCP), a key task in the DPLL algorithm. We evaluate the new solver on a large collection of Boolean circuit benchmarks obtained from formal verification problems. The new solver outperforms the top solvers of the SAT 2007 competition and SAT-Race 2008 in terms of run time on a large majority of the benchmarks.

Improving Coq propositional reasoning using a lazy CNF conversion scheme

Abstract: In an attempt to improve automation capabilities in the Coq proof assistant, we develop a tactic for the propositional fragment based on the DPLL procedure. Although formulas naturally arising in interactive proofs do not require a state-of-the-art SAT solver, the conversion to clausal form required by DPLL strongly damages the performance of the procedure. In this paper, we present a reflexive DPLL algorithm formalized in Coq which outperforms the existing tactics. It is tightly coupled with a lazy CNF conversion scheme which, unlike Tseitin-style approaches, does not disrupt the procedure. This conversion relies on a lazy mechanism which requires slight adaptations of the original DPLL. As far as we know, this is the first formal proof of this mechanism and its Coq implementation raises interesting challenges.

Variable Influences in Conjunctive Normal Forms

Abstract: We provide an upper bound on the total influence of Boolean functions defined by k -cnfs Our bound is nearly optimal We achieve it by an extension and appropriate use of an algorithm of Paturi, Pudlak, and Zane We also discuss applications to prove and compute lower bounds for the maximum clause width k

A Refined Resolution Calculus for CTL

Abstract: In this paper, we present a refined resolution-based calculus for Computation Tree Logic (CTL). The calculus requires a polynomial time computable transformation of an arbitrary CTL formula to an equi-satisfiable clausal normal form formulated in an extension of CTL with indexed existential path quantifiers. The calculus itself consists of a set of resolution rules which can be used as the basis for an EXPTIME decision procedure for the satisfiability problem of CTL. We prove soundness and completeness of the calculus. In addition, we introduce CTL-RP, our implementation of the calculus as well as some experimental results.

Understanding Space in Resolution: Optimal Lower Bounds and Exponential Trade-offs

Abstract: For current state-of-the-art satisfiability algorithms ba sed on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. Understanding time and memory consumption, and how they are related to one another, is therefore a question of considerable practical importance. In the field of proof complexity, thes e resources correspond to the length and space of resolution proofs for formulas in conjunctive normal form (CNF). There has been a long line of research investigating these proof complexity measures, but while strong results have been established for length, our understanding of space and how it relates to length has remained quite poor. In particular, the question whether resolution proofs can be optimized for length and space simultaneously, or whether there are trade-offs between these two measures, has remained essentially open apart from a few results in very limited settings suffering from various technical r estrictions. In this paper, we remedy this situation by proving a host of length-space trade-off results for resolution in a completely general setting. Our collection of tr ade-offs cover space ranging over the whole interval from constant to O(n/loglogn), and most of them are superpolynomial or even exponential. Our key technical contribution is the following, somewhat surprising, theorem: Any CNF formulaF can be transformed by simple substitution into a new formula F 0 such that if F has the right properties, F 0 can be proven in essentially the same length as F while the minimal space needed for F 0 is lowerbounded by the number of variables mentioned simultaneously in any proof forF . Applying this theorem to so-called pebbling formulas defined in terms of pebble gam es on directed acyclic graphs, and then using known results from the pebbling literature as well as a proving a couple of new ones, we obtain our resolution trade-off theorems.

Deriving all minimal consistency-based diagnosis sets using SAT solvers

Abstract: In this paper, a novel method is proposed for judging whether a component set is a consistency-based diagnostic set, using SAT solvers. Firstly, the model of the system to be diagnosed and all the observations are described with conjunctive normal forms (CNF). Then, all the related clauses in the CNF files to the components other than the considered ones are extracted, to be used for satisfiability checking by SAT solvers. Next, all the minimal consistency-based diagnostic sets are derived by the CSSE-tree or by other similar algorithms. We have implemented four related algorithms, by calling the gold medal SAT solver in SAT07 competition – RSAT. Experimental results show that all the minimal consistency-based diagnostic sets can be quickly computed. Especially our CSSE-tree has the best efficiency for the single- or double-fault diagnosis.

Speeding up SAT-Based ATPG Using Dynamic Clause Activation

Abstract: SAT-based ATPG turned out to be a robust alternative to classical structural ATPG algorithms such as FAN. The number of unclassified faults can be significantly reduced using a SAT-based ATPG approach. In contrast to structural ATPG, SAT solvers work on a Boolean formula in Conjunctive Normal Form (CNF). This results in some disadvantages for SAT solvers when applied to ATPG, e.g. CNF transformation time and loss of structural knowledge. As a result, SAT-based ATPG algorithms are very robust for hard-to-test faults, but suffer from the overhead for easy-to-test faults. We propose the SAT technique Dynamic Clause Activation (DCA) in order to reduce the run time gap between structural and SAT-based ATPG algorithms and, at the same time, retain the high level of robustness. Using DCA, the SAT solver works on a partial formula of a logic circuit which is dynamically extended during the search process using structural knowledge. Furthermore, efficient dynamic learning techniques can be easily integrated within the proposed technique. The approach is evaluated on large industrial circuits.

Using SAT-Solvers to compute inference-proof database instances

Abstract: An inference-proof database instance is a published, secure view of an input instance containing secret information with respect to a security policy and a user profile. In this paper, we show how the problem of generating an inference-proof database instance can be represented by the partial maximum satisfiability problem. We present a prototypical implementation that relies on highly efficient SAT-solving technology and study its performance in a number of test cases.

A declarative encoding of telecommunications feature subscription in SAT

Abstract: This paper describes the encoding of a telecommunications feature subscription configuration problem to propositional logic and its solution using a state-of-the-art Boolean satisfaction solver. The transformation of a problem instance to a corresponding propositional formula in conjunctive normal form is obtained in a declarative style. An experimental evaluation indicates that our encoding is considerably faster than previous approaches based on the use of Boolean satisfaction solvers. The key to obtaining such a fast solver is the careful design of the Boolean representation and of the basic operations in the encoding. The choice of a declarative programming style makes the use of complex circuit designs relatively easy to incorporate into the encoder and to fine tune the application.

On Some SAT-Variants over Linear Formulas

Abstract: We investigate the computational complexity of some prominent variants of the propositional satisfiability problem (SAT), namely not-all-equal SAT (NAE-SAT) and exact SAT (XSAT) restricted to the class of linear conjunctive normal form (CNF) formulas. Clauses of a linear formula pairwise have at most one variable in common. We prove that NAE-SAT and XSAT both remain NP-complete when restricted to linear formulas. Since the corresponding reduction is not valid when input formulas are not allowed to have 2-clauses, we also prove that NAE-SAT and XSAT still behave NP-complete on formulas only containing clauses of length at least k, for each fixed integer k ≥ 3. Moreover, NP-completeness proofs for NAE-SAT and XSAT restricted to monotone linear formulas are presented. We also discuss the length restricted monotone linear formula classes regarding NP-completeness where a difficulty arises for NAE-SAT, when all clauses are k-uniform, for k ≥ 4. Finally, we show that NAE-SAT is polynomial-time decidable on exact linear formulas, where each pair of distinct clauses has exactly one variable in common. And, we give some hints regarding the complexity of XSAT on the exact linear class.

Does This Set of Clauses Overlap with at Least One MUS

Abstract: This paper is concerned with the problem of checking whether a given subset Γ of an unsatisfiable Boolean CNF formula Σ takes part in the basic causes of the inconsistency of Σ. More precisely, an original approach is introduced to check whether Γ overlaps with at least one minimally unsatisfiable subset (MUS) of Σ. In the positive case, it intends to compute and deliver one such MUS. The approach re-expresses the problem within an evolving coarser-grained framework where clusters of clauses of Σ are formed and examined according to their levels of mutual conflicts when they are interpreted as basic interacting entities. It then progressively refines the framework and the solution by splitting most promising clusters and pruning the useless ones until either some maximal preset computational resources are exhausted, or a final solution is discovered. The viability and the usefulness of the approach are illustrated through benchmarks experimentations.