Ground Literals Research Articles

SMELS: Satisfiability Modulo Equality with Lazy Superposition

We consider the problem of checking satisfiability of quantified formulae in First Order Logic with Equality. We propose a new procedure for combining SAT solvers with Superposition Theorem Provers to handle quantified formulae in an efficient and complete way. In our procedure, the input formula is converted into CNF as in traditional first order logic theorem provers. The ground clauses are given to the SAT solver, which runs a DPLL method to build partial models. The partial model is reduced, and then passed to a Superposition procedure, along with justifications of literals. The Superposition procedure then performs an inference rule, which we call Justified Superposition, between the ground literals and the nonground clauses, plus usual Superposition rules with the nonground clauses. Any resulting ground clauses are provided to the DPLL engine. We prove the completeness of our procedure, using a nontrivial modification of the Bachmair and Ganzinger's model generation technique. We have implemented a theorem prover based on this idea by reusing state-of-the-art SAT solver and Superposition Theorem Prover. Our theorem prover inherits the best of both worlds: a SAT solver to handle ground clauses efficiently, and a Superposition theorem prover which uses powerful orderings to handle the nonground clauses. Experimental results are promising, and hereby confirm the viability of our method.

A prototype implementation of a distributed satisfiability modulo theories solver in the ToolBus framework

An increasing number of verification tools (e.g., software model-checkers) require the use of Satisfiability Modulo Theories (SMT) solvers to implement the back-ends for the automatic analysis of specifications and properties. The most prominent approach to build SMT solvers consists in integrating an efficient Boolean solver with decision procedures capable of checking the satisfiability of sets of ground literals in selected theories. Although the problem of checking the satisfiability of arbitrary Boolean combinations of atoms modulo a background theory is NP-hard, there is a strong demand for high-performance SMT-solvers. In this paper, we describe the design and prototype implementation of-to the best of our knowledge-the first distributed SMT solver. The emphasis is on providing ways to reduce the implementation effort and to make the system easily extensible. This is achieved in two ways: (a) we re-use as much as possible the code of an available sequential SMT solver and (b) we adopt the TOOLBUS architecture for rapid prototyping. The behavior of the distributed SMT solver was tested on a set of problems which are representative of those generated by software verification techniques. The experiments show the possibility to obtain super-linear speed-ups of the distributed SMT solver with respect to its sequential version.

A prototype implementation of a distributed Satisfiability Modulo Theories solver in the ToolBus framework

Abstract An increasing number of verification tools (e.g., software model-checkers) require the use of Satisfiability Modulo Theories (SMT) solvers to implement the back-ends for the automatic analysis of specifications and properties. The most prominent approach to build SMT solvers consists in integrating an efficient Boolean solver with decision procedures capable of checking the satisfiability of sets of ground literals in selected theories. Although the problem of checking the satisfiability of arbitrary Boolean combinations of atoms modulo a background theory is NP-hard, there is a strong demand for high-performance SMT-solvers. In this paper, we describe the design and prototype implementation of—to the best of our knowledge—the first distributed SMT solver. The emphasis is on providing ways to reduce the implementation effort and to make the system easily extensible. This is achieved in two ways: (a) we re-use as much as possible the code of an available sequential SMT solver and (b) we adopt theToolBus architecture for rapid prototyping. The behavior of the distributed SMT solver was tested on a set of problems which are representative of those generated by software verification techniques. The experiments show the possibility to obtain super-linear speed-ups of the distributed SMT solver with respect to its sequential version.

On the Expressiveness of Levesque's Normal Form

Levesque proposed a generalization of a database called a proper knowledge base (KB), which is equivalent to a possibly infinite consistent set of ground literals. In contrast to databases, proper KBs do not make the closed-world assumption and hence the entailment problem becomes undecidable. Levesque then proposed a limited but efficient inference method V for proper KBs, which is sound and, when the query is in a certain normal form, also logically complete. He conjectured that for every first-order query there is an equivalent one in normal form. In this note, we show that this conjecture is false. In fact, we show that any class of formulas for which V is complete must be strictly less expressive than full first-order logic. Moreover, in the propositional case it is very unlikely that a formula always has a polynomial-size normal form.

On Variable-inactivity and Polynomial Formula-Satisfiability Procedures

Verification problems require to reason in theories of data structures and fragments of arithmetic. Thus, decision procedures for such theories are needed, to be embedded in, or interfaced with, proof assistants or software model checkers. Such decision procedures ought to be sound and complete, to avoid false negatives and false positives, efficient, to handle large problems, and easy to combine, because most problems involve multiple theories. The rewrite-based approach to decision procedures aims at addressing these sometimes conflicting issues in a uniform way, by harnessing the power of general first-order theorem proving. In this article, we generalize the rewrite-based approach from deciding the satisfiability of sets of ground literals to deciding that of arbitrary ground formulae in the theory. Next, we present polynomial rewrite-based satisfiability procedures for the theories of records with extensionality and integer offsets. The generalization of the rewrite-based approach to arbitrary ground formulae and the polynomial satisfiability procedure for the theory of records with extensionality use the same key property—termed variable-inactivity—that allows one to combine theories in a simple way in the rewrite-based approach.

Decision procedures for extensions of the theory of arrays

The theory of arrays, introduced by McCarthy in his seminal paper "Towards a mathematical science of computation," is central to Computer Science. Unfortunately, the theory alone is not sufficient for many important verification applications such as program analysis. Motivated by this observation, we study extensions of the theory of arrays whose satisfiability problem (i.e., checking the satisfiability of conjunctions of ground literals) is decidable. In particular, we consider extensions where the indexes of arrays have the algebraic structure of Presburger arithmetic and the theory of arrays is augmented with axioms characterizing additional symbols such as dimension, sortedness, or the domain of definition of arrays. We provide methods for integrating available decision procedures for the theory of arrays and Presburger arithmetic with automatic instantiation strategies which allow us to reduce the satisfiability problem for the extension of the theory of arrays to that of the theories decided by the available procedures. Our approach aims to re-use as much as possible existing techniques so as to ease the implementation of the proposed methods. To this end, we show how to use model-theoretic, rewriting-based theorem proving (i.e., superposition), and techniques developed in the Satisfiability Modulo Theories communities to implement the decision procedures for the various extensions.

Rewrite-Based Satisfiability Procedures for Recursive Data Structures

If a rewrite-based inference system is guaranteed to terminate on the axioms of a theory T and any set of ground literals, then any theorem-proving strategy based on that inference system is a rewrite-based decision procedure for T-satisfiability. In this paper, we consider the class of theories defining recursive data structures, that might appear out of reach for this approach, because they are defined by an infinite set of axioms. We overcome this obstacle by designing a problem reduction that allows us to prove a general termination result for all these theories. We also show that the theorem-proving strategy decides satisfiability problems in any combination of these theories with other theories decided by the rewrite-based approach.

Answering Queries Addressed to Several Databases According to a Majority Merging Approach

The general context of this work is the problem of merging data provided by several sources which can be contradictory. Focusing on the case when the information sources do not contain any disjunction, this paper first defines a propositional modal logic for reasoning with data obtained by merging several information sources according to a majority approach. Then it defines a theorem prover to automatically deduce these merged data. Finally, it shows how to use this prover to implement a query evaluator which answers queries addressed to several databases. This evaluator is such that the answer to a query is the one that could be computed by a classical evaluator if the query was addressed to the merged databases. The databases we consider are made of an extensional part, i.e. a set of positive or negative ground literals, and an intensional part i.e. a set of first order function-free clauses. A restriction is imposed to these databases in order to avoid disjunctive data.

Cooperation of Background Reasoners in Theory Reasoning by Residue Sharing

We propose a general way of combining background reasoners in theory reasoning. Using a restricted version of the Craig interpolation lemma, we show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of existing combination methods for decision procedures. In this case, constraint information is propagated across reasoners eexchanging residues that are, in essence, disjunctions of ground literals over a common signature. As an application of our approach, we describe a multitheory version of the semantic tableau calculus, and we prove it sound and complete.

Proving quantified literals in defeasible logic

Defeasible Logic is a nonmonotonic reasoning approach which has an efficient implementation. Currently Defeasible Logic can only prove ground literals. We describe a version of Defeasible Logic which is capable of proving existentially and universally closed literals, as well as ground literals. The intuition motivating the formalism is presented, and some of its properties are proved. We also discuss and justify the claim that existentially and universally closed literals can be proved.

A note on the stable model semantics for logic programs

The stable model semantics for logic programs is extended from ground literals onto open literals by augmenting the program language with an infinite set of new constants. This, in turn, leads to a natural translation of logic programs into open default theories.

The computational complexity of propositional STRIPS planning

I present several computational complexity results for propositional STRIPS planning, i.e., STRIPS planning restricted to ground formulas. Different planning problems can be defined by restricting the type of formulas, placing limits on the number of pre-and postconditions, by restricting negation in pre- and postconditions, and by requiring optimal plans. For these types of restrictions, I show when planning is tractable (polynomial) and intractable (NP-hard). In general, it is PSPACE-complete to determine if a given planning instance has any solutions. Extremely severe restrictions on both the operators and the formulas are required to guarantee polynomial time or even NP-completeness. For example, when only ground literals are permitted, determining plan existence is PSPACE-complete even if operators are limited to two preconditions and two postconditions. When definite Horn ground formulas are permitted, determining plan existence is PSPACE-complete even if operators are limited to zero preconditions and one postcondition. One of the interesting tractable problems is if each operator is restricted to positive preconditions and one postcondition (only ground literals). The blocks-world problem, slightly modified, is a subproblem of this restricted planning problem. These results in combination with previous analyses are not encouraging for domain-independent planning.

Negation as failure: a comparison of Clark's completed data base and Reiter's closed world assumption

Clark shows that a query evaluation procedure for data base deductions using a “negation as failure” inference rule can be regarded as making deductions from the “completed data base” (CDB) obtained by replacing the “if” clauses of the data base by “iff” clauses. We show they can also be regarded as deductions from the “closed world assumption” (CWA) of Reiter. Usually these deduction systems are incomplete and the CDB and CWA differ; one may be consistent and the other not, and when both are separately consistent they may be incompatible. However when the data base is Horn and definite they are compatible and when Clark's query evaluation procedure is complete for ground literals they are essentially equivalent. When the query evaluation procedure is not complete it may lack some basic closure properties. The conditions given by Clark for the completeness of his query evaluation procedure are not quite sufficient, and when made so are rather restrictive.