Hantao Zhang

SATO: An Efficient Propositional Prover

SATO (Satissability Testing Optimized) is a propositional prover based on the Davis-Putnam method 3], which is is one of the major practical methods for the satissability (SAT) problem of propositional logic. The rst report of SATO appeared in 12]. Since then, we constantly add new techniques into SATO to make it more eecient 14, 13]. One of the major motivations to develop SATO was to attack open Latin square problems. While SATO works well on Latin square problems, its previous versions did not work well on many classes of the SAT problem. In the fall of 1996, we made an eeort to improve SATO so that it works well on a large set of the SAT problem. In the following, we discuss brieey two techniques that we found eeective to improve SATO performance. One is about splitting rules; the other is about connict analysis. While these two techniques are known in the community, the real challenge is how to integrate these techniques without weakening each other. We are happy to report here that the two techniques integrated very well with the techniques previously implemented in SATO. In the following discussions, we assume that the reader is familiar with propositional logic and the Davis-Putnam method 3]. One important place where heuristics may be inserted in the Davis-Putnam method is in the choice of a literal for splitting. It is well-known that diierent splitting rules make the performance of the Davis-Putnam algorithm diierent by a magnitude of several orders. While SATO provides several popular splitting rules, each rule works well only for a particular class of SAT instances. For instance, in our study of quasigroup problems, one rule seems better than the others: choose one literal in one of the shortest positive clauses (a positive clause is a clause where all the literals are positive). On the other hand, a proved eeective splitting rule is to choose a variable x such that the value f 2 (x)f 2 (:x) is maximal, where f 2 (L) is one plus the number of occurrences of literal L in binary clauses 2, 5]. We tried to combine the above two rules into one as follows: Let 0 < a 1 and n be the number of shortest non-Horn clauses in the current set. At rst, we collect all the variable names appearing in the rst da ne shortest positive clauses. Then we choose x in …

PSATO: a distributed propositional prover and its application to quasigroup problems

Abstract We present a distributed/parallel prover for propositional satisfiability (SAT), called PSATO, for networks of workstations. PSATO is based on the sequential SAT prover SATO, which is an efficient implementation of the Davis –Putnam algorithm. The master–slave model is used for communication. A simple and effective workload balancing method distributes the workload among workstations. A key property of our method is that the concurrent processes explore disjoint portions of the search space. In this way, we use parallelism without introducing redundant search. Our approach provides solutions to the problems of (i) cumulating intermediate results of separate runs of reasoning programs; (ii) designing highly scalable parallel algorithms and (iii) supporting “fault-tolerant” distributed computing. Several dozens of open problems in the study of quasigroups have been solved using PSATO. We also show how a useful technique called the cyclic group construction has been coded in propositional logic.

An overview of Rewrite Rule Laboratory (RRL)

Abstract RRL ( Rewrite Rule Laboratory ) was originally developed as an environment for experimenting with automated reasoning algorithms for equational logic based on rewrite techniques. It has now matured into a full-fledged theorem prover which has been used to solve hard and challenging mathematical problems in automated reasoning literature as well as a research tool for investigating the use of formal methods in hardware and software design. We provide a brief historical account of development of RRL and its descendants, give an overview of the main capabilities of RRL and conclude with a discussion of applications of RRL .

On sufficient-completeness and related properties of term rewriting systems

SummaryThe decidability of the sufficient completeness property of equational specifications satisfying certain conditions is shown. In addition, the decidability of the related concept of quasi-reducibility of a term with respect to a set of rules is proved. Other results about irreducible ground terms of a term rewriting system also follow from a key technical lemma used in these decidability proofs; this technical lemma states that there is a finite bound on the substitutions of ground terms that need to be considered in order to check for a given term, whether the result obtained by any substitution of ground terms into the term is irreducible. These results are first shown for untyped systems and are subsequently extended to typed systems.

A Mechanizable Induction Principle for Equational Specifications

Automating proofs of properties of functions defined on inductively constructed data structures is important in many computer science and artificial intelligence applications, in particular in program verification and specification systems. A new induction principle based on a constructor model of a data structure is developed. This principle along with a given function definition as a set of equations is used to construct automatically an induction scheme suitable for proving inductive properties of the function. The proposed induction principle thus gives different induction schema for different function definitions, just as Boyer and Moores prover does. A novel feature of this approach is that it can also be used for proving properties by induction for data structures such as integers, finite sets, whose values cannot be freely constructed, i.e., constructors for such data structures are related to each other. This method has been implemented in RRL, a rewrite-rule based theorem prover. More than a hundred theorems in number theory including the unique prime factorization theorem, have been proved using the method.

Implementing the Davis–Putnam Method

The method proposed by Davis, Putnam, Logemann, and Loveland for propositional reasoning, often referred to as the Davis–Putnam method, is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the Davis–Putnam method efficiently using the trie data structure for propositional clauses. A new technique of indexing only the first and last literals of clauses yields a unit propagation procedure whose complexity is sublinear to the number of occurrences of the variable in the input. We also show that the Davis–Putnam method can work better when unit subsumption is not used. We illustrate the performance of our programs on some quasigroup problems. The efficiency of our programs has enabled us to solve some open quasigroup problems.

Sufficient-completeness, ground-reducibility and their complexity

SummaryThe sufficient-completeness property of equational algebraic specifications has been found useful in providing guidelines for designing abstract data type specifications as well as in proving inductive properties using the induction-less-induction method. The sufficient-completeness property is known to be undecidable in general. In an earlier paper, it was shown to be decidable for constructor-preserving, complete (canonical) term rewriting systems, even when there are relations among constructor symbols. In this paper, the complexity of the sufficient-completeness property is analyzed for different classes of term rewriting systems. A number of results about the complexity of the sufficient-completeness property for complete (canonical) term rewriting systems are proved: (i) The problem is co-NP-complete for term rewriting systems with free constructors (i.e., no relations among constructors are allowed), (ii) the problem remains co-NP-complete for term rewriting systems with unary and nullary constructors, even when there are relations among constructors, (iii) the problem is provably in “almost” exponential time for left-linear term rewriting systems with relations among constructors, and (iv) for left-linear complete constructor-preserving rewriting systems, the problem can be decided in steps exponential innlogn wheren is the size of the rewriting system. No better lower-bound for the complexity of the sufficient-completeness property for complete (canonical) term rewriting system with nonlinear left-hand sides is known. An algorithm for left-linear complete constructor-preserving rewriting systems is also discussed. Finally, the sufficient-completeness property is shown to be undecidable for non-linear complete term rewriting systems with associative functions. These complexity results also apply to the ground-reducibility property (also called inductive-reducibility) which is known to be directly related to the sufficient-completeness property.

Proof by induction using test sets

A new method for proving an equational formula by induction is presented. This method is based on the use of the Knuth-Bendix completion procedure for equational theories, and it does not suffer from limitations imposed by the inductionless induction methods proposed by Musser and Huet and Hullot. The method has been implemented in RRL, a Rewrite Rule Laboratory. Based on extensive experiments, the method appears to be more practical and efficient than a recently proposed method by Jouannaud and Kounalis. Using ideas developed for this method, it is also possible to check for sufficient completeness of equational axiomatizations.

First-Order Theorem Proving Using Conditional Rewrite Rules

A method based on superposition on maximal literals in clauses and conditional rewriting is discussed for automatically proving theorems in first-order predicate calculus with equality. First-order formulae (clauses) are represented as conditional rewrite rules which turn out to be an efficient representation. The use of conditional rewriting for reducing search space is discussed. The method has been implemented in RRL, a Rewrite Rule Laboratory, a theorem proving environment based on rewriting techniques. It has been tried on a number of examples with considerable success. Its performance on bench-mark examples, including Schuberts Steamroller problem, SAMs lemma, and examples from set theory, compared favorably with the performance of other methods reported in the literature.

Opening the AC-unification race

This note reports about the implementation of AC-unification algorithms, based on the variable-abstraction method of Stickel and on the constant-abstraction method of Livesey, Siekmann, and Herold. We give a set of 105 benchmark examples and compare execution times for implementations of the two approaches. This documents for other researchers what we consider to be the state-of-the-art performance for elementary AC-unification problems.