Connection Tableaux with Lazy Paramodulation

It is well known that the connection refinement of clause tableaux with paramodulation is incomplete (even with weak connections). In this paper, we present a new connection tableau calculus for logic with equality. This calculus is based on a lazy form of paramodulation where parts of the unification step become auxiliary subgoals in a tableau and may be subjected to subsequent paramodulations. Our calculus uses ordering constraints and a certain form of the basicness restriction.     

Analytic Tableaux for Higher-Order Logic with Choice

While many higher-order interactive theorem provers include a choice operator, higher-order automated theorem provers so far have not. In order to support automated reasoning in the presence of a choice operator, we present a cut-free ground tableau calculus for Church’s simple type theory with choice. The tableau calculus is designed with automated search in mind. In particular, the rules only operate on the top level structure of formulas. Additionally, we restrict the instantiation terms for quantifiers to a universe that depends on the current branch. At base types the universe of instantiations is finite. Both of these restrictions are intended to minimize the number of rules a corresponding search procedure is obligated to consider. We prove completeness of the tableau calculus relative to Henkin models.     

Tableaux for logic programming

We present a logic programming language, which we call Proflog, with an operational semantics based on tableaux and a denotational semantics based on supervaluations. We show the two agree. Negation is well behaved, and semantic noncomputability issues do not arise. This is accomplished essentially by dropping a domain closure requirement. The cost is that intuitions developed through the use of classical logic may need modification, though the system is still classical at a level once removed. Implementation problems are discussed very briefly; the thrust of the paper is primarily theoretical.Research partly supported by NSF Grant CCR-9104015.     

Invariant-Free Clausal Temporal Resolution

Resolution is a well-known proof method for classical logics that is well suited for mechanization. The most fruitful approach in the literature on temporal logic, which was started with the seminal paper of M. Fisher, deals with Propositional Linear-time Temporal Logic (PLTL) and requires to generate invariants for performing resolution on eventualities. The methods and techniques developed in that approach have also been successfully adapted in order to obtain a clausal resolution method for Computation Tree Logic (CTL), but invariant handling seems to be a handicap for further extension to more general branching temporal logics. In this paper, we present a new approach to applying resolution to PLTL. The main novelty of our approach is that we do not generate invariants for performing resolution on eventualities. Hence, we say that the approach presented in this paper is invariant-free. Our method is based on the dual methods of tableaux and sequents for PLTL that we presented in a previous paper. Our resolution method involves translation into a clausal normal form that is a direct extension of classical CNF. We first show that any PLTL-formula can be transformed into this clausal normal form. Then, we present our temporal resolution method, called trs-resolution, that extends classical propositional resolution. Finally, we prove that trs-resolution is sound and complete. In fact, it finishes for any input formula deciding its satisfiability, hence it gives rise to a new decision procedure for PLTL.     

Approximability of Clausal Constraints

We study a family of problems, called Maximum Solution (Max Sol), where the objective is to maximise a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints. When the domain is Boolean (i.e. restricted to {0,1}), the maximum solution problem is identical to the well-studied Max Ones problem, and the complexity and approximability is completely understood for all restrictions on the underlying constraints. We continue this line of research by considering the Max Sol problem for relations defined by regular signed logic over finite subsets of the natural numbers; the complexity of the corresponding decision problem has recently been classified by Creignou et al. (Theory Comput. Syst. 42(2):239–255, 2008). We give sufficient conditions for when such problems are polynomial-time solvable and we prove that they are APX-hard otherwise. Similar dichotomies are also obtained for variants of the Max Sol problem.     

Ordered Semantic Hyper Tableaux

A family of tableau methods, called ordered semantic hyper (OSH) tableau methods for first-order theories with function symbols, is presented. These methods permit semantic information to guide the search for a proof. They also may make use of orderings on literals, clauses, and interpretations to guide the search. In a typical tableau, the branches represent conjunctions of literals, and the tableau represents the disjunction of the branches. An OSH tableau is as usual except that each branch B has an interpretation I ₀[B] associated with it, where I ₀ is an interpretation supplied at the beginning and I ₀[B] is the interpretation most like I ₀ that satisfies B. Only clauses that I ₀[B] falsifies may be used to expand the branch B, thus restricting the kinds of tableau that can be constructed. This restriction guarantees the goal sensitivity of these methods if I ₀ is properly chosen. Certain choices of I ₀ may produce a purely bottom-up tableau construction, while others may result in goal-oriented evaluation for a given query. The choices of which branch is selected for expansion and which clause is used to expand this branch are examined and their effects on the OSH tableau methods considered. A branch reordering method is also studied, as well as a branch pruning technique called complement modification, that adds additional literals to branches in a soundness-preserving manner. All members of the family of OSH tableaux are shown to be sound, complete, and proof convergent for refutations. Proof convergence means that any allowable sequence of operations will eventually find a proof, if one exists. OSH tableaux are powerful enough to be treated as a generalization of several classes of tableau discussed in the literature, including forward chaining and backward chaining procedures. Therefore, they can be used for efficient query processing.     

Database Repairs and Analytic Tableaux

In this article, we characterize in terms of analytic tableaux the repairs of inconsistent relational databases, that is databases that do not satisfy a given set of integrity constraints. For this purpose we provide closing and opening criteria for branches in tableaux that are built for database instances and their integrity constraints. We use the tableaux based characterization as a basis for consistent query answering, that is for retrieving from the database answers to queries that are consistent with respect to the integrity constraints.     

Complexity of Clausal Constraints Over Chains

We investigate the complexity of the satisfiability problem of constraints over finite totally ordered domains. In our context, a clausal constraint is a disjunction of inequalities of the form x≥d and x≤d. We classify the complexity of constraints based on clausal patterns. A pattern abstracts away from variables and contains only information about the domain elements and the type of inequalities occurring in a constraint. Every finite set of patterns gives rise to a (clausal) constraint satisfaction problem in which all constraints in instances must have an allowed pattern. We prove that every such problem is either polynomially decidable or NP-complete, and give a polynomial-time algorithm for recognizing the tractable cases. Some of these tractable cases are new and have not been previously identified in the literature.     

First-Order Glue

Glue has evolved significantly during the past decade. Although the recent move to type-theoretic notation was a step in the right direction, basing the current Glue system on System F (second-order λ-calculus) was an unfortunate choice. An extension to two sorts and ad hoc restrictions were necessary to avoid inappropriate composition of meanings. As a result, the current system is unnecessarily complicated. A first-order Glue system is hereby proposed as its replacement. This new system is not only simpler and more elegant, as it captures the exact requirements for Glue-style compositionality without ad hoc improvisations, but it also turns out to be more powerful than the current two-sorted (pseudo-) second-order system. First-order Glue supports all existing Glue analyses as well as more elegant alternatives. It also supports new, more demanding analyses.     

Confirmation-Guided Discovery of First-Order Rules with Tertius

This paper deals with learning first-order logic rules from data lacking an explicit classification predicate. Consequently, the learned rules are not restricted to predicate definitions as in supervised inductive logic programming. First-order logic offers the ability to deal with structured, multi-relational knowledge. Possible applications include first-order knowledge discovery, induction of integrity constraints in databases, multiple predicate learning, and learning mixed theories of predicate definitions and integrity constraints. One of the contributions of our work is a heuristic measure of confirmation, trading off novelty and satisfaction of the rule. The approach has been implemented in the Tertius system. The system performs an optimal best-first search, finding the k most confirmed hypotheses, and includes a non-redundant refinement operator to avoid duplicates in the search. Tertius can be adapted to many different domains by tuning its parameters, and it can deal either with individual-based representations by upgrading propositional representations to first-order, or with general logical rules. We describe a number of experiments demonstrating the feasibility and flexibility of our approach.     

Superposition-based Equality Handling for Analytic Tableaux

We present a variant of the basic ordered superposition rules to handle equality in an analytic free-variable tableau calculus. We prove completeness of this calculus by an adaptation of the model generation technique commonly used for completeness proofs of superposition in the context of resolution calculi. The calculi and the completeness proof are compared to earlier results of Degtyarev and Voronkov. Some variations and refinements are discussed.     

Single Step Tableaux for Modal Logics

Single Step Tableaux (SST) are the basis of a calculus for modal logics that combines different features of sequent and prefixed tableaux into a simple, modular, strongly analytic, and effective calculus for a wide range of modal logics.The paper presents a number of the computational results about SST (confluence, decidability, space complexity, modularity, etc.) and compares SST with other formalisms such as translation methods, modal resolution, and Gentzen-type tableaux. For instance, it discusses the feasibility and infeasibility of deriving decision procedures for SST and translation-based methods by replacing loop checking techniques with simpler termination checks.The complexity of searching for validity and logical consequence with SST and other methods is discussed. Minimal conditions on SST search strategies are proven to yield Pspace (and NPtime for S5 and KD45) decision procedures. The paper also presents the methodology underlying the construction of the correctness and completeness proofs.     

Clausal Presentation of Theories in Deduction Modulo

Resolution modulo is an extension of first-order resolution in which rewrite rules are used to rewrite clauses during the search. In the first version of this method, clauses are rewritten to arbitrary propositions. These propositions are needed to be dynamically transformed into clauses. This unpleasant feature can be eliminated when the rewrite system is clausal, i.e., when it rewrites clauses to clauses. We show in this paper how to transform any rewrite system into a clausal one, preserving the existence of cut free proofs of any sequent.