Beagle – A Hierarchic Superposition Theorem Prover

We present a type system for linear constraints over the reals intended for reasoning about the input-output directionality of variables. Types model the properties of definiteness, range width or approximation, lower and upper bounds of variables in a linear constraint. Several proof procedures are presented for inferring the type of a variable and for checking validity of type assertions. We rely on theory and tools for linear programming problems, linear algebra, parameterized polyhedra and negative constraints. An application of the type system is proposed in the context of the static analysis of constraint logic programs. Type assertions are at the basis of the extension of well-moding from pure logic programming. The proof procedures (both for type assertion validity and for well-moding) are implemented and their computational complexity is discussed. We report experimental results demonstrating the efficiency in practice of the proposed approach.

This paper deals with automated deduction for classical and partial logics, especially for the three-valued logic L3, which has been introduced, for example, in the study of natural language semantics. Based on ideas from a Plaisted's Gentzen style system for classical two-valued logic, we present a new tree-structured proof procedure (TMPR) together with a new completeness proof using proof transformation techniques and some improvements including the generation and use of lemmata. TMPR extends SLD-resolution with a Prolog-style backward chaining to full first-order logic by a controlled use of case analysis. This is done without having to extend negative goals needed, for example, for model elimination. A classification of TMPR, model elimination and related calculi in a common tableau framework is given. Thereafter, we present our extension of the TMPR proof procedure to L3 and show its soundness and completeness. As a side result, a TMPR proof system for the four-valued logic L4 is given. Finally, the restriction of TMPR to ‘L3-Horn clauses’ is considered, and, additionally, an idea for similarly extending model elimination and related systems to L3 (and L4) is illustrated.

The Inst-Gen-Eq method is an instantiation-based calculus which is complete for first-order clause logic modulo equality. Its distinctive feature is that it combines first-order reasoning with efficient ground satisfiability checking which is delegated in a modular way to any state-of-the-art ground SMT solver. The first-order reasoning modulo equality employs a superposition-style calculus which generates the instances needed by the ground solver to refine a model of a ground abstraction or to witness unsatisfiability. In this paper we present and compare different labelling mechanisms in the unit superposition calculus that facilitates finding the necessary instances. We demonstrate and evaluate how different label structures such as sets, AND/OR trees and OBDDs affect the interplay between the proof procedure and blocking mechanisms for redundancy elimination.

First-order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like Frege's morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its formal semantics, and a proof procedure using tableaus. Naturally the tableau rules are not complete, but they are with respect to a Henkinization of the true semantics. We demonstrate the use of the tableau rules by proving one of the theorems involved in Godel's ontological argument, one of the rare instances in the literature where higher-order modal constructs have appeared. A fuller treatment of the material presented here is in preparation

Automated theorem provers for first-order logic usually operate on sets of first-order clauses. It is well-known that the translation of a formula in full first-order logic to a clausal normal form (CNF) can crucially affect performance of a theorem prover. In our recent work we introduced a modification of first-order logic extended by the first class boolean sort and syntactical constructs that mirror features of programming languages. We called this logic FOOL. Formulas in FOOL can be translated to ordinary first-order formulas and checked by first-order theorem provers. While this translation is straightforward, it does not result in a CNF that can be efficiently handled by state-of-the-art theorem provers which use superposition calculus. In this paper we present a new CNF translation algorithm for FOOL that is friendly and efficient for superposition-based first-order provers. We implemented the algorithm in the Vampire theorem prover and evaluated it on a large number of problems coming from formalisation of mathematics and program analysis. Our experimental results show an increase of performance of the prover with our CNF translation compared to the naive translation.

We define an automatic proof procedure for finitely many-valued logics given by truth tables. The proof procedure is based on the inverse method. To define this procedure, we introduce so-called introduction-based sequent calculi. By studying proof-theoretic properties of these calculi we derive efficient validity- and satisfiability-checking procedures based on the inverse method. We also show how to translate the validity problem for a formula to unsatisfiability checking of a set of propositional clauses.KeywordsInference RuleTheorem ProveInverse MethodAutomate ReasoningPropositional VariableThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The principle of distributed amplification has recently been proposed as a means for obtaining amplification with conventional vacuum tubes over very great bandwidths. Unlike conventional circuits, distributed amplifiers have an attainable gain-bandwidth product which is not limited by shunt capacitance associated with the vacuum tubes and circuit wiring. This paper cites experimental results which essentially corroborate the predictions based on first-order theory. It is pointed out that when extreme bandwidths are sought, these predicted characteristics are modified by several factors which are difficult to control in actual practice. Corrective methods are available, however, which permit a limited control over these modifying effects. These methods are discussed and their applications are illustrated by measurements on actual amplifiers with pass-bands on the order of 200 to 300 Mc.

This paper offers a new practical approach toward automated commonsense reasoning with first-order logic (FOL) SUMO-based ontologies. We propose a new black-box evaluation framework for SUMO-based ontologies, which exploits the world knowledge encoded in WordNet and its mapping into SUMO. Our proposal consists of both a novel semi-automatic method for the creation of a large set of commonsense problems and a new procedure that enables its automatic evaluation by using automated theorem provers (ATPs). The application of our method enables the creation of a very large benchmark consisting of more than 15 000 problems from a small set of manually built question patterns that exploit the WordNet semantic relations. By means of the resulting benchmark, we successfully evaluate the competency of different translations of SUMO into FOL and the performance of various state-of-the-art FOL ATPs according to several quality criteria. A general analysis of our experimental results demonstrates that the proposed commonsense problems are heterogeneous and non-trivial. Furthermore, a fine-grained analysis of the experimental results obtained for a sample of our benchmark enables the detection of some mapping errors and some discrepancies between the knowledge of WordNet and SUMO. The evaluation benchmark and all the resources that have been used and developed during this work are released in a single package.

First-order logic augments the expressive power of propositional logic as it links the logical assertions to properties of objects of some non-empty universe: the domain of discourse. This is achieved by allowing the propositional symbols to take arguments that range over elements of the domain of discourse. These are now called predicate symbols and are interpreted as relations on the domain. Elements of the domain of discourse are denoted by terms built up from variables, constants, and functions applied to other terms. First-order logic also expands the lexicon of propositional logic with the quantifiers “for all” and “there exists” that are interpreted consistently with their natural language meaning.This chapter is devoted to classical first-order logic. Our presentation is similar to the one conducted for propositional logic. We first define the syntax of first-order logic, followed by its semantics. Next we define a proof system for it and present the fundamental theoretical results of soundness and completeness. We also discuss the decision problems related to this logic. The remaining sections of the chapter cover variations and extensions of first-order logic, as well as first-order theories.

It is known that one can extract Craig interpolants from so-called localproofs. An interpolant extracted from such a proof is a booleancombination of formulas occurring in the proof. However, standard completeproof systems, such as superposition, for theories having the interpolationproperty are not necessarily complete for local proofs: there are formulashaving non-local proofs but no local proof.In this paper we investigate interpolant extraction from non-local refutations(proofs of contradiction) in the superposition calculus and prove a numberof general results about interpolant extraction and complexity of extractedinterpolants. In particular, we prove that the number of quantifier alternationsin first-order interpolants of formulas without quantifier alternations isunbounded. This result has far-reaching consequences for using local proofsas a foundation for interpolating proof systems: any such proof system shoulddeal with formulas of arbitrary quantifier complexity.To search for alternatives for interpolating proof systems, we consider severalvariations on interpolation and local proofs. Namely, we give an algorithm forbuilding interpolants from resolution refutations in logic without equality anddiscuss additional constraints when this approach can be also used for logicwith equality. We finally propose a new direction related to interpolation vialocal proofs in first-order theories.

We start from the algebraic method of theorem-proving based on the translation of logic formulas into polynomials over finite fields, and adapt the case of first-order formulas by employing certain rings equipped with infinitary operations. This paper defines the notion of M-ring, a kind of polynomial ring that can be naturally associated to each first-order structure and each first-order theory, by means of generators and relations. The notion of M-ring allows us to operate with some kind of infinitary version of Boolean sums and products, in this way expressing algebraically first-order logic with a new gist. We then show how this polynomial representation of first-order sentences can be seen as a legitimate algebraic semantics for first-order logic, an alternative to cylindric and polyadic algebras and closer to the primordial forms of algebraization of logic. We suggest how the method and its generalization could be lifted successfully to n-valued logics and to other non-classical logics helping to reconcile some lost ties between algebra and logic.

We study how to efficiently combine satisfiability procedures built by using a superposition calculus with satisfiability procedures for theories, for which the superposition calculus may not apply (e.g., for various decidable fragments of Arithmetic). Our starting point is the Nelson-Oppen combination method, where satisfiability procedures cooperate by exchanging entailed (disjunction of) equalities between variables. We show that the superposition calculus deduces sufficiently many such equalities for convex theories (e.g., the theory of equality and the theory of lists) and disjunction of equalities for non-convex theories (e.g., the theory of arrays) to guarantee the completeness of the combination method. Experimental results on proof obligations extracted from the certification of auto-generated aerospace software confirm the efficiency of the approach. Finally, we show how to make satisfiability procedures built by superposition both incremental and resettable by using a hierarchic variant of the Nelson-Oppen method.

Hydraulic and tracer tests were conducted in a flow cell containing a mixture of sediments designed to mimic a two-dimensional, log-normally distributed, second-order stationary, exponentially correlated random conductivity field. With 60 integral scales in the direction of mean flow and 25 integral scales perpendicular to this direction, behavior of flow and transport in the interior of the flow cell can be compared directly with stochastic solutions for flow and transport. Using 144 piezometers and 361 platinum electrodes, the distribution of hydraulic head and the concentrations of an ionic tracer could be monitored in substantial detail. The present discussion presents the details of the experimental equipment. Results and initial analysis of hydraulic measurements and characterization of a two-dimensional tracer plume are also presented. Analysis using first-order hydraulic theory shows that the flow through the medium was consistent with an effective conductivity equal to the geometric mean of the conductivity distribution. Further, the semivariogram of head increments as observed in the experimental results was consistent with the semivariogram predicted by theory. The chemical transport experiments are here compared with the early solutions presented by Dagan (1984, 1987). The observed rate of longitudinal spread of two tracer plumes was slightly less than that predicted using this theory. Further, the spread in the transverse dimension was observed to decline from the initial plume dimensions and then remain constant or increase slightly, but at a rate lower than predicted by the theory. The difference between the hydraulic and transport results is believed to be related to the fact that the hydraulic results were averaged over a very large portion of the flow cell such that ergodic conditions could be assumed. In contrast, the initial geometry of the plume covered only approximately five integral scales in the transverse direction such that the validity of the assumption of ergodic conditions must be questioned in the analysis of results for the chemical transport.

A scaling law for K′ 0 for silicates with constant mean atomic mass

The thermodynamic properties of fluorocarbon+hydrocarbon mixtures 3. Gas-liquid critical temperatures and a comparison with the theory of Rowlinson and Sutton