Saturation-based decision procedures for extensions of the guarded fragment

We apply the framework of Bachmair and Ganzinger for saturation-based theorem proving to derive a range of decision procedures for logical formalisms, starting with a simple terminological language EL, which allows for conjunction and existential restrictions only, and ending with extensions of the guarded fragment with equality, constants, functionality, number restrictions and compositional axioms of form S ◦ T ⊆ H. Our procedures are derived in a uniform way using standard saturation-based calculi enhanced with simplification rules based on the general notion of redundancy. We argue that such decision procedures can be applied for reasoning in expressive description logics, where they have certain advantages over traditionally used tableau procedures, such as optimal worst-case complexity and direct correctness proofs.Wir wenden das Framework von Bachmair und Ganzinger für saturierungsbasiertes Theorembeweisen an, um eine Reihe von Entscheidungsverfahren für logische Formalismen abzuleiten, angefangen von einer simplen terminologischen Sprache EL, die nur Konjunktionen und existentielle Restriktionen erlaubt, bis zu Erweiterungen des Guarded Fragment mit Gleichheit, Konstanten, Funktionalität, Zahlenrestriktionen und Kompositionsaxiomen der Form S ◦ T ⊆ H. Unsere Verfahren sind einheitlich abgeleitet unter Benutzung herkömmlicher saturierungsbasierter Kalküle, verbessert durch Simplifikationsregeln, die auf dem Konzept der Redundanz basieren. Wir argumentieren, daß solche Entscheidungsprozeduren für das Beweisen in ausdrucksvollen Beschreibungslogiken angewendet werden können, wo sie gewisse Vorteile gegenüber traditionell benutzten Tableauverfahren besitzen, wie z.B. optimale worst-case Komplexität und direkte Korrektheitsbeweise

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Disproving in First-Order Logic with Definitions, Arithmetic and Finite Domains

This thesis explores several methods which enable a first-order reasoner to conclude satisfiability of a formula modulo an arithmetic theory. The most general method requires restricting certain quantifiers to range over finite sets; such assumptions are common in the software verification setting. In addition, the use of first-order reasoning allows for an implicit representation of those finite sets, which can avoid scalability problems that affect other quantified reasoning methods. These new techniques form a useful complement to existing methods that are primarily aimed at proving validity. The Superposition calculus for hierarchic theory combinations provides a basis for reasoning modulo theories in a first-order setting. The recent account of ‘weak abstraction’ and related improvements make an mplementation of the calculus practical. Also, for several logical theories of interest Superposition is an effective decision procedure for the quantifier free fragment. The first contribution is an implementation of that calculus (Beagle), including an optimized implementation of Cooper’s algorithm for quantifier elimination in the theory of linear integer arithmetic. This includes a novel means of extracting values for quantified variables in satisfiable integer problems. Beagle won an efficiency award at CADE Automated theorem prover System Competition (CASC)-J7, and won the arithmetic non-theorem category at CASC-25. This implementation is the start point for solving the ‘disproving with theories’ problem. Some hypotheses can be disproved by showing that, together with axioms the hypothesis is unsatisfiable. Often this is relative to other axioms that enrich a base theory by defining new functions. In that case, the disproof is contingent on the satisfiability of the enrichment. Satisfiability in this context is undecidable. Instead, general characterizations of definition formulas, which do not alter the satisfiability status of the main axioms, are given. These general criteria apply to recursive definitions, definitions over lists, and to arrays. This allows proving some non-theorems which are otherwise intractable, and justifies similar disproofs of non-linear arithmetic formulas. When the hypothesis is contingently true, disproof requires proving existence of a model. If the Superposition calculus saturates a clause set, then a model exists, but only when the clause set satisfies a completeness criterion. This requires each instance of an uninterpreted, theory-sorted term to have a definition in terms of theory symbols. The second contribution is a procedure that creates such definitions, given that a subset of quantifiers range over finite sets. Definitions are produced in a counter-example driven way via a sequence of over and under approximations to the clause set. Two descriptions of the method are given: the first uses the component solver modularly, but has an inefficient counter-example heuristic. The second is more general, correcting many of the inefficiencies of the first, yet it requires tracking clauses through a proof. This latter method is shown to apply also to lists and to problems with unbounded quantifiers. Together, these tools give new ways for applying successful first-order reasoning methods to problems involving interpreted theories

Logical Foundations of Object-Oriented and Frame-Based Languages

We propose a novel logic, called Frame Logic (abbr., F-logic), that accounts in a clean, declarative fashion for most of the structural aspects of object-oriented and frame-based languages. These features include object identity, complex objects, inheritance, polymorphic types, methods, encapsulation, and others. In a sense, F-logic stands in the same relationship to the object-oriented paradigm as classical predicate calculus stands to relational programming. The syntax of F-logic is higher-order, which, among other things, allows the user to explore data and schema using the same declarative language. F-logic has a model-theoretic semantics and a sound and complete resolution-based proof procedure. This paper also discusses various aspects of programming in declarative object-oriented languages based on F-logic