A Decision Procedure for Herbrand Formulas without Skolemization

This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with A -||- A & A (= &I) as the sole rule that increases the complexity of given FOLDNFs. The described algorithm illustrates how, in the case of Herbrand formulae, decision problems can be solved through a systematic search for proofs that reduce the number of applications of the rule &I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying &I. Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a &I-minimal proof strategy in the case of a general search for proofs of contradiction

Proceedings of the Joint Automated Reasoning Workshop and Deduktionstreffen: As part of the Vienna Summer of Logic – IJCAR 23-24 July 2014

Preface For many years the British and the German automated reasoning communities have successfully run independent series of workshops for anybody working in the area of automated reasoning. Although open to the general public they addressed in the past primarily the British and the German communities, respectively. At the occasion of the Vienna Summer of Logic the two series have a joint event in Vienna as an IJCAR workshop. In the spirit of the two series there will be only informal proceedings with abstracts of the works presented. These are collected in this document. We have tried to maintain the informal open atmosphere of the two series and have welcomed in particular research students to present their work. We have solicited for all work related to automated reasoning and its applications with a particular interest in work-in-progress and the presentation of half-baked ideas. As in the previous years, we have aimed to bring together researchers from all areas of automated reasoning in order to foster links among researchers from various disciplines; among theoreticians, implementers and users alike, and among international communities, this year not just the British and German communities

Combining Enumeration and Deductive Techniques in order to Increase the Class of Constructible Infinite Models

AbstractA new method for building infinite models for first-order formulae is presented. The method combines enumeration techniques with existing deductive (in a broad sense) ones. Its soundness and completeness w.r.t. the class of models that can be represented by equational constraints are proven. This shows that the use of enumeration techniques strictly increases the power of existing methods for building Herbrand models that are not complete in this sense. Some strategies are proposed to reduce the search space. We give examples and show how to use this approach for building interactively a model of a formula introduced by Goldfarb in his proof of the undecidability of the Gödel class with identity. This formula is satisfiable but has no finite model

On Generalizing Decidable Standard Prefix Classes of First-Order Logic

Recently, the separated fragment (SF) of first-order logic has been introduced. Its defining principle is that universally and existentially quantified variables may not occur together in atoms. SF properly generalizes both the Bernays-Sch\"onfinkel-Ramsey (BSR) fragment and the relational monadic fragment. In this paper the restrictions on variable occurrences in SF sentences are relaxed such that universally and existentially quantified variables may occur together in the same atom under certain conditions. Still, satisfiability can be decided. This result is established in two ways: firstly, by an effective equivalence-preserving translation into the BSR fragment, and, secondly, by a model-theoretic argument. Slight modifications to the described concepts facilitate the definition of other decidable classes of first-order sentences. The paper presents a second fragment which is novel, has a decidable satisfiability problem, and properly contains the Ackermann fragment and---once more---the relational monadic fragment. The definition is again characterized by restrictions on the occurrences of variables in atoms. More precisely, after certain transformations, Skolemization yields only unary functions and constants, and every atom contains at most one universally quantified variable. An effective satisfiability-preserving translation into the monadic fragment is devised and employed to prove decidability of the associated satisfiability problem.Comment: 34 page

Semantically-guided goal-sensitive reasoning: decision procedures and the Koala prover

The main topic of this article are SGGS decision procedures for fragments of first-order logic without equality. SGGS (Semantically-Guided Goal-Sensitive reasoning) is an attractive basis for decision procedures, because it generalizes to first-order logic the Conflict-Driven Clause Learning (CDCL) procedure for propositional satisfiability. As SGGS is both refutationally complete and model-complete in the limit, SGGS decision procedures are model-constructing. We investigate the termination of SGGS with both positive and negative results: for example, SGGS decides Datalog and the stratified fragment (including Effectively PRopositional logic) that are relevant to many applications. Then we discover several new decidable fragments, by showing that SGGS decides them. These fragments have the small model property, as the cardinality of their SGGS-generated models can be upper bounded, and for most of them termination tools can be applied to test a set of clauses for membership. We also present the first implementation of SGGS - the Koala theorem prover - and we report on experiments with Koala