Saturation-based Boolean conjunctive query answering and rewriting for the guarded quantification fragments

Query answering is an important problem in AI, database and knowledge representation. In this paper, we develop saturation-based Boolean conjunctive query answering and rewriting procedures for the guarded, the loosely guarded and the clique-guarded fragments. Our query answering procedure improves existing resolution-based decision procedures for the guarded and the loosely guarded fragments and this procedure solves Boolean conjunctive query answering problems for the guarded, the loosely guarded and the clique-guarded fragments. Based on this query answering procedure, we also introduce a novel saturation-based query rewriting procedure for these guarded fragments. Unlike mainstream query answering and rewriting methods, our procedures derive a compact and reusable saturation, namely a closure of formulas, to handle the challenge of querying for distributed datasets. This paper lays the theoretical foundations for the first automated deduction decision procedures for Boolean conjunctive query answering and the first saturation-based Boolean conjunctive query rewriting in the guarded, the loosely guarded and the clique-guarded fragments

Proceedings of the Automated Reasoning Workshop (ARW 2019)

Preface This volume contains the proceedings of ARW 2019, the twenty sixths Workshop on Automated Rea- soning (2nd{3d September 2019) hosted by the Department of Computer Science, Middlesex University, England (UK). Traditionally, this annual workshop which brings together, for a two-day intensive pro- gramme, researchers from different areas of automated reasoning, covers both traditional and emerging topics, disseminates achieved results or work in progress. During informal discussions at workshop ses- sions, the attendees, whether they are established in the Automated Reasoning community or are only at their early stages of their research career, gain invaluable feedback from colleagues. ARW always looks at the ways of strengthening links between academia, industry and government; between theoretical and practical advances. The 26th ARW is affiliated with TABLEAUX 2019 conference. These proceedings contain forteen extended abstracts contributed by the participants of the workshop and assembled in order of their presentations at the workshop. The abstracts cover a wide range of topics including the development of reasoning techniques for Agents, Model-Checking, Proof Search for classical and non-classical logics, Description Logics, development of Intelligent Prediction Models, application of Machine Learning to theorem proving, applications of AR in Cloud Computing and Networking. I would like to thank the members of the ARW Organising Committee for their advice and assis- tance. I would also like to thank the organisers of TABLEAUX/FroCoS 2019, and Andrei Popescu, the TABLEAUX Conference Chair, in particular, for the enormous work related to the organisation of this affiliation. I would also like to thank Natalia Yerashenia for helping in preparing these proceedings. London Alexander Bolotov September 201

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

Inductive logic programming as satisfiability modulo theories

At the intersection of machine learning, program synthesis and automated reasoning lies the field of Inductive Logic Programming (ILP). The aim of ILP is to automatically learn relational programs from input/output examples, typically through logic-based techniques. Inspired by Karl Popper’s falsification perspective on science, this dissertation sets out a new approach to ILP: Learning From Failures (LFF). In science, starting from a huge set of a priori viable hypotheses, we select a hypothesis to test. This hypothesis typically gets falsified due to failing in some specific way. By examining the failure we learn that an entire space of related hypotheses is now ruled out. Having thus reduced our set of viable hypotheses, we subsequently select from just those that remain. LFF applies this methodology to program induction, codifying it as a three-stage loop. The generate stage maintains a formula whose satisfying assignments correspond to the set of viable hypotheses. The test stage takes a satisfying assignment, interprets it as a logic program and tests it against training examples – imperfect fit is considered a failure. The constrain stage turns a failure into constraints to add to the generate stage’s formula, thereby eliminating a class of hypotheses which will fail for the same reason. The thesis of this dissertation is three-fold. The first claim is that LFF frames the ILP problem as one of Satisfiability Modulo Theories (SMT). With the search for viable hypotheses handed-off to a SAT-solver, the test stage can be regarded as a theory solver collaborating with the SAT-solver, effectively making ILP’s notion of background knowledge into a (Horn) background theory. The second claim is that LFF’s three-stage loop is an effective basis for falsification-based program induction. Chapter 4 develops the above correspondence into a feature-rich and flexible three-stage ILP system. Experimental evidence is provided for this system going beyond the state-of-the-art in ILP, e.g., by supporting large hypothesis spaces and large domains. The third claim is that the LFF-as-SMT-perspective helps apply satisfiability solving techniques to ILP, in particular to reduce hypothesis space exploration. In Chapter 5, we show that SMT-based techniques for explaining conflicts have a natural analog in terms of explaining which parts of a hypothesis are responsible for its failure. In Chapter 6, we incorporate other theory solvers alongside the test stage to reason about the (satisfiability of) over-approximating properties of hypotheses. We show that both of these techniques can significantly reduce the number of iterations of the three-stage loop

Automated Theorem Proving with Extensions of First-Order Logic

Automated theorem provers are computer programs that check whether a logical conjecture follows from a set of logical statements. The conjecture and the statements are expressed in the language of some formal logic, such as first-order logic. Theorem provers for first-order logic have been used for automation in proof assistants, verification of programs, static analysis of networks, and other purposes. However, the efficient usage of these provers remains challenging. One of the challenges is the complexity of translating domain problems to first-order logic. Not only can such translation be cumbersome due to semantic differences between the domain and the logic, but it might inadvertently result in problems that provers cannot easily handle.The work presented in the thesis addresses this challenge by developing an extension of first-order logic named FOOL. FOOL contains syntactical features of programming languages and more expressive logics, is friendly for translation of problems from various domains, and can be efficiently supported by existing theorem provers. We describe the syntax and semantics of FOOL and present a simple translation from FOOL to plain first-order logic. We describe an efficient clausal normal form transformation algorithm for FOOL and based on it implement a support for FOOL in the Vampire theorem prover. We illustrate the efficient use of FOOL for program verification by describing a concise encoding of next state relations of imperative programs in FOOL. We show a usage of features of FOOL in problems of static analysis of networks. We demonstrate the efficiency of automated theorem proving in FOOL with an extensive set of experiments. In these experiments we compare the performance of Vampire on a large collection of problems from various sources translated to FOOL and ordinary first-order logic. Finally, we fix the syntax for FOOL in TPTP, the standard language of first-order theorem provers

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