Large language models (LLMs) have achieved significant performance in various natural language reasoning tasks. However, they still struggle with performing first-order logic reasoning over formal logical theories expressed in natural language. This is because the previous LLMs-based reasoning systems have the theoretical incompleteness issue. As a result, it can only address a limited set of simple reasoning problems, which significantly decreases their generalization ability. To address this issue, we propose a novel framework, named Generalizable and Faithful Reasoner (GFaiR), which introduces the paradigm of resolution refutation. Resolution refutation has the capability to solve all first-order logic reasoning problems by extending reasoning rules and employing the principle of proof by contradiction, so our system's completeness can be improved by introducing resolution refutation. Experimental results demonstrate that our system outperforms previous works by achieving state-of-the-art performances in complex scenarios while maintaining performances in simple scenarios. Besides, we observe that GFaiR is faithful to its reasoning process.

Current Max-SAT solvers are able to efficiently compute the optimal value of an input instance but they do not provide any certificate of its validity. In this paper, we present a tool, called MS-Builder, which generates certificates for the Max-SAT problem in the particular form of a sequence of equivalence-preserving transformations. To generate a certificate, MS-Builder iteratively calls a SAT oracle to get a SAT resolution refutation which is handled and adapted into a sound refutation for Max-SAT. In particular, we prove that the size of the computed Max-SAT refutation is linear with respect to the size of the initial refutation if it is semi-read-once, tree-like regular, tree-like or semi-tree-like. Additionally, we propose an extendable tool, called MS-Checker, able to verify the validity of any Max-SAT certificate using Max-SAT inference rules. Both tools are evaluated on the unweighted and weighted benchmark instances of the 2020 Max-SAT Evaluation.

Ontological query answering is the problem of answering queries in the presence of schema constraints representing the domain of interest. Datalog+/- is a common family of languages for schema constraints, including tuple-generating dependencies (TGDs) and equality-generating dependencies (EGDs). The interplay of TGDs and EGDs leads to undecidability or intractability of query answering when adding EGDs to tractable Datalog+/- fragments, like Warded Datalog+/-, for which, in the sole presence of TGDs, query answering is PTIME in data complexity. There have been attempts to limit the interaction of TGDs and EGDs and guarantee tractability, in particular with the introduction of separable EGDs, to make EGDs irrelevant for query answering as long as the set of constraints is satisfied. While being tractable, separable EGDs have limited expressive power. We propose a more general class of EGDs, which we call ``harmless'', that subsume separable EGDs and allow to model a much broader class of problems. Unlike separable EGDs, harmless EGDs, besides enforcing ground equality constraints, specialize the query answer by grounding or renaming the labelled nulls introduced by existential quantification in the TGDs. Harmless EGDs capture the cases when the answer obtained in the presence of EGDs is less general than the one obtained with TGDs only. We conclude that the theoretical problem of deciding whether a set of constraints contains harmless EGDs is undecidable. We contribute a sufficient syntactic condition characterizing harmless EGDs, broad and useful in practice. We focus on Warded Datalog+/- with harmless EGDs and argue that, in such fragment, query answering is decidable and PTIME in data complexity. We study chase-based techniques for query answering in Warded Datalog+/- with harmless EGDs, conducive to an efficient algorithm to be implemented in state-of-the-art reasoners.

This paper proposes the integration of the resolution rule for Max-SAT with unsatisfiability-based Max-SAT solvers. First, we show that the resolution rule for Max-SAT can be safely applied as dictated by the resolution proof associated with an unsatisfiable core when such proof is read-once, that is, each clause is used at most once in the resolution process. Second, we study how this property can be integrated in an unsatisfiability-based solver. In particular, the resolution rule for Max-SAT is applied to read-once proofs or to read-once subparts of a general proof. Finally, we perform an empirical investigation on structured instances from recent Max-SAT evaluations. Preliminary results show that the use of read-once resolution substantially improves the performance of the solver.

INRA Toulouse, France Abstract Weighted Max-SA T is the optimization version of SA T and many important problems can be naturally encoded as such. Solving weighted Max-SA T is an important problem from both a theoretical and a practical point of view. In recent ye ars, there has been considerable interest in finding efficient solving techniques. Most of thi s work focus on the computation of good quality lower bounds to be used within a branch and bou nd DPLL-like algorithm. Most often, these lower bounds are described in a procedural way. Because of that, it is difficult to realize the logic that is behind. In this paper we introduce an original framework for Max-SA T that stresses the parallelism with classical SA T. Then, we extend the two basic SA T s olving techniques: search and inference. We show that many algorithmic tricks used in state-of-the-art Max-SA T solvers are easily expressable in logic terms with our framework in a unified manner. Besides, we introduce an original search algorithm that per forms a restricted amount of weighted resolution at each visited node. We empirically compare our algorithm w ith a variety of solving alternatives on several benchmarks. Our experiments, which constitute to the best of our knowledge the most comprehensive Max-sat eva luation ever reported, show that our algorithm is generally orders of magnitude faster t han any competitor. Preprint submitted to Elsevier Science 11 September 2018 1 Introduction Weighted Max-SA T is the optimization version of the SA T prob lem and many important problems can be naturally expressed as such. In recent years, there has been a considerable effort in finding efficient exact algorithms. A common drawback of all these alg orithms is that albeit the close relationship between SA T and Max-SA T, they cannot be easily described with logic terminology. For instance, the contributions of [11,12,13,14] are good quality lower bounds to be incorporated into a depth-first branch and bound procedure. These lower bounds are mostly defined in a procedural way and it is very difficult to see the logic that is behind the execution of the procedure. This is in contrast with SA T algorithms where the solving process can b e easily decomposed into atomic logical steps. In this paper we introduce an original framework for (weight ed) Max-SA T in which the notions of upper and lower bound are incorporated into the problem definition. Under this framework classical SA T is just a particular case of Max-SA T, and the main SA T solving techniques can be naturally extended. In pa rticular, we extend the basic simplification rules (for example, idempotency, absorption, unit clause reduction, etc) and introduce a new one, hardening, that does not make sense in the SA T context.

This paper considers the problem of expressing predicate calculus in connectionist networksthat are based on energy minimization. Given a firstorder-logic knowledgebase and a bound k, a symmetric network is constructed (like a Boltzman machine or a Hopfield network) that searches for a proof for a given query. If a resolution-based proof of length no longer than k exists, then the global minima of the energy function that is associated with the network represent such proofs. The network that is generated is of size cubic in the bound k and linear in the knowledge size. There are no restrictions on the type of logic formulas that can be represented.

This paper considers the problem of expressing predicate calculus in connectionist networks that are based on energy minimization. Given a firstorder-logic knowledge base and a bound k, a symmetric network is constructed (like a Boltzman machine or a Hopfield network) that searches for a proof for a given query. If a resolution-based proof of length no longer than k exists, then the global minima of the energy function that is associated with the network represent such proofs. The network that is generated is of size cubic in the bound k and linear in the knowledge size. There are no restrictions on the type of logic formulas that can be represented.

This paper considers the problem of expressing predicate calculus in connectionist networks that are based on energy minimization. Given a firstorder-logic knowledge base and a bound k, a symmetric network is constructed (like a Boltzman machine or a Hopfield network) that searches for a proof for a given query. If a resolution-based proof of length no longer than k exists, then the global minima of the energy function that is associated with the network represent such proofs. The network that is generated is of size cubic in the bound k and linear in the knowledge size. There are no restrictions on the type of logic formulas that can be represented.