Nowadays is critical to complement Artificial Intelligence (AI) systems, such as those based on Deep Learning and Generative AI, by robust and trustworthy methodologies like different useful mathematical tools, such as (fuzzy) logic, formal concept analysis, rough set theory, etc. Multi-adjoint algebras are flexible structures considered in many of these mathematical tools, which are fundamental for obtaining traceable and reliable information from data sets. Along this line, this paper aims at introducing an algebraizable logic having the quasi-variety of these algebras as its associated equivalent algebraic semantics. To do so, we first introduce a logic associated with bounded lattices with an implication and show it is algebraizable in the sense of Blok-Pigozzi. We then expand this base logic to another algebraizable logic able to properly capture the multi-adjoint framework. Furthermore, we also consider different extensions of the logics and their properties analysed.

Abstract In this work, we show that both logic programming and abstract argumentation frameworks can be interpreted in terms of Nelson’s constructive logic N4. We do so by formalising, in this logic, two principles that we call noncontradictory inference and strengthened closed world assumption: the first states that no belief can be held based on contradictory evidence while the latter forces both unknown and contradictory evidence to be regarded as false. Using these principles, both logic programming and abstract argumentation frameworks are translated into constructive logic in a modular way and using the object language. Logic programming implication and abstract argumentation supports become, in the translation, a new implication connective following the noncontradictory inference principle. Attacks are then represented by combining this new implication with strong negation. Under consideration in Theory and Practice of Logic Programming (TPLP).

Abstract Epistemic logic programs constitute an extension of the stable model semantics to deal with new constructs called subjective literals. Informally speaking, a subjective literal allows checking whether some objective literal is true in all or some stable models. As it can be imagined, the associated semantics has proved to be non-trivial, since the truth of subjective literals may interfere with the set of stable models it is supposed to query. As a consequence, no clear agreement has been reached and different semantic proposals have been made in the literature. Unfortunately, comparison among these proposals has been limited to a study of their effect on individual examples, rather than identifying general properties to be checked. In this paper, we propose an extension of the well-known splitting property for logic programs to the epistemic case. We formally define when an arbitrary semantics satisfies the epistemic splitting property and examine some of the consequences that can be derived from that, including its relation to conformant planning and to epistemic constraints. Interestingly, we prove (through counterexamples) that most of the existing approaches fail to fulfill the epistemic splitting property, except the original semantics proposed by Gelfond 1991 and a recent proposal by the authors, called Founded Autoepistemic Equilibrium Logic.

In this paper, we propose a generalisation of Dung's abstract argumentation framework that allows representing higher-order attacks and supports, that is attacks or supports whose targets are other attacks or supports. We follow the necessary interpretation of the support, based on the intuition that the acceptance of an argument requires the acceptance of each supporter. We propose semantics accounting for acceptability of arguments and validity of interactions, where the standard notion of extension is replaced by a triple of a set of arguments, a set of attacks and a set of supports. Our framework is a conservative generalisation of Argumentation Frameworks with Necessities (AFN). When supports are ignored, Argumentation Frameworks with Recursive Attacks are recovered.

Abstract In this paper, we propose a variant of Answer Set Programming (ASP) with evaluable functions that extends their application to sets of objects, something that allows a fully logical treatment of aggregates. Formally, we start from the syntax of First Order Logic with equality and the semantics of Quantified Equilibrium Logic with evaluable functions ( ${\rm QEL}^=_{\cal F}$ ). Then, we proceed to incorporate a new kind of logical term,intensional set(a construct commonly used to denote the set of objects characterised by a given formula), and to extend ${\rm QEL}^=_{\cal F}$ semantics for this new type of expression. In our extended approach, intensional sets can be arbitrarily used as predicate or function arguments or even nested inside other intensional sets, just as regular first-order logical terms. As a result, aggregates can be naturally formed by the application of some evaluable function (count,sum,maximum, etc) to a set of objects expressed as an intensional set. This approach has several advantages. First, while other semantics for aggregates depend on some syntactic transformation (either via a reduct or a formula translation), the ${\rm QEL}^=_{\cal F}$ interpretation treats them as regular evaluable functions, providing a compositional semantics and avoiding any kind of syntactic restriction. Second, aggregates can be explicitly defined now within the logical language by the simple addition of formulas that fix their meaning in terms of multiple applications of some (commutative and associative) binary operation. For instance, we can use recursive rules to definesumin terms of integer addition. Last, but not least, we prove that the semantics we obtain for aggregates coincides with the one defined by Gelfond and Zhang for the ${\cal A}\mathit{log}$ language, when we restrict to that syntactic fragment.

The standard method to retrieve information can be formally defined as follows. To ask a query, one gives the properties of the entities to be retrieved, and the answer is the set of all the entities that satisfy the query. Another method, is to ask the overall information about a given entity, and the answer is the corresponding information. An example of the first kind of query is: ‘who are the persons who have had a car accident?’, an example of the second kind is: ‘what is the overall information about a given person?’. This latter method has deserved very few researches though it has great potential practical applications. However, it raises many non-trivial issues that are investigated here. The first one is to find a precise definition of the fact that a piece of information ‘is about’ a given entity. We propose a new formal definition of this notion of aboutness for query languages in first-order logic with function symbols, and the main properties that follow from this definition are presented. The second one is to define a bridge between this abstract semantic definition and automated deduction methods based on Resolution Principle. Deduction strategies are defined in this direction and it is proved that they are complete.

In this paper we define the notion of a block algebra, which is based upon a spatial application of Allen's interval algebra. In the p‐dimensional Euclidean space, where p ≥ 1, we consider only blocks whose sides are parallel to the axes of some orthogonal basis. The block algebra consists of a set of relations (the block relations) together with the fundamental operations of composition, converse and intersection. The 13p basic relations of this algebra constitute the exhaustive list of the relations possibly holding between two blocks. We are interested in the problem of testing the consistency of a set of spatial constraints between blocks, i.e. a block network. The consistency question for block networks is NP‐complete. We first extend the notions of convexity and preconvexity to the block algebra. Similarly to the interval algebra case, convexity leads to a tractable set whereas, contrary to the interval algebra case, preconvexity leads to an intractable set. Nevertheless we characterize a tractable subset of the preconvex relations: the strongly preconvex relations. Moreover we show that strong preconvexity and ORD‐Horn representability are the same.

The Duration Calculus is an interval logic with an additional notion of duration. It became one of the main references of real-time system specification for which it was introduced. From a practical point of view an important challenge is to define automated proof procedures for this calculus. Since the propositional Duration Calculus is undecidable, in this paper we isolate a fragment and we define a mixed decision method combining standard tableau techniques with temporal constraint network resolution algorithms. This method gives a natural procedure to decide whether a given formula is satisfiable. This fragment is strong enough to embed Allen's Interval Algebra.

In this paper we present a decision procedures for modal logics with transitive models together with additionnal properties like confluence and density. These procedures are tableaux-based, and we show how to handle them in this well-known framework by generalizing usual tableaux (which are trees) to richer structures: directed acyclic graphs.

In this paper we generalize the existing tableau methods for modal logics. First of all, while usual modal tableaux are based on trees, our basic structures are rooted directed acyclic graphs (RDAG). This allows natural tableau rules for some modal logics that are difficult to capture in the usual way (such as those having an accessibility relation that is dense or confluent). Second, tableau rules rewrite patterns, which are (schemas of) parts of a RDAG. A particular case of these rules are the single-step rules recently proposed by Massacci. This allows in particular tableau rule presentations for K5, KD5, K45, KD45, and S5 that respect the subformula property. Third, we divide modal tableau rules into propagation rules and structural rules. Structural rules construct new edges and nodes (without adding formulas to nodes), while propagation rules add formulas to nodes. This distinction allows to prove completeness in a modular way.

Incidence geometry is based on two-sorted structures consisting of ‘points’ and ‘lines’ together with an intersort binary relation called incidence. We introduce an equivalent one-sorted geometrical structure, called incidence frame, which is suitable for modal considerations. Incidence frames constitute the semantical basis of MIG, the modal logic of incidence geometry. A completeness theorem for MIG is proved: a modal formula is a theorem of MIG if and only if it is valid in all incidence frames. Extensions to projective and affine geometries are also considered.

We investigate the notion of interference between formulas as a basis for change operations. Such a notion permits us to enrich conditional logics with a frame axiom. This new logic allows us to solve in a natural way some of the problems appearing in the model based approach to change. © 1994 John Wiley & Sons, Inc.