The logical reasoning skills of Large Language Models (LLMs) is poorly understood and often overstated. Current evaluation suites rely on algebraic or commonsense puzzles that mix reasoning with symbolic manipulation and/or provide static datasets that quickly saturate or leak into pretraining corpora. In purely logical terms, the most relevant reasoning skill is the meta-mathematical task of valid formula recognition, which is at the foundation of higher-level reasoning tasks (including deduction and minimization of assertions, to name just a few). In the current landscape of LLMs benchmarking, puzzles are most often stated in propositional or first-order logic, with a few exceptions for point-based temporal logic, such as LTL; yet, in the real world, event-based temporal statements are prevalent, and they are more naturally expressed in interval-based temporal logic. Interval temporal logic offers a much richer (w.r.t. point-based temporal logic, for example) variety of problems, and not only do different languages present different expressive powers, but also the computational complexity of the validity problem can vary widely. In this paper, we tackle the problem of assessing the ability of LLMs to reason about interval-based statements in the form of validity recognition. We explore whether their accuracy is sensible to the underlying language, the computational complexity of the associated validity problem, and the intrinsic hardness of the problem in terms of formula length and modal depth of the problem. We benchmark several frontier LLMs (Gemma 3 27b It, Llama 4 Maverick, DeepSeek Chat V3 release 0324, Qwen 3 32b, and Qwen 3 235b) and show that, despite apparently impressive performance on algebraic or commonsense benchmarks, they falter on logically rigorous tasks.

Machine learning from multivariate time series is a common task, and countless different approaches to typical learning problems have been proposed in recent years. In this talk, we review some basic ideas towards logic-based learning methods, and we sketch a general framework.

Multivariate time series classification is an ubiquitous and widely studied problem. Due to their strong generalization capability, neural networks are suitable for this problem, but their intrinsic black-box nature often limits their applicability. Temporal decision trees are a relevant alternative to neural networks for the same task regarding classification performances while attaining higher levels of transparency and interpretability. In this work, we approach the problem of hybridizing these two techniques, and present three independent, natural hybridization solutions to study if, and in what measure, both the ability of neural networks to capture complex temporal patterns and the transparency and flexibility of temporal decision trees can be leveraged. To this end, we provide initial experimental results for several tasks in a binary classification setting, showing that our proposed neural-symbolic hybridization schemata may be a step towards accurate and interpretable models.

Symbolic learning is the subfield of machine learning concerned with learning predictive models with knowledge represented in logical form, such as decision tree and decision list models. Ensemble learning methods, such as random forests, are usually deployed to improve the performance of decision trees; unfortunately, interpreting tree ensembles is challenging. In order to deal with unstructured (e.g., temporal or spatial) data, moreover, decision trees and random forests have been recently generalized to the use of modal logics, which are harder to interpret than their propositional counterpart. Recently, a methodology for extracting simple rules from propositional random forests, based on a sequence of optimization steps, was proposed. In this work, we generalize this approach along two directions: from propositional to modal logic and from a sequence of optimization steps to a single multi-objective optimization problem. Even if confined to the temporal domain, our experimental results, based on open-source implementations and public data, show that our method is robust and able to extract small, accurate, and informative decision lists even for complex classification problems.

The large amount of data that is produced today with new technologies is an impediment for machine learning algorithms to work correctly, both due to the memory requirements and the necessary execution times. That is why the processes of reducing both the quantity and the size of the data are increasingly important. One of these processes is the so-called instance selection. In this paper we propose three-objective constrained optimization models to formulate instance selection wrapper and filter methods (separately) for classification problems, which are solved with multi-objective evolutionary algorithms and multi-objective differential evolution. In the proposed instance selection wrapper method, an objective is added to the usual ones to minimize the generalization error of the classifier. The proposed instance selection filter method simultaneously optimizes the correlation, redundancy and consistency of the datasets. Instance retention constraints are imposed on optimization models to retain a maximum percentage of samples, established by the decision maker, in big data scenarios. The experiments have been designed to compare (1) the NSGA-II and MODE algorithms, (2) two- and three-objective optimization models, (3) two different constraint handling techniques, and (4) the proposed evolutionary approaches and other 12 non-evolutionary approaches used in literature. The proposed wrapper and filter instance selection methods have been used in a real-world business engineering application, and have also been validated using three public datasets to facilitate the replicability of the research results. The results of the experiments show the superiority of the three-objective constrained evolutionary techniques proposed in this paper over the non-evolutionary techniques and over the two-objective evolutionary approaches used in the literature.

Symbolic learning is the logic-based approach to machine learning. The mission of symbolic learning is to provide algorithms and methodologies to extract logical information from data and express it in an interpretable way. In the context of temporal data, interval temporal logic has been recently proposed as a suitable tool for symbolic learning, specifically via the design of an interval temporal logic decision tree extraction algorithm. Building on it, we study here its natural generalization to interval temporal random forests, mimicking the corresponding schema at the propositional level. Interval temporal random forests turn out to be a very performing multivariate time series classification method, which, despite the introduction of a functional component, are still logically interpretable to some extent. We apply this method to the problem of diagnosing COVID-19 based on the time series that emerge from cough and breath recording of positive versus negative subjects. Our experiment show that our models achieve very high accuracies and sensitivities, often superior to those achieved by classical methods on the same data. Although other recent approaches to the same problem (based on different and more numerous data) show even better statistical results, our solution is the first logic-based, interpretable, and explainable one.

Temporal reasoning based on intervals is nowadays ubiquitous in artificial intelligence, and the most representative interval temporal logic, called HS, was introduced by Halpern and Shoham in the eighties. There has been a great effort in the past in studying the expressive power and computational properties of the satisfiability problem for HS and its fragments, but only recently HS has been proposed as a suitable formalism for artificial intelligence applications. Such applications highlighted some of the intrinsic limits of HS: Sometimes, when dealing with real-life data one is not able to express temporal relations and propositional labels in a definite, crisp way. In this paper, following the seminal ideas of Fitting and Zadeh, among others, we present a fuzzy generalization of HS that partially solves such problems of expressive power, and we prove that, as in the crisp case, its satisfiability problem is generally undecidable.

Branching Algebra is the natural branching-time generalization of Allen’s Interval Algebra. As in the linear case, the consistency problem for Branching Algebra is NP-hard. Being relatively new, however, not much is known about the computational behaviour of the consistency problem of its sub-algebras, except in the case of the recently found subset of convex branching relations, for which the consistency of a network can be tested via path consistency and it is therefore deterministic polynomial. In this paper, following Nebel and Bürckert, we define the Horn fragment of Branching Algebra, and prove that it is a sub-algebra of the latter, being closed under inverse, intersection, and composition, that it strictly contains both the convex fragment of Branching Algebra and the Horn fragment of Interval Algebra, and that its consistency problem can be decided via path consistency. Finally, we experimentally prove that the Horn fragment of Branching Algebra can be used as an heuristic for checking the consistency of a generic network with a considerable improvement over the convex subset.

Multivariate temporal, or time, series classification is, in a way, the temporal generalization of (numeric) classification, as every instance is described by multiple time series instead of multiple values. Symbolic classification is the machine learning strategy to extract explicit knowledge from a data set, and the problem of symbolic classification of multivariate temporal series requires the design, implementation, and test of ad-hoc machine learning algorithms, such as, for example, algorithms for the extraction of temporal versions of decision trees. One of the most well-known algorithms for decision tree extraction from categorical data is Quinlan’s ID3, which was later extended to deal with numerical attributes, resulting in an algorithm known as C4.5, and implemented in many open-sources data mining libraries, including the so-called Weka, which features an implementation of C4.5 called J48. ID3 was recently generalized to deal with temporal data in form of timelines, which can be seen as discrete (categorical) versions of multivariate time series, and such a generalization, based on the interval temporal logic HS, is known as Temporal ID3. In this paper we introduce Temporal C4.5, that allows the extraction of temporal decision trees from undiscretized multivariate time series, describe its implementation, called Temporal J48, and discuss the outcome of a set of experiments with the latter on a collection of public data sets, comparing the results with those obtained by other, classical, multivariate time series classification methods.

A Conditional Simple Temporal Network with Uncertainty and Decisions (CSTNUD) is a formalism that tackles controllable and uncontrollable durations as well as controllable and uncontrollable choices simultaneously. In the classic top-down model-based engineering approach, a designer builds a CSTNUD to model, validate and execute some temporal plan of interest. Instead, in this paper, we investigate the bottom-up approach by providing a deterministic polynomial time algorithm to mine a CSTNUD from a set of execution traces (i.e., a log). This paper paves the way for the design of controllable temporal networks mined from traces that also contain information on uncontrollable events.

Abstract In this work, a data set describing phone interactions arising in a multichannel and multiskill contact centre is considered with the aim of classifying inbound sessions into those that will be eventually managed by an agent and those that, instead, will be abandoned before. More precisely, the goal of the work is to extract interpretable pieces of information that allow us to predict whether a user will or will not abandon a call, which may turn out to be very useful for the purpose of contact centre managing. To this end, the performance of two well‐known, state‐of‐the‐art evolutionary algorithms for feature selection (evolutionary nondominated radial slots based algorithm and nondominated sorted genetic algorithm) is compared for the task of feature selection, under the criteria of accuracy and cardinality of the selection, as well as for the task of fuzzy rule extraction, under the criteria of interpretability, accuracy, and hypervolume test. The best obtained fuzzy classifier, chosen after a decision making process, is validated and interpreted by domain experts.

The interpretability of classification systems refers to the ability of these to express their behavior in a way that is easily understandable by a user. Interpretable classification models allow for external validation by an expert and, in certain disciplines, such as medicine or business, providing information about decision making is essential for ethical and human reasons. Fuzzy rule based classification systems are consolidated powerful classification tools based on fuzzy logic and designed to produce interpretable models; however, in presence of a large number of attributes, even rule-based models tend to be too complex to be easily interpreted. In this paper, we propose a novel multivariate feature selection method in which both search strategy and classifier are based on multiobjective evolutionary computation. We designed a set of experiments to establish an acceptable setting with respect to the number of evaluations required by the search strategy and by the classifier. We tested our strategy on a real-life dataset and compared the results against a wide range of feature selection methods that includes filter, wrapper, multivariate, and univariate methods, with deterministic and probabilistic search strategies, and with evaluators of diverse nature. Finally, the fuzzy rule based classification model obtained with the proposed method has been evaluated with standard performance metrics and compared with other well-known fuzzy rule based classifiers. We have used two real-life datasets extracted from a contact center; in one case, with the proposed method, we obtained an accuracy of 0.7857 with eight rules, while the best fuzzy classifier compared obtained 0.7679 with eight rules, and in the second case, we obtained an accuracy of 0.7403 with five rules, while the best fuzzy classifier compared obtained 0.6364 with four rules.

Discovering association rules is a classical data mining task with a wide range of applications that include the medical, the financial, and the planning domains, among others. Modern rule extraction algorithms focus on static rules, typically expressed in the language of Horn propositional logic, as opposed to temporal ones, which have received less attention in the literature. Since in many application domains temporal information is stored in form of intervals, extracting interval-based temporal rules seems the natural choice. In this paper we extend the well-known algorithm APRIORI for rule extraction to discover interval temporal rules written in the Horn fragment of Halpern and Shoham's interval temporal logic.

Allen's Interval Algebra (IA) is one of the most prominent formalisms in the area of qualitative temporal reasoning; however, its applications are naturally restricted to linear flows of time. When dealing with nonlinear time, Allen's algebra can be extended in several ways, and, as suggested by Ragni and Wölfl [M. Ragni and S. Wölfl, 2004], a possible solution consists in defining the Branching Algebra (BA) as a set of 19 basic relations (13 basic linear relations plus 6 new basic nonlinear ones) in such a way that each basic relation between two intervals is completely defined by the relative position of the endpoints on a tree-like partial order. While the problem of deciding the consistency of a network of IA-constraints is well-studied, and every subset of the IA has been classified with respect to the tractability of its consistency problem, the fragments of the BA have received less attention. In this paper, we first define the notion of convex BA-relation, and, then, we prove that the consistency of a network of convex BA-relations can be decided via path consistency, and is therefore a polynomial problem. This is the first non-trivial tractable fragment of the BA; given the clear parallel with the linear case, our contribution poses the bases for a deeper study of fragments of BA towards their complete classification.

Evaluating in a correct, fair, systematic and reliable way the quality of the work is a central problem in modern business. Both from the psychological and the social point of view, this problem is very far away from being solved, let alone from being managed by a (semi-) automatic decision support system. In this paper we consider the case study of evaluating the operators’ work quality in a medium-sized contact center, and, in particular, the problem of selecting the correct variables to be used in such an evaluation. Starting from a data set representative of the company’s range and size of activities, that allowed no usable predictive model for evaluating the skills of the agents, we were able to devise a reproducible methodology, along with an a posteriori optimization process, to select the essential variables that should be used to objectively evaluate the quality of the agents’ work. These results may be used in a support system helping the supervisors in evaluating the agents’ performances. Moreover, we believe that our methodology may be extrapolated and reused in other comparable contexts characterized by the measurability of the human operators’ performance.

In this paper, we consider the well-known modal logics $\mathbf{K}$, $\mathbf{T}$, $\mathbf{K4}$, and $\mathbf{S4}$, and we study some of their sub-propositional fragments, namely the classical Horn fragment, the Krom fragment, the so-called core fragment, defined as the intersection of the Horn and the Krom fragments, plus their sub-fragments obtained by limiting the use of boxes and diamonds in clauses. We focus, first, on the relative expressive power of such languages: we introduce a suitable measure of expressive power, and we obtain a complex hierarchy that encompasses all fragments of the considered logics. Then, after observing the low expressive power, in particular, of the Horn fragments without diamonds, we study the computational complexity of their satisfiability problem, proving that, in general, it becomes polynomial.

There are two natural and well-studied approaches to temporal ontology and reasoning: point-based and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. A recent result by Balbiani, Goranko, and Sciavicco presented an explicit two-sorted point-interval temporal framework in which time instants (points) and time periods (intervals) are considered on a par, allowing the perspective to shift between these within the formal discourse. We consider here two-sorted first-order languages based on the same principle, and therefore including relations, as first studied by Reich, among others, between points, between intervals, and inter-sort. We give complete classifications of its sub-languages in terms of relative expressive power, thus determining how many, and which, are the intrinsically different extensions of two-sorted first-order logic with one or more such relations. This approach roots out the classical problem of whether or not points should be included in a interval-based semantics.

Clausal forms of logics are of great relevance in Artificial Intelligence, because they couple a high expressivity with a low complexity of reasoning problems. They have been studied for a wide range of classical, modal and temporal logics to obtain tractable fragments of intractable formalisms. In this paper we show that such restrictions can be exploited to lower the complexity of interval temporal logics as well. In particular, we show that for the Horn fragment of the interval logic AAbar (that is, the logic with the modal operators for Allen’s relations meets and met by) without diamonds the complexity lowers from NEXPTIME-complete to P-complete. We prove also that the tractability of the Horn fragments of interval temporal logics is lost as soon as other interval temporal operators are added to AAbar, in most of the cases.

Within the timeline-based framework, planning problems are modeled as sets of independent, but interacting, components whose behavior over time is described by a set of temporal constraints. Timeline-based planning is being used successfully in a number of complex tasks, but its theoretical properties are not so well studied. In particular, while it is known that Linear Temporal Logic (LTL) can capture classical action-based planning, a similar logical characterization was not available for timeline-based planning formalisms. This paper shows that timeline-based planning with bounded temporal constraints can be captured by a bounded version of Timed Propositional Temporal Logic, augmented with past operators, which is an extension of LTL originally designed for the verification of real-time systems. As a byproduct, we get that the proposed logic is expressive enough to capture temporal action-based planning problems.

The problem of temporal dataset evaluation consists in establishing to what extent a set of temporal data (histories) complies with a given temporal condition. It presents a strong resemblance with the problem of model checking enhanced with the ability of rating the compliance degree of a model against a formula. In this paper, we solve the temporal dataset evaluation problem by suitably combining the outcomes of model checking an interval temporal logic formula against sets of histories (finite interval models), possibly taking into account domain-dependent measures/criteria, like, for instance, sensitivity, specificity, and accuracy. From a technical point of view, the main contribution of the paper is a (deterministic) polynomial time algorithm for interval temporal logic model checking over finite interval models. To the best of our knowledge, this is the first application of a (truly) interval temporal logic model checking in the area of temporal databases and data mining rather than in the formal verification setting.

Abstract In this paper, we consider a data set taken from the administration of the Behavior Assessment System for Children test to 157 subjects, and we approach the problem of clustering and classify the subjects in an interpretable fashion. Because the Behavior Assessment System for Children test is originally composed of 149 questions (152 in the particular version used for this experiment), we first propose a feature selection wrapper model composed by a multi‐objective evolutionary algorithm, the iterative clustering method expectation–maximization, and the classifier C4.5 for the unsupervised feature selection towards the classification of the data with two objectives: maximizing the likelihood of the clustering model and maximizing the accuracy of the obtained classifier. We propose a methodology to integrate feature selection for unsupervised classification, model evaluation, decision‐making (to choose the most satisfactory model according to ana posterioriprocess in a multi‐objective context), and testing. The selected data set that is the result of this process, where each instance is labeled with its class, is then used for supervised learning via both C4.5 and a novel evolutionary computation‐based fuzzy classifier to obtain interpretable rules. We discuss and compare the behavior of two different evolutionary algorithms (ENORA (Evolutionary NOn‐dominated Radial slots based Algorithm) and NSGA‐II (Non‐dominated Sorted Genetic Algorithm)) at different levels: as search strategies for feature selection, as search strategies for fuzzy classification, and in terms of quality of the results. It turns out that ENORA behaves better in terms of quality of the result in the feature selection phase (obtaining a selection that shows higher accuracy under C4.5 after cross‐validation), and again in the fuzzy classification phase, from both points of view: hypervolume evolution and interpretability of results. During the entire process, the solutions are validated by the psychologists who collected the data.

We investigate the satisfiability problem for Horn fragments of the Halpern-Shoham interval temporal logic depending on the type (box or diamond) of the interval modal operators, the type of the underlying linear order (discrete or dense), and the type of semantics for the interval relations (reflexive or irreflexive). For example, we show that satisfiability of Horn formulas with diamonds is undecidable for any type of linear orders and semantics. On the contrary, satisfiability of Horn formulas with boxes is tractable over both discrete and dense orders under the reflexive semantics and over dense orders under the irreflexive semantics but becomes undecidable over discrete orders under the irreflexive semantics. Satisfiability of binary Horn formulas with both boxes and diamonds is always undecidable under the irreflexive semantics.

Modal logic is a paradigm for several useful and applicable formal systems in computer science, and, in particular, for temporal logics of various kinds. It generally retains the low complexity of classical propositional logic, but notable exceptions exist that present higher complexity or are even undecidable. In search of computationally well-behaved fragments, clausal forms and other sub-propositional restrictions of temporal and description logics have been recently studied. It is known that the Horn fragments of the modal logics between K and S4 are PSPACE-complete, keeping the same complexity of the the full propositional versions. In this paper, inspired by similar results in the temporal case, we sharpen the above result by showing that if we allow only box modalities in the language the Horn fragments of the modal logics between K and S4 become P-complete. Exploring the innermost reasons for the tractability of sub-Horn modal logics is a necessary condition to understand the behaviour of more expressive temporal and spatial languages under similar restrictions.

Allen's Interval Algebra is one of the most prominent formalisms in the area of qualitative temporal (and, by extension, spatial) reasoning. However, its applications are naturally restricted to linear flows of time. While there is some recent work focused on studying relations between intervals (and also between intervals and points) on branching structures, there is no rigorous study of the first-order theory of branching time. In this paper, we approach this problem under a very general definition of time structures as tree-like lattices. Allen's representation theorem shows that "meets" is expressively complete for the class of all unbounded linear orders, and it is easy to see that it is also complete for the class of all linear orders. Here we prove that, surprisingly, "meets" remains complete for the class of all unbounded tree-like lattices, and we provide an easy axiomatization of the class of all unbounded tree-like lattices in the branching language. Then, we show that "meets" becomes incomplete in the class of all tree-like lattices, we give a minimal complete set of three relations for this case along with an axiomatization, which turns out to be particularly challenging to obtain.

The chop operator C is a binary modality that plays an important role in interval temporal logics. Such an operator, which is not definable in Halpern and Shoham's modal logic of time intervals HS, allows one to split an interval into two parts and to specify what is true over them. C appears both in Moszkowski's PITL (that pairs it with a modal constant Pi which is true on all and only the intervals with coincident endpoints) and in Venema's CDT (that also features the binary modalities D and T, and Pi). Without the so-called locality principle, which restricts the semantics of proposition letters, the satisfiability problem for both PITL and CDT turns out to be undecidable over all meaningful classes of linear orders. The problem has been shown to be undecidable also for the fragment C, that is, PITL without Pi, over infinite linear orders. In this paper, we prove that the same holds for C over finite linear orders. To this end, we exploit the close relation between C and the reflexive version of the HS fragment BE, whose modalities correspond to Allen's relations starts and finishes: we prove that the satisfiability problem for reflexive BE is undecidable, undecidability of the same problem for C comes as a corollary.

Description logics of the DL-Lite family are widely used in knowledge representation because of their low computational complexity and rather good expressivity sufficient to capture important conceptual modelling constructs and the OWL2 QL profile of the Ontology Web Language (OWL). Recently, various point-based temporal extensions of DL-Lite have been investigated. Here, we propose to extend DL-Lite with fragments of Halpern and Shoham's interval logic of Allen's relations (ℋ𝒮). We formally define such extensions and show how they can be successfully used in knowledge representation. In the quest for a decidable logic, we discuss the challanges in combining decidable fragments of ℋ𝒮 with DL-Lite.

Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. Despite being relevant for a broad spectrum of application domains, ranging from temporal databases to artificial intelligence and verification of reactive systems, interval temporal logics still misses algorithms and tools capable of supporting them in an efficient way. In this paper, we approach the finite satisfiability problem for the simplest meaningful interval temporal logic (which is NEXPTIME-complete), namely A (also known as Right Propositional Neighborhood Logic), by means of a multi-objective combinatorial optimization model solved with three different multi-objective evolutionary algorithms. As a result we obtain a decision procedure that, although incomplete, turns out to be unexpectedly suitable and easy to implement with respect to classical complete algorithms. Moreover, this approach allows one to effectively search for the minimal model that satisfy a set of A-formulas without using any kind of normal form.

Interval temporal logics are temporal logics that take time intervals, instead of time instants, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham's modal logic of time intervals (HS), which has a distinct modality for each binary relation between intervals over a linear order. As HS turns out to be undecidable over most classes of linear orders, the study of HS fragments, featuring a proper subset of HS modalities, is a major item in the research agenda for interval temporal logics. A characterization of HS fragments in terms of their relative expressive power has been given for the class of all linear orders. Unfortunately, there is no easy way to directly transfer such a result to other meaningful classes of linear orders. In this paper, we provide a complete classification of the expressiveness of HS fragments over the class of (all) dense linear orders.

There are two natural and well-studied approaches to temporal ontology and reasoning, that is, point-based and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. Recently, a two-sorted point-interval temporal logic in a modal framework in which time instants (points) and time periods (intervals) are considered on a par has been presented. We consider here two-sorted first-order languages, interpreted in the class of all linear orders, based on the same principle, with relations between points, between intervals, and inter-sort. First, for those languages containing only interval-interval, and only inter-sort relations we give complete classifications of their sub-fragments in terms of relative expressive power, determining how many, and which, are the different two-sorted first-order languages with one or more such relations. Then, we consider the full two-sorted first-order logic with all the above mentioned relations, restricting ourselves to identify all expressively complete fragments and all maximal expressively incomplete fragments, and posing the basis for a forthcoming complete classification.

We compare the expressiveness of the fragments of Halpern and Shoham’s interval logic (HS), i.e., of all interval logics with modal operators associated with Allen’s relations between intervals in linear orders. We establish a complete set of interdefinability equations between these modal operators, and thus obtain a complete classification of the family of 2^12 fragments of HS with respect to their expressiveness. Using that result and a computer program, we have found that there are 1347 expressively different such interval logics over the class of all linear orders.

The introduction of Halpern and Shoham's modal logic of intervals (later on called HS) dates back to 1986. Despite its natural semantics, this logic is undecidable over all interesting classes of temporal structures. This discouraged research in this area until recently, when a number of non trivial decidable fragments have been found. This paper is a contribution toward the complete classification of HS fragments. Different combinations of Allen's interval relations begins (B), meets (A), and later (L), and their inverses A̅, B̅, and L̅, have been considered in the literature. We know from previous work that the combination ABB̅A̅ is decidable over finite linear orders and undecidable everywhere else. We extend these results by showing that ABB̅L̅ is decidable over the class of all (resp., dense, discrete) linear orders, and that it is maximal with respect to decidability over these classes: adding any other interval modality immediately leads to undecidability.

Unlike the Moon, the dark side of interval temporal logics is the one we usually see: their ubiquitous undesirability. Identifying minimal undecidable interval logics is thus a natural and important issue in the research agenda in the area. The decidability status of a logic often depends on the class of models (in our case, the class of interval structures)in which it is interpreted. In this paper, we have identified several new minimal undecidable logics amongst the fragments of Halpern-Shoham logic HS, including the logic of the overlaps relation, over the classes of all and finite linear orders, as well as the logic of the meet and subinterval relations, over the class of dense linear orders. Together with previous undecid ability results, this work contributes to delineate the border of the dark side of interval temporal logics quite sharply.

Decidability and complexity of the satisfiability problem for the logics of time intervals have been extensively studied in the last years. Even though most interval logics turnout to be undecidable, meaningful exceptions exist, such as the logics of temporal neighborhood and (some of) the logics of the subinterval relation. In this paper, we explore a different path to decidability: instead of restricting the set of modalities or imposing suitable semantic restrictions, we take the most expressive interval temporal logic studied so far, namely, Venema's CDT, and we suitably limit the nesting degree of modalities. The decidability of the satisfiability problem for the resulting CDT fragment is proved by embedding it into a well-known decidable prefix quantifier class of first-order logic, namely, the Bernays-Schonfinkel's class. In addition, we show that such a fragment is in fact NP-complete (theBernays-Schonfinkel's class is NEXPTIME-complete), and that any natural extension of it is undecidable.

Interval temporal logics formalize reasoning about interval structures over (usually) linearly ordered domains, where time intervals are the primitive ontological entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we introduce and study Metric Propositional Neighborhood Logic (MPNL) over natural numbers. MPNL features two modalities referring, respectively, to an interval that is “met by” the current one and to an interval that “meets” the current one, plus an infinite set of length constraints, regarded as atomic propositions, to constrain the lengths of intervals. We argue that MPNL can be successfully used in different areas of artificial intelligence to combine qualitative and quantitative interval temporal reasoning, thus providing a viable alternative to well-established logical frameworks such as Duration Calculus. We show that MPNL is decidable in double exponential time and expressively complete with respect to a well-defined subfragment of the two-variable fragment FO2[N, =, <, s] of first-order logic for linear orders with successor function, interpreted over natural numbers. Moreover, we show that MPNL can be extended in a natural way to cover full FO2[N, =, <, s], but, unexpectedly, the latter (and hence the former) turns out to be undecidable.

In this paper, we focus our attention on the interval temporal logic of the Allen's relations ``meets'', ``begins'', and ``begun by'' ($\ABB$ for short), interpreted over natural numbers. We first introduce the logic and we show that it is expressive enough to model distinctive interval properties, such as accomplishment conditions, to capture basic modalities of point-based temporal logic, such as the until operator, and to encode relevant metric constraints. Then, we prove that the satisfiability problem for $\ABB$ over natural numbers is decidable by providing a small model theorem based on an original contraction method. Finally, we prove the EXPSPACE-completeness of the problem.

Temporal reasoning plays an important role in artificial intelligence. Temporal logics provide a natural framework for its formalization and implementation. A standard way of enhancing the expressive power of temporal logics is to replace their unidimensional domain by a multidimensional one. In particular, such a dimensional increase can be exploited to obtain spatial counterparts of temporal logics. Unfortunately, it often involves a blow up in complexity, possibly losing decidability. In this paper, we propose a spatial generalization of the decidable metric interval temporal logic RPNL+INT, called Directional Area Calculus (DAC). DAC features two modalities, that respectively capture (possibly empty) rectangles to the north and to the east of the current one, and metric operators, to constrain the size of the current rectangle. We prove the decidability of the satisfiability problem for DAC, when interpreted over frames built on natural numbers, and we analyze its complexity. In addition, we consider a weakened version of DAC, called WDAC, which is expressive enough to capture meaningful qualitative and quantitative spatial properties and computationally better.

Interval temporal logics are based on interval structures over linearly (or partially) ordered domains, where time intervals, rather than time instants, are the primitive ontological entities. In this paper we introduce and study Right Propositional Neighborhood Logic over natural numbers with integer constraints for interval lengths, which is a propositional interval temporal logic featuring a modality for the 'right neighborhood' relation between intervals and explicit integer constraints for interval lengths. We prove that it has the bounded model property with respect to ultimately periodic models and is therefore decidable. In addition, we provide an EXP SPACE procedure for satisfiability checking and we prove EXPSPACE-hardness by a reduction from the exponential corridor tiling problem.

We investigate fragments of Halpern-Shoham's interval logic HS involving the modal operators for the relations of left or right overlap of intervals. We prove that most of these fragments are undecidable, by employing a non-trivial reduction from the octant tiling problem.

Interval-based temporal logics are natural frameworks for modeling a number of problems from various areas of computer science such as artificial intelligence, natural language processing, temporal databases and formal specification. Quite a few interval-based temporal logics became popular in recent years, such as Venema's CDT logic, Halpern and Shoham's HS logic, Moszkowski's ITL and its prepositional version, and Goranko, Montanari, and Sciavicco 's PNL. In this work we introduce a new prepositional interval-based temporal logic called CW, which can be considered an extension of the prepositional fragment of Moszkowski's ITL evaluated over different (parallel) lines, and which is particularly adapt for expressing natural language sentences. We study the logic CW and develop a (non-terminating) sound and complete deduction system based on tableaux for it.

It is widely accepted that spatial reasoning plays a central role in artificial intelligence, for it has a wide variety of potential applications, e.g., in robotics, geographical information systems, medical analysis and diagnosis. As noticed by many authors, spatial and temporal reasoning have a close connection. In this paper we propose a new, semi-decidable, modal logic for spatial reasoning through directional relations, which is able to express meaningful spatial statements. Spatial propositional neighborhood logic can be polynomially reduced to a decidable temporal logic based on time intervals preserving, at least, valid formulas. Thanks to such a reduction, we are able to reuse a sound and complete tableaux method in order to reason with spatial propositional neighborhood logic; to the best of our knowledge, there are practically no previous attempts of devising automatic reasoning methods for spatial reasoning

In this paper, we study the definability and decidability of binary predicates for time granularity with respect to monadic theories over finitely and infinitely layered structures. We focus our attention on the equi-level (resp. equi-column) predicate constraining two time points to belong to the same layer (resp. column) and on the horizontal (resp. vertical) successor predicate relating a time point to its successor within a given layer (resp. column). We give a number of positive and negative results by reduction to/from a wide spectrum of decidable/undecidable problems.