Acta Informatica manuscript No. (will be inserted by the editor) A Complete Classification of the Expressiveness of Interval Logics of Allen’s Relations The General and the Dense Cases

Abstract Interval temporal logics 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 associates a modal operator with each binary relation between intervals over a linear order (the so-called Allen’s interval relations). In this paper, we compare and classify the expressiveness of all fragments of HS on the class of all linear orders and on the subclass of all dense linear orders. For each of these classes, we identify a complete set of definabilities between HS modalities, valid in that class, thus obtaining a complete classification of the family of all 4096 fragments of HS with respect to their expressiveness. We show that on the class of all linear orders there are exactly 1347 expressively different fragments of HS, while on the class of dense linear orders there are exactly 966 such expressively different fragments

Temporal Information in Data Science: An Integrated Framework and its Applications

Data science is a well-known buzzword, that is in fact composed of two distinct keywords, i.e., data and science. Data itself is of great importance: each analysis task begins from a set of examples. Based on such a consideration, the present work starts with the analysis of a real case scenario, by considering the development of a data warehouse-based decision support system for an Italian contact center company. Then, relying on the information collected in the developed system, a set of machine learning-based analysis tasks have been developed to answer specific business questions, such as employee work anomaly detection and automatic call classification. Although such initial applications rely on already available algorithms, as we shall see, some clever analysis workflows had also to be developed. Afterwards, continuously driven by real data and real world applications, we turned ourselves to the question of how to handle temporal information within classical decision tree models. Our research brought us the development of J48SS, a decision tree induction algorithm based on Quinlan's C4.5 learner, which is capable of dealing with temporal (e.g., sequential and time series) as well as atemporal (such as numerical and categorical) data during the same execution cycle. The decision tree has been applied into some real world analysis tasks, proving its worthiness. A key characteristic of J48SS is its interpretability, an aspect that we specifically addressed through the study of an evolutionary-based decision tree pruning technique. Next, since a lot of work concerning the management of temporal information has already been done in automated reasoning and formal verification fields, a natural direction in which to proceed was that of investigating how such solutions may be combined with machine learning, following two main tracks. First, we show, through the development of an enriched decision tree capable of encoding temporal information by means of interval temporal logic formulas, how a machine learning algorithm can successfully exploit temporal logic to perform data analysis. Then, we focus on the opposite direction, i.e., that of employing machine learning techniques to generate temporal logic formulas, considering a natural language processing scenario. Finally, as a conclusive development, the architecture of a system is proposed, in which formal methods and machine learning techniques are seamlessly combined to perform anomaly detection and predictive maintenance tasks. Such an integration represents an original, thrilling research direction that may open up new ways of dealing with complex, real-world problems.Data science is a well-known buzzword, that is in fact composed of two distinct keywords, i.e., data and science. Data itself is of great importance: each analysis task begins from a set of examples. Based on such a consideration, the present work starts with the analysis of a real case scenario, by considering the development of a data warehouse-based decision support system for an Italian contact center company. Then, relying on the information collected in the developed system, a set of machine learning-based analysis tasks have been developed to answer specific business questions, such as employee work anomaly detection and automatic call classification. Although such initial applications rely on already available algorithms, as we shall see, some clever analysis workflows had also to be developed. Afterwards, continuously driven by real data and real world applications, we turned ourselves to the question of how to handle temporal information within classical decision tree models. Our research brought us the development of J48SS, a decision tree induction algorithm based on Quinlan's C4.5 learner, which is capable of dealing with temporal (e.g., sequential and time series) as well as atemporal (such as numerical and categorical) data during the same execution cycle. The decision tree has been applied into some real world analysis tasks, proving its worthiness. A key characteristic of J48SS is its interpretability, an aspect that we specifically addressed through the study of an evolutionary-based decision tree pruning technique. Next, since a lot of work concerning the management of temporal information has already been done in automated reasoning and formal verification fields, a natural direction in which to proceed was that of investigating how such solutions may be combined with machine learning, following two main tracks. First, we show, through the development of an enriched decision tree capable of encoding temporal information by means of interval temporal logic formulas, how a machine learning algorithm can successfully exploit temporal logic to perform data analysis. Then, we focus on the opposite direction, i.e., that of employing machine learning techniques to generate temporal logic formulas, considering a natural language processing scenario. Finally, as a conclusive development, the architecture of a system is proposed, in which formal methods and machine learning techniques are seamlessly combined to perform anomaly detection and predictive maintenance tasks. Such an integration represents an original, thrilling research direction that may open up new ways of dealing with complex, real-world problems

On Begins, Meets and Before

Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous temporal logic for intervals studied so far is probably Halpern and Shoham’s HS, which is the logic of the thirteen Allen’s interval relations. Unfortunately, HS and most of its fragments are undecidable. This discouraged the research in this area until recently, when a number non-trivial decidable ITLs have been discovered. This paper is a contribution towards the complete classification of all different fragments of HS. We consider here different combinations of the interval relations begins(B), meets (A), later(L) and their inverses (Abar, Bbar, and Lbar). We know from previous work that the combination ABBbarAbar is decidable only when finite domains are considered (and undecidable elsewhere), and that ABBbar is decidable over the natural numbers. In the present paper we show that, over strongly discrete linear models (e.g. finite orders, the naturals, the integers), decidability of ABBbar can be further extended to capture the language ABBbarLbar, which lies strictly in between ABBbar and ABBbarAbar. This logic turns out to be maximal w.r.t decidability over the considered classes, and its satisfiability problem is EXPSPACE-complete. In this paper we also provide an optimal non-deterministic decision procedure, and we show that the language is powerful enough to polynomially encode metric constraints on the length of the current interval