Time series distance measures are fundamental in numerous domains, including finance, healthcare, and signal processing, enabling crucial tasks such as pattern recognition, anomaly detection, and predictive modeling. However, many applications require computing distances between all pairs of time series in large datasets, a computationally intensive task that can become a significant bottleneck in analysis pipelines. The tsdistances library is a high-performance Python package designed for computing distances between time series, with GPU support for accelerated processing. This article introduces tsdistances and its key features, focusing on the implementation of elastic distance algorithms and their optimizations. We present both CPU and GPU implementations, highlighting the use of dynamic programming techniques and GPU-specific optimizations such as warp-based parallelization. The performance of tsdistances is compared with existing alternatives in the literature, demonstrating significant speed improvements, especially for large-scale time series analysis tasks.

Abstract This paper presents , a novel distance between time series based on Random Forests (RFs). We extend to the time-series domain concepts and tools of RF distances, a recent class of robust data-dependent distances defined for vectorial representations, thus proposing the first RF distance for time series. The distance is determined by (i) creating an RF to model a set of time series, and (ii) exploiting the trained RF to quantify the similarity between time series. As for the first step, we introduce in this paper the Extremely Randomized Canonical Interval Forest (ERCIF), a novel extension of Canonical Interval Forests that can model time series and can be trained without labels. We then exploit three different schemes, following ideas already employed in the vectorial case. The proposed distance, in different variants, has been thoroughly evaluated with 128 datasets from the archive, showing promising results compared with literature alternatives.

Business Process Compliance is a family of methods to evaluate Business Processes in terms of the existence of one execution (one trace) that does not violate constraints superimposed on the process itself. The dual version is formulated as the superimposition of a set of constraints and consequent evaluation of the process for all the executions . These problems are relevant to a large part of actual applications, especially those in the context of regulatory compliance where we aim at verifying the process against a normative background (including, for instance, soft ones, such as guidelines, product specification, and product standards) or goals fixed by the owner of the process. In this paper we discuss one new type of compliance, that is impact compliance , devised to verify when a process respects a set of constraints, to establish that certain amounts, measuring the undesired effects of the tasks executed to implement the process, are below given limits . In the current literature on Business Process Management, Business Process Analysis, and Business Process Compliance, this type of compliance checking process has not yet been addressed. As we demonstrate in this paper, this problem is significant and complex to address. In particular, we show that the checking problems described above are, under certain structural conditions, polynomially solvable on deterministic machines. In general, however, the first problem is NP-complete whilst the second is polynomially solvable on deterministic machines.

In this paper, we propose a methodology for deriving a new kind of approximate temporal functional dependencies, called Approximate Predictive Functional Dependencies (APFDs), based on a three-window framework and on a multi-temporal relational model. Different features are proposed for the Observation Window (OW), where we observe predictive data, for the Waiting Window (WW), and for the Prediction Window (PW), where the predicted event occurs. We then consider the concept of approximation for such APFDs, introduce new error measures, and discuss different strategies for deriving APFDs. We discuss the quality, i.e., the informative content, of the derived AFDs by considering their entropy and information gain. Moreover, we outline the results in deriving APFDs focusing on the Acute Kidney Injury (AKI). We use real clinical data contained in the MIMIC III dataset related to patients from Intensive Care Units to show the applicability of our approach to real-world data.

A classic result by Stockmeyer gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of chop under the homogeneity assumption. In this paper, we study the complexity of the satisfiability problem for suitable weakenings of the chop interval temporal logic, that can be equivalently viewed as fragments of Halpern and Shoham interval logic. We first consider the logic $\mathsf{BD}_{hom}$ featuring modalities $B$, for \emph{begins}, corresponding to the prefix relation on pairs of intervals, and $D$, for \emph{during}, corresponding to the infix relation. The homogeneous models of $\mathsf{BD}_{hom}$ naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations. Such a fragment has been recently shown to be PSPACE-complete . In this paper, we study the extension $\mathsf{BD}_{hom}$ with the temporal neighborhood modality $A$ (corresponding to the Allen relation \emph{Meets}), and prove that it increases both its expressiveness and complexity. In particular, we show that the resulting logic $\mathsf{BDA}_{hom}$ is EXPSPACE-complete.

In this paper, we study the finite satisfiability problem for the logic BE under the homogeneity assumption. BE is the cornerstone of Halpern and Shoham’s interval temporal logic, and features modal operators corresponding to the prefix (a.k.a. "Begins") and suffix (a.k.a. "Ends") relations on intervals. In terms of complexity, BE lies in between the "Chop" logic C, whose satisfiability problem is known to be non-elementary, and the PSpace-complete interval logic D of the sub-interval (a.k.a. "During") relation. BE was shown to be ExpSpace-hard, and the only known satisfiability procedure is primitive recursive, but not elementary. Our contribution consists of tightening the complexity bounds of the satisfiability problem for BE, by proving it to be ExpSpace-complete. We do so by devising an equi-satisfiable normal form with boundedly many nested modalities. The normalization technique resembles Scott’s quantifier elimination, but it turns out to be much more involved due to the limitations enforced by the homogeneity assumption.

In this paper, we prove Pspace-completeness of the finite satisfiability and model checking problems for the fragment of Halpern and Shoham interval logic with modality , for the “suffix” relation on pairs of intervals, and modality , for the “sub-interval” relation, under the homogeneity assumption. The result significantly improves the Expspace upper bound recently established for the same fragment, and proves the rather surprising fact that the complexity of the considered problems does not change when we add either the modality for suffixes () or, symmetrically, the modality for prefixes () to the logic of sub-intervals (featuring only ).

The expressive power of interval temporal logics (ITLs) makes them one of the most natural choices in a number of application domains, ranging from the specification and verification of complex reactive systems to automated planning. However, for a long time, because of their high computational complexity, they were considered not suitable for practical purposes. The recent discovery of several computationally well-behaved ITLs has finally changed the scenario. In this paper, we investigate the finite satisfiability and model checking problems for the ITL D, that has a single modality for the sub-interval relation, under the homogeneity assumption (that constrains a proposition letter to hold over an interval if and only if it holds over all its points). We first prove that the satisfiability problem for D, over finite linear orders, is PSPACE-complete, and then we show that the same holds for its model checking problem, over finite Kripke structures. In such a way, we enrich the set of tractable interval temporal logics with a new meaningful representative.

In this paper, we establish Pspace-completeness of the finite satisfiability and model checking problems for the fragment of Halpern and Shoham interval logic with modality ⟨E⟩, for the "suffix" relation on pairs of intervals, and modality ⟨D⟩, for the "sub-interval" relation, under the homogeneity assumption. The result significantly improves the Expspace upper bound recently established for the same fragment, and proves the rather surprising fact that the complexity of the considered problems does not change when we add either the modality for suffixes (⟨E⟩) or, symmetrically, the modality for prefixes (⟨B⟩) to the logic of sub-intervals (featuring only ⟨D⟩).

Extracting association rules from large datasets has been widely studied in many variants in the last two decades; they allow to extract relations between values that occur more “often” in a database. With temporal association rules the concept has been declined to temporal databases. In this context the “most frequent” patterns of evolution of one or more attribute values are extracted. In the temporal setting, especially where the interference betweeen temporal patterns cannot be neglected (e.g., in medical domains), there may be the case that we are looking for a set of temporal association rules for which a “significant” portion of the original database represents a consistent model for all of them. In this work, we introduce a simple and intuitive form for temporal association rules, called pure evolving association rules (PE-ARs for short), and we study the complexity of checking a set of PE-ARs over an instance of a temporal relation under approximation (i.e., a percentage of tuples that may be deleted from the original relation). As a by-product of our study we address the complexity class for a general problem on Directed Acyclic Graphs which is theoretically interesting per se.

A classic result by Stockmeyer [Stockmeyer, 1974] gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of the chop operator under the homogeneity assumption [Halpern et al., 1983]. In this paper, we study the complexity of the satisfiability problem for a suitable weakening of the chop interval temporal logic, that can be equivalently viewed as a fragment of Halpern and Shoham interval logic featuring the operators B, for "begins", corresponding to the prefix relation on pairs of intervals, and D, for "during", corresponding to the infix relation. The homogeneous models of the considered logic naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations.

BPMN (Business Process Model and Notation) is widely used standard modeling technique for representing Business Processes by using diagrams, but lacks in some aspects. Representing execution-dependent and time-dependent decisions in BPMN Diagrams may be a daunting challenge [Carlo Combi et al., 2017]. In many cases such constraints are omitted in order to preserve the simplicity and the readability of the process model. However, for purposes such as compliance checking, process mining, and verification, formalizing such constraints could be very useful. In this paper, we propose a novel approach for annotating BPMN Diagrams with Temporal Synchronization Rules borrowed from the timeline-based planning field. We discuss the expressivity of the proposed approach and show that it is able to capture a lot of complex temporally-related constraints without affecting the structure of BPMN diagrams. Finally, we provide a mapping from annotated BPMN diagrams to timeline-based planning problems that allows one to take advantage of the last twenty years of theoretical and practical developments in the field.

In recent years, model checking with interval temporal logics is emerging as a viable alternative to model checking with standard point-based temporal logics, such as LTL, CTL, CTL*, and the like. The behavior of the system is modeled by means of (finite) Kripke structures, as usual. However, while temporal logics which are interpreted “point-wise” describe how the system evolves state-by-state, and predicate properties of system states, those which are interpreted “interval-wise” express properties of computation stretches, spanning a sequence of states. A proposition letter is assumed to hold over a computation stretch (interval) if and only if it holds over each component state (homogeneity assumption). A natural question arises: is there any advantage in replacing points by intervals as the primary temporal entities, or is it just a matter of taste? In this article, we study the expressiveness of Halpern and Shoham’s interval temporal logic (HS) in model checking, in comparison with those of LTL, CTL, and CTL*. To this end, we consider three semantic variants of HS: the state-based one, introduced by Montanari et al. in [30, 34], that allows time to branch both in the past and in the future, the computation-tree-based one, that allows time to branch in the future only, and the trace-based variant, that disallows time to branch. These variants are compared among themselves and to the aforementioned standard logics, getting a complete picture. In particular, we show that HS with trace-based semantics is equivalent to LTL (but at least exponentially more succinct), HS with computation-tree-based semantics is equivalent to finitary CTL*, and HS with state-based semantics is incomparable with all of them (LTL, CTL, and CTL*).

In this paper, we investigate the finite satisfiability and model checking problems for the logic D of the sub-interval relation under the homogeneity assumption, that constrains a proposition letter to hold over an interval if and only if it holds over all its points. First, we prove that the satisfiability problem for D, over finite linear orders, is PSPACE-complete; then, we show that its model checking problem, over finite Kripke structures, is PSPACE-complete as well.

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.

In the last years, model checking with interval temporal logics is emerging as a viable alternative to model checking with standard point-based temporal logics, such as LTL, CTL, CTL*, and the like. The behavior of the system is modeled by means of (finite) Kripke structures, as usual. However, while temporal logics which are interpreted "point-wise" describe how the system evolves state-by-state, and predicate properties of system states, those which are interpreted "interval-wise" express properties of computation stretches, spanning a sequence of states. A proposition letter is assumed to hold over a computation stretch (interval) if and only if it holds over each component state (homogeneity assumption). A natural question arises: is there any advantage in replacing points by intervals as the primary temporal entities, or is it just a matter of taste? In this paper, we study the expressiveness of Halpern and Shoham's interval temporal logic (HS) in model checking, in comparison with those of LTL, CTL, and CTL*. To this end, we consider three semantic variants of HS: the state-based one, introduced by Montanari et al., that allows time to branch both in the past and in the future, the computation-tree-based one, that allows time to branch in the future only, and the trace-based variant, that disallows time to branch. These variants are compared among themselves and to the aforementioned standard logics, getting a complete picture. In particular, we show that HS with trace-based semantics is equivalent to LTL (but at least exponentially more succinct), HS with computation-tree-based semantics is equivalent to finitary CTL*, and HS with state-based semantics is incomparable with all of them (LTL, CTL, and CTL*).

Temporal Functional Dependencies (TFDs for short) are functional dependencies that predicate on temporal databases characterized by a special temporal dimension called valid time (VT). In [1] Combi et al. proposed a uniform framework that subsumes many of the TFDs proposed in literature and, by the combination of them, allow us to express finer constraints. Some interesting constraints are the Temporally Mixed Functional Dependencies (TMFD for short) that allow one to write constraints on the evolution of the data in the database. The problem of checking a TMFD against an instance of a temporal schema is polynomial. We will show that when approximation comes into play (i.e., we look for TMFD holding for almost all database tuples) the problem turns out to be NP-Complete. Moreover we introduce a type of association rules build over TMFD called Temporally Mixed Association Rule (TMAR). We prove that verifying TMAR under approximation is still NP-Complete, by reducing it to a novel problem on directed acyclic graphs.

The propositional interval logic of temporal neighborhood (PNL for short) features two modalities that make it possible to access intervals adjacent to the right (modality xAy) and to the left (modality xAy) of the current interval. PNL stands at a central position in the realm of interval temporal logics, as it is expressive enough to encode meaningful temporal conditions and decidable (undecidability rules over interval temporal logics, while PNL is NEXPTIME-complete). Moreover, it is expressively complete with respect to FO2|<;|. Various extensions of PNL have been studied in the literature, including metric, hybrid, and first-order ones. Here, we study the effects of the addition of an equivalence relation ~ to Metric PNL (MPNL~). We first show that finite satisfiability for PNL extended with ~ is still NEXPTIME-complete. Then, we prove that finite satisfiability for MPNL~ can be reduced to the decidable 0-0 reach ability problem for vector addition systems and vice versa (EXPSPACE-hardness immediately follows).

Temporal functional dependencies (TFDs) add valid time to classical functional dependencies (FDs) in order to express data integrity constraints over the flow of time. If the temporal dimension adopted is an interval, we have to deal with interval-based temporal functional dependencies (ITFDs for short), which consider different interval relations between valid times of related tuples. The related approximate problem is when we want to check if our data satisfy, without any constraint for the schema, a given ITFD under a given error threshold 0 ≤ d ≤ 1. This can be rephrased as: given a relation instance r, is it possible to delete at most c · |r| tuples from it in such a way that the resulting instance satisfies the given ITFD? This optimization problem, ITFD-Approx for short, may represent a way to discover (data mining) important dependencies among attribute values in a database as well as a way to control data consistency. In this paper we analyze the complexity of problem ITFD-Approx restricting ourselves to Allen's interval relations: we will see how the complexity of such a problem may significantly change, depending on the considered interval relation.

Interval temporal logics provide a general framework for temporal representation and reasoning, where classical (point-based) linear temporal logics can be recovered as special cases. In this paper, we study the effects of the addition of an equivalence relation to one of the most representative interval temporal logics, namely, the logic ABB̅ of Allen's relations meets, begun by, and begins. We first prove that the satisfiability problem for the resulting logic ABB̅ ℕ remains decidable over finite linear orders, but it becomes nonprimitive recursive, while decidability is lost over N. We also show that decidability over can be recovered by restricting to a suitable subset of models. Then, we show that ABB̅ ℕ is expressive enough to define ωS-regular languages, thus establishing a promising connection between interval temporal logics and extended ω-regular languages.

The propositional logic of temporal neighborhood (PNL) features two modalities that make it possible to access intervals adjacent to the right and to the left of the current one. PNL has been extensively studied in the last years. In particular, decidability and complexity of its satisfiability problem have been systematically investigated, and optimal decision procedures have been developed, for various (classes of) linear orders, including N, Z, and Q. The only missing piece is that for R. It is possible to show that PNL is expressive enough to separate Q and R. Unfortunately, there is no way to reduce the satisfiability problem for PNL over R to that over Q. In this paper, we first prove the NEXPTIME-completeness of the satisfiability problem for PNL over R, and then we devise an optimal tableau system for it.

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.

In the last years the representation and management of temporal information has become crucial for several computer applications. In the temporal database literature, every fact stored into a database may be equipped with two temporal dimensions: the valid time, that describes the time when the fact is true in the modeled reality, and the transaction time, that describes the time when the fact is current in the database and it can be retrieved. Temporal functional dependencies (TFDs) add (transaction) valid time to classical functional dependencies (FDs) in order to express database integrity constraints over the flow of time. Currently, proposals dealing with TFDs adopt a point-based approach, where tuples hold at specific time points. Moreover, TFDs may involve the use of different granularities (i.e., partitions of the time domain), to express integrity constraints as "for each month, the salary of an employee depends only on his role". At the best of our knowledge, there are no proposals dealing with interval-based temporal functional dependencies (ITFDs for short) where the associated valid time is represented by an interval. In this paper, we propose a set of ITFDs based on the Allen's interval relations, we analyze their expressive power with respect to other TFDs proposed in the literature and we propose an algorithm for verifying ITFDs in a database system.

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.

An interval temporal logic is a propositional, multi-modal logic interpreted over interval structures of partial orders. The semantics of each modal operator are given in the standard way with respect to one of the natural accessibility relations defined on such interval structures. In this paper, we consider the modal operators based on the (reflexive) sub-interval relation and the (reflexive) super-interval relation. We show that the satisfiability problems for the interval temporal logics featuring either or both of these modalities, interpreted over interval structures of finite linear orders, are all PSPACE-complete. These results fill a gap in the known complexity results for interval temporal logics.

In this article, we develop tableau-based decision procedures for the logics of subinterval structures over dense linear orderings. In particular, we consider the two difficult cases: the relation of strict subintervals (with both endpoints strictly inside the current interval) and the relation of proper subintervals (that can share one endpoint with the current interval). For each of these logics, we establish a small pseudo-model property and construct a sound, complete and terminating tableau that searches systematically for existence of such a pseudo-model satisfying the input formulas. Both constructions are non-trivial, but the latter is substantially more complicated because of the presence of beginning and ending subintervals which require special treatment. We prove PSPACE completeness for both procedures and implement them in the generic tableau-based theorem prover Lotrec.

Propositional interval temporal logics come into play in many areas of artificial intelligence and computer science. Unfortunately, most of them turned out to be (highly) undecidable. Some positive exceptions, belonging to the classes of neighborhood logics and of logics of subinterval relations, have been recently identified. In this paper, we address the decision problem for the future fragment of Propositional Neighborhood Logic (Right Propositional Neighborhood Logic) interpreted over trees and we positively solve it by providing a tableau-based decision procedure that works in exponential space. Moreover, we prove that the decision problem for the logic is EXPSPACE-hard, thus showing the optimality of the proposed procedure.