Decidability of the interval temporal logic ABBar over the natural numbers

In this paper, we focus our attention on the interval temporal logic of the Allen's relations "meets", "begins", and "begun by" (ABBar 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 ABBar 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 proble

Begin, After, and Later: a Maximal Decidable Interval Temporal Logic

Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous ITL studied so far is Halpern and Shoham's HS, which is the logic of the thirteen Allen's interval relations. Unfortunately, HS and most of its fragments have an undecidable satisfiability problem. 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 different combinations of the interval relations Begins, After, Later and their inverses Abar, Bbar, and Lbar. We know from previous works that the combination ABBbarAbar is decidable only when finite domains are considered (and undecidable elsewhere), and that ABBbar is decidable over the natural numbers. We extend these results by showing that decidability of ABBar can be further extended to capture the language ABBbarLbar, which lays in between ABBar and ABBbarAbar, and that turns out to be maximal w.r.t decidability over strongly discrete linear orders (e.g. finite orders, the naturals, the integers). We also prove that the proposed decision procedure is optimal with respect to the complexity class

Interval-based Synthesis

We introduce the synthesis problem for Halpern and Shoham's modal logic of intervals extended with an equivalence relation over time points, abbreviated HSeq. In analogy to the case of monadic second-order logic of one successor, the considered synthesis problem receives as input an HSeq formula phi and a finite set Sigma of propositional variables and temporal requests, and it establishes whether or not, for all possible evaluations of elements in Sigma in every interval structure, there exists an evaluation of the remaining propositional variables and temporal requests such that the resulting structure is a model for phi. We focus our attention on decidability of the synthesis problem for some meaningful fragments of HSeq, whose modalities are drawn from the set A (meets), Abar (met by), B (begins), Bbar (begun by), interpreted over finite linear orders and natural numbers. We prove that the fragment ABBbareq is decidable (non-primitive recursive hard), while the fragment AAbarBBbar turns out to be undecidable. In addition, we show that even the synthesis problem for ABBbar becomes undecidable if we replace finite linear orders by natural numbers.Comment: In Proceedings GandALF 2014, arXiv:1408.556

An Optimal Decision Procedure for MPNL over the Integers

Interval temporal logics provide a natural framework for qualitative and quantitative temporal reason- ing over interval structures, where the truth of formulae is defined over intervals rather than points. In this paper, we study the complexity of the satisfiability problem for Metric Propositional Neigh- borhood Logic (MPNL). MPNL features two modalities to access intervals "to the left" and "to the right" of the current one, respectively, plus an infinite set of length constraints. MPNL, interpreted over the naturals, has been recently shown to be decidable by a doubly exponential procedure. We improve such a result by proving that MPNL is actually EXPSPACE-complete (even when length constraints are encoded in binary), when interpreted over finite structures, the naturals, and the in- tegers, by developing an EXPSPACE decision procedure for MPNL over the integers, which can be easily tailored to finite linear orders and the naturals (EXPSPACE-hardness was already known).Comment: In Proceedings GandALF 2011, arXiv:1106.081

Enigmatic trace fossils from the Aeolian lower Jurassic Clarens formation, Southern Africa

The Lower Jurassic aeolienites of the Clarens Formation in southern Africa contain unique sedimentary structures that are unlikely to be non-biogenic. They are also unlike any known modern or ancient trace fossils. Here, some nigmatic, horizontal,regularly-oriented sedimentary structures are described, which occur in association with other trace fossils as well as features that were previously nterpreted as nests of termites or termite-like ancient social nsects. These spectacular structures are exposed in enormous profusion as straight, ~5 mm cylinders with strong compass orientation, in parallel alignment with one another and to ancient horizontal bedding planes. Their fill is identical to that of the host rock: clean, well-sorted, very fine- to finegrained quartz-arenite. In cross-section, each structure is defined by a subtle, ~0.1 mm thin, concentric gap. Without comparable modern biogenic structures, the biological origin of the structures is uncertain. Their strong compass orientations are, however, also inconsistent with an inorganic origin, even though they may resemble pipey concretions generated by flowing groundwater. Nonetheless, this paper, based on spatiotemporal distribution patterns of the oriented structures, their locally high abundance and association with obvious trace fossils, as well as other sedimentological and palaeontological lines of evidence, argues that the compass structures may be products of ancient social invertebrates living in a resource-limited, semi-arid to arid environment. Furthermore, the compass structures as well as the accompanying structures of the predominantly aeolian Clarens Formation collectively imply the recurrence of favourable ecological parameters (e.g., moist ubstrates) related to episodic climate fluctuations in the Early Jurassic of southern Pangaea (i.e., southern Gondwana)

Interval-based temporal functional dependencies: specification and verification

In the temporal database literature, every fact stored in a database may beequipped with two temporal dimensions: the valid time, which describes the time whenthe fact is true in the modeled reality, and the transaction time, which describes the timewhen the fact is current in the database and can be retrieved. Temporal functional dependencies(TFDs) add valid time to classical functional dependencies (FDs) in order to expressdatabase integrity constraints over the flow of time. Currently, proposals dealing with TFDsadopt a point-based approach, where tuples hold at specific time points, to express integrityconstraints such as \u201cfor each month, the salary of an employee depends only on his role\u201d. Tothe best of our knowledge, there are no proposals dealing with interval-based temporal functionaldependencies (ITFDs), where the associated valid time is represented by an intervaland there is the need of representing both point-based and interval-based data dependencies.In this paper, we propose ITFDs based on Allen\u2019s interval relations and discuss theirexpressive power with respect to other TFDs proposed in the literature: ITFDs allow us toexpress interval-based data dependencies, which cannot be expressed through the existingpoint-based TFDs. ITFDs allow one to express constraints such as \u201cemployees starting towork the same day with the same role get the same salary\u201d or \u201cemployees with a given roleworking on a project cannot start to work with the same role on another project that willend before the first one\u201d. Furthermore, we propose new algorithms based on B-trees to efficientlyverify the satisfaction of ITFDs in a temporal database. These algorithms guaranteethat, starting from a relation satisfying a set of ITFDs, the updated relation still satisfies thegiven ITFDs

The addition of temporal neighborhood makes the logic of prefixes and sub-intervals EXPSPACE-complete

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.Comment: arXiv admin note: substantial text overlap with arXiv:2109.0832

Satisfiability and Model Checking for the Logic of Sub-Intervals under the Homogeneity Assumption

The expressive power of interval temporal logics (ITLs) makes them really fascinating, and one of the most natural choices as specification and planning language. However, for a long time, due to 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 featuring 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. The paper enrich the set of tractable interval temporal logics with a meaningful representative.Comment: arXiv admin note: text overlap with arXiv:1901.0388

Maximal decidable fragments of Halpern and Shoham's modal logic of intervals

In this paper, we focus our attention on the fragment of Halpern and Shoham's modal logic of intervals (HS) that features four modal operators corresponding to the relations ``meets'', ``met by'', ``begun by'', and ``begins'' of Allen's interval algebra (AAbarBBbar logic). AAbarBBbar properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BBbar of Allen's ``begun by/begins'' relations and propositional neighborhood logic AAbar, in its many variants (including metric ones). We prove that the satisfiability problem for AAbarBBbar, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AAbarBBbar turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AAbarBBbar is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows N, Z, Q, and R

Satisfiability and model checking for the logic of sub-intervals under the homogeneity assumption

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