Right Propositional Neighborhood Logic over Natural Numbers with Integer Constraints for Interval Lengths

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.

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).

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.

Interval temporal logics are expressive formalisms for temporal representation and reasoning, which use time intervals as primitive temporal entities. They have been extensively studied for the past two decades and successfully applied in AI and computer science. Unfortunately, they lack the ability of expressing promptness conditions, as it happens with the commonly-used temporal logics, e.g., LTL: whenever we deal with a liveness request, such as “something good eventually happens”, there is no way to impose a bound on the delay with which it is fulfilled. In the last years, such an issue has been addressed in automata theory, game theory, and temporal logic. In this paper, we approach it in the interval temporal logic setting. First, we introduce PROMPT- PNL, a prompt extension of the well-studied interval temporal logic PNL, and we prove the undecidability of its satisfiability problem; then, we show how to recover decidability (NEXPTIME-completeness) by imposing a natural syntactic restriction on it.

Decision trees are simple, yet powerful, classification models used to classify categorical and numerical data, and, despite their simplicity, they are commonly used in operations research and management, as well as in knowledge mining. From a logical point of view, a decision tree can be seen as a structured set of logical rules written in propositional logic. Since knowledge mining is rapidly evolving towards temporal knowledge mining, and since in many cases temporal information is best described by interval temporal logics, propositional logic decision trees may evolve towards interval temporal logic decision trees. In this paper, we define the problem of interval temporal logic decision tree learning, and propose a solution that generalizes classical decision tree learning.

Interval logics are very expressive temporal formalisms, but reasoning with them is often undecidable or has high computational complexity. As a result, a vast number of approaches limiting their expressive power—in order to obtain better computational behaviour—have been introduced. Unfortunately, due to such restrictions, interval logics often lose referentiality, that is, the capacity to refer to specific time intervals, which is crucial for temporal representation and reasoning. The computational price that needs to be paid in order to regain referentiality is not well studied and our research aims to fill this gap. In particular we study the main interval temporal logic, called the Halpern-Shoham logic, and its low complexity modifications. To regain referentiality in these modifications, we extend the language with the hybrid machinery—nominals and satisfaction operators—and classify the obtained logics according to their computational complexity. We show that such a hybridisation often makes tractable logics intractable but not undecidable. This allows us to construct hybrid interval temporal logics which are referential as well as maintain a good compromise between expressiveness and complexity; it makes them valuable formalisms for temporal knowledge representation. We also introduce a class of models which, due to a specific interplay between the interpretation of modal operators and a structure of time, makes reasoning in interval logics computationally hard even in the absence of the hybrid machinery.

The primary characteristic of interval temporal logic is that intervals, rather than points, are taken as the primitive ontological entities. Their computational behaviour is generally bad, and several restrictions have been considered in order to define decidable and computationally affordable temporal logics based on intervals. In this paper we take inspiration from Golumbic and Shamir’s coarser interval algebras, which generalize the classical Allen’s Interval Algebra, in order to define two previously unknown variants of Halpern and Shoham’s logic (\(\mathrm{HS} \)). We prove that one of them (denoted here by \(\mathrm{HS_7} \)) is still generally undecidable, while the other one (\(\mathrm{HS_3} \)) becomes, perhaps surprisingly, PSpace-complete, at least in the finite case.KeywordsInterval Temporal LogicCoarse IntervalsSatisfiability ProblemInterval Algebra (IA)Finite CaseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper introduces a tableau method for propositional interval temporal logic (ITL) [14]. Beyond the usual operators of linear temporal logic, ITL contains sequencing and iterative operators, ‘;’ and proj akin to programming combinators. Central to our approach is a normal form for the formulas of ITL, particularly ‘;’ and proj, in terms of the ‘◯’ operator of the logic.

Decidability and complexity of the fragments of the modal logic of Allen's relations over the rationals

A transformation from Propositional Projection Temporal Logic (PPTL) as well as Propositional Interval Temporal Logic (PITL) with infinite models to monadic second order logic with one successor (S1S) is presented in this paper. To this end, intervals where PPTL and PITL formulas are interpreted over are represented as I-structures. Further, the semantics of PPTL and PITL formulas are redefined over I-structures. Moreover, according to I-structure semantics, a PPTL or PITL formula is translated to a formula in S1S. As a result, many mature theoretical and technical results, such as decidability etc. for S1S can be easily inherited by PPTL and PITL.

Temporal logic has been extensively utilized in academia and industry to formally specify and verify behavioural properties of numerous kinds of hardware and software. We present a novel way to apply temporal logic to the study of a version of itself, namely, propositional linear-time temporal logic (PTL). This involves a hierarchical framework for obtaining standard results for PTL, including a small model property, decision procedures and axiomatic completeness. A large number of the steps involved are expressed in a propositional version of Interval Temporal Logic (ITL) which is referred to as PITL. It is a natural generalization of PTL and includes operators for reasoning about periods of time and sequential composition. Versions of PTL with finite time and infinite time are both considered and one benefit of the framework is the ability to systematically reduce infinite-time reasoning to finite-time reasoning. The treatment of PTL with the operator until and past time naturally reduces to that for PTL without either one. The interval-oriented methodology differs from other analyses of PTL which typically use sets of formulas and sequences of such sets for canonical models. Instead, we represent models as time intervals expressible in PITL. The analysis furthermore relates larger intervals with smaller ones. Being an interval-based formalism, PITL is well suited for sequentially combining and decomposing the relevant formulas. Consequently, we can articulate issues of equal significance in more conventional analyses of PTL but normally only considered at the metalevel. A good example of this is the existence of bounded models with periodic suffixes for PTL formulas which are satisfiable in infinite time. We also describe decision procedures based on binary decision diagrams and exploit some links with finite-state automata. Beyond the specific issues involving PTL, the research is a significant application of ITL and interval-based reasoning and illustrates a general approach to formally reasoning about sequential and parallel behaviour in discrete linear time. The work also includes some interesting representation theorems. In addition, it has relevance to hardware description and verification since the specification languages PSL/Sugar (IEEE Standard 1850) and ‘temporal e’ (part of IEEE Standard 1647) both contain temporal constructs concerning intervals of time as does the related SystemVerilog Assertion language contained in SystemVerilog (IEEE Standard 1800), an extension of the IEEE 1364–2001 Verilog language.

The first results in proof theory were obtained using axiomatic systems of logical deduction. The chapter develops propositional and predicate logic in such an axiomatic system. It shows how reasoning under assumption can be regained in an axiomatic system by proving the so-called “deduction theorem” propositional logic (and later, predicate logic as well). The deduction theorem provides the opportunity to introduce one of the main tools needed for proving properties of formal systems, namely, proof by induction on the natural numbers. The differences between minimal, intuitionistic, and classical systems of logic according to the properties of negation characterizing each of the systems are outlined. Independence of theorems such as the law of excluded middle are here given model-theoretically (in later chapters, these are also presented proof-theoretically). After presenting classical predicate logic, the systems of classical and intuitionistic arithmetic are presented. Gentzen’s first substantial result, the so-called Gödel-Gentzen translation, is discussed in the final section where it is proved that if intuitionistic arithmetic is consistent, so is classical arithmetic.

We show that the class of properties of programs expressible in propositional temporal logic can be substantially extended if we assume the programs to be data-independent. Basically, a program is data-independent if its behavior does not depend on the specific data it operates upon. Our results significantly extend the applicability of program verification and synthesis methods based on propositional temporal logic.

We investigate logical consequence in temporal logics in terms of logical consecutions, i.e., inference rules. First, we discuss the question: what does it mean for a logical consecution to be ‘correct’ in a propositional logic. We consider both valid and admissible consecutions in linear temporal logics and discuss the distinction between these two notions. The linear temporal logicLDTL, consisting of all formulas valid in the frame 〈L≤, ≥〉 of all integer numbers, is the prime object of our investigation. We describe consecutions admissible inLDTLin a semantic way—via consecutions valid in special temporal Kripke/Hintikka models. Then we state that any temporal inference rule has a reduced normal form which is given in terms of uniform formulas of temporal degree 1. Using these facts and enhanced semantic techniques we construct an algorithm, which recognizes consecutions admissible inLDTL. Also, we note that using the same technique it follows that the linear temporal logicL(N) of all natural numbers is also decidable w.r.t. inference rules. So, we prove that both logicsLDTLandL(N) are decidable w.r.t. admissible consecutions. In particular, as a consequence, they both are decidable (known fact), and the given deciding algorithms are explicit.

Summary form only given. We briefly present a rule-based framework, called EAGLE, shown to be capable of defining and implementing finite trace monitoring logics, including future and past time temporal logic, extended regular expressions, real-time and metric temporal logics (MTL), interval logics, forms of quantified temporal logics, and so on. In this paper we focus on a linear temporal logic (LTL) specialisation of EAGLE. For an initial formula of size m, we establish upper bounds of O(m/sup 2/2/sup m/logm) and O(m/sup 4/2/sup 2m/log/sup 2/m) for the space and time complexity, respectively, of single step evaluation over an input trace. EAGLE has been successfully used, in both LTL and metric LTL forms, to test a real-time controller of an experimental NASA planetary rover.