Propositional Interval Temporal Logic Research Articles

A separation theorem for discrete-time interval temporal logic

Gabbay's separation theorem about linear temporal logic with past has proved to be one of the most useful theoretical results in temporal logic. In this paper, we establish an analogous statement about Moszkowski's discrete-time propositional Interval Temporal Logic () with two sets of expanding modalities, namely the unary neighbourhood modalities and the binary weak inverses of 's chop operator. We prove that separation holds for both with and without its loop construct chop-star.

Detection of Back Attack based on Interval Temporal Logic

Contrasted with other detection methods, based on temporal logic the intrusion detection method can deal with complex network attacks actively. However, due to the lack of network attacks temporal logic formula, common back attack cannot use the above methods of detection. Therefore, the Propositional Interval Temporal Logic (ITL) is used to establish temporal logic formulas for back attack. Firstly, the attack principle of back attack was analyzed. Secondly, the critical steps of the attack were break up into atomic actions and the atomic proposition was defined. Finally, the temporal logic formula for the attack was established based on the logical relationship between the atomic propositions. According to the principle of model checking, the obtained temporal logic formula can be used as the input for the model checker (intrusion detector). The automaton was utilized to model the log library as another input to the model checker. And the intrusion detection method for these three attacks was given. and it proved the validityof the novel method by some Simulation experiments

An application of temporal projection to interleaving concurrency

Abstract We revisit the earliest temporal projection operator Π in discrete-time Propositional Interval Temporal Logic (PITL) and use it to formalise interleaving concurrency. The logical properties of Π as a normal modality and a way to eliminate it in both PITL and conventional point-based Linear-Time Temporal Logic (LTL), which can be viewed as a PITL subset, are examined, as are stutter-invariant formulas. Striking similarities between the expressiveness of Π and the standard LTL operator U (‘until’) are briefly illustrated. We also formalise concurrent imperative programming constructs with and without Π , and relate the two approaches. Peterson’s mutual exclusion algorithm is used to illustrate reasoning with Π about a concrete programming example. Projection with fairness and non-fairness assumptions are both discussed. This all illustrates an approach to the analysis of such concurrent interleaving finite-state systems using temporal logic formulas with projection constructs to reason about correctness properties. Unlike conventional LTL formulas about concurrency which normally largely focus on global time, properties expressed in LTL combined with Π help to reveal and analyse important differing viewpoints involving global time and the local projected time seen by each individual process. Links between Π and another standard PITL projection operator, both suitable for reasoning about different time granularities, are demonstrated by showing the two operators to be interdefinable. We briefly look at other (mostly interval-based) temporal logics with similar forms of projection, as well as some related applications and industrial standards.

Two-Variable Separation Logic and Its Inner Circle

Separation logic is a well-known assertion language for Hoare-style proof systems. We show that first-order separation logic with a unique record field restricted to two quantified variables and no program variables is undecidable. This is among the smallest fragments of separation logic known to be undecidable, and this contrasts with the decidability of two-variable first-order logic. We also investigate its restriction by dropping the magic wand connective, known to be decidable with nonelementary complexity, and we show that the satisfiability problem with only two quantified variables is not yet elementary recursive. Furthermore, we establish insightful and concrete relationships between two-variable separation logic and propositional interval temporal logic (PITL), data logics, and modal logics, providing an inner circle of closely related logics.

Model Checking Multirate Hybrid Systems with Dense Timed Interval Temporal Logic

This paper investigates the model checking issue of multirate hybrid systems. To this end, multirate automata are used to represent the possible behavior of multirate hybrid systems, and a dense timed interval temporal logic (DTITL) is defined to describe the desirable property. To check whether a multirate automaton satisfies a DTITL formula, a corresponding region automaton and a propositional interval temporal logic (PITL) formula are constructed. After each vertex of the region automaton being labeled with propositions appearing in the corresponding PITL formula, the model checking problem for mutirate hybrid systems is reduced to the same issue for PITL, which can be solved readily.

A Complete Axiom System for Propositional Interval Temporal Logic with Infinite Time

Interval Temporal Logic (ITL) is an established temporal formalism for reasoning about time periods. For over 25 years, it has been applied in a number of ways and several ITL variants, axiom systems and tools have been investigated. We solve the longstanding open problem of finding a complete axiom system for basic quantifier-free propositional ITL (PITL) with infinite time for analysing nonterminating computational systems. Our completeness proof uses a reduction to completeness for PITL with finite time and conventional propositional linear-time temporal logic. Unlike completeness proofs of equally expressive logics with nonelementary computational complexity, our semantic approach does not use tableaux, subformula closures or explicit deductions involving encodings of omega automata and nontrivial techniques for complementing them. We believe that our result also provides evidence of the naturalness of interval-based reasoning.

Undecidability of the Logic of Overlap Relation over Discrete Linear Orderings

The validity/satisfiability problem for most propositional interval temporal logics is (highly) undecidable, under very weak assumptions on the class of interval structures in which they are interpreted. That, in particular, holds for most fragments of Halpern and Shoham's interval modal logic HS. Still, decidability is the rule for the fragments of HS with only one modal operator, based on an Allen's relation. In this paper, we show that the logic O of the Overlap relation, when interpreted over discrete linear orderings, is an exception. The proof is based on a reduction from the undecidable octant tiling problem. This is one of the sharpest undecidability result for fragments of HS.

John Buridan’s Sophismata and Interval Temporal Semantics

In this paper we look at the suitability of modern interval-based temporal logic for modeling John Buridan’s treatment of tensed sentences in his Sophismata. Building on the paper (Øhrstrøm 1984), we develop Buridan’s analysis of temporal logic, paying particular attention to his notions of negation and the absolute/relative nature of the future and the past.We introduce a number of standard modern propositional interval temporal logics (ITLs) to illustrate where Buridan’s interval-based temporal analysis differs from the standard modern approaches. We give formal proofs of some claims in (Øhrstrøm 1984), and sketch how the standard modern systems could be defined in terms of Buridan’s proposals, showing that his logic can be taken as more basic. In diesem Aufsatz betrachten wir das Geeignetsein einer modernen intervallbasierten temporalen Logik für die Modellierung von John Buridans Behandlung zeitabhängiger Sätze in seinen Sophismata. Aufbauend auf den Aufsatz (Øhrstrøm 1984) stellen wir Buridans Analyse der temporalen Logik dar, mit besonderer Berücksichtigung seiner Begriffe der Negation und der absoluten/relativen Natur der Zukunft und der Vergangenheit. Wir führen einige standardmäßige moderne intervallbasierte temporale Aussagenlogiken (ITLs) ein, um zu illustrieren, wo Buridans intervallbasierte temporale Analyse von den standardmäßigen modernen Zugangsweisen abweicht. Wir geben formale Beweise einiger Behauptungen in (Øhrstrøm 1984) an und skizzieren, wie die standardmäßigen modernen Systeme im Sinne der Vorschläge Buridans definiert werden könnten; wir zeigen damit, dass seine Logik als die grundlegendere aufgefasst werden kann.

A decision procedure for propositional projection temporal logic with infinite models

This paper investigates the satisfiability of Propositional Projection Temporal Logic (PPTL) with infinite models. A decision procedure for PPTL formulas is given. To this end, Normal Form (NF) and Labeled Normal Form Graph (LNFG) for PPTL formulas are defined, and algorithms for transforming a formula to its normal form and constructing the LNFG for the given formula are presented. Further, the finiteness of LNFGs is proved in details. Moreover, the decision procedure is extended to check the satisfiability of the formulas of Propositional Interval Temporal Logic. In addition, examples are also given to illustrate how the decision procedure works.

An Optimal Decision Procedure for Right Propositional Neighborhood Logic

Propositional interval temporal logics are quite expressive temporal logics that allow one to naturally express statements that refer to time intervals. Unfortunately, most such logics turn out to be (highly) undecidable. In order to get decidability, severe syntactic or semantic restrictions have been imposed to interval-based temporal logics to reduce them to point-based ones. The problem of identifying expressive enough, yet decidable, new interval logics or fragments of existing ones that are genuinely interval-based is still largely unexplored. In this paper, we focus our attention on interval logics of temporal neighborhood. We address the decision problem for the future fragment of Neighborhood Logic (Right Propositional Neighborhood Logic, RPNL for short), and we positively solve it by showing that the satisfiability problem for RPNL over natural numbers is NEXPTIME-complete. Then, we develop a sound and complete tableau-based decision procedure, and we prove its optimality.

A general tableau method for propositional interval temporal logics: Theory and implementation

In this paper, we focus our attention on tableau methods for propositional interval temporal logics. These logics provide a natural framework for representing and reasoning about temporal properties in several areas of computer science. However, while various tableau methods have been developed for linear and branching time point-based temporal logics, not much work has been done on tableau methods for interval-based ones. We develop a general tableau method for Venema's CDT logic interpreted over partial orders ( BCDT + for short). It combines features of the classical tableau method for first-order logic with those of explicit tableau methods for modal logics with constraint label management, and it can be easily tailored to most propositional interval temporal logics proposed in the literature. We prove its soundness and completeness, and we show how it has been implemented.

A Hierarchical Completeness Proof for Propositional Interval Temporal Logic with Finite Time

We present a completeness proof for Propositional Interval Temporal Logic (PITL) with finite time which avoids certain difficulties of conventional methods. It is more gradated than previous efforts since we progressively reduce reasoning within the original logic to simpler reasoning in sublogics. Furthermore, our approach benefits from being less constructive since it is able to invoke certain theorems about regular languages over finite words without the need to explicitly describe the associated intricate proofs. A modified version of regular expressions called Fusion Expressions is used as part of an intermediate logic called Fusion Logic. Both have the same expressiveness as PITL but are lower-level notations which play an important role in the hierarchical structure of the overall completeness proof. In particular, showing completeness for PITL is reduced to showing completeness for Fusion Logic. This in turn is shown to hold relative to completeness for conventional linear-time temporal logic with finite time. Logics based on regular languages over finite words and !-words offer a promising but elusive framework for formal specification and verification. A number of such logics and decision procedures have been proposed. In addition, various researchers have obtained complete axiom systems by embedding and expressing the decision procedures directly within the logics. The work described here contributes to this topic by showing how to exploit some interesting links between regular languages and interval-based temporal logics.

A Decision Procedure and Complete Axiomatization of Finite Interval Temporal Logic with Projection

This paper presents a complete axiomatization for propositional interval temporal logic (PITL) with projection. The axiomatization is based on a tableau decision procedure for the logic, which in turn is founded upon a normal form for PITL formulae. The construction of the axiomatization provides a general mechanism for generating axiomatizations thus: given a normal form for a new connective, axioms can be generated for the connective from the tableau construction using that normal form. The paper concludes with a discussion of aspects of compositionality for PITL with projection.