Propositional Linear Temporal Logic Research Articles

More efficient proof-search for sequents of temporal logic

The present paper deals with efficiency improvement of backward proof-search of sequents of propositional linear temporal logic, using a loop-type sequent calculus. The improvement is achieved by syntactic transformation of sequents into equivalent to them simpler ones. It is proved that some formulas can be removed from sequents with no impact on their derivability.

A dynamic soft sensor of industrial fuzzy time series with propositional linear temporal logic

The fuzzy time series (FTS) model is widely used to forecast time series data. However, the predicted results of FTS are poor for industrial time series data, especially when data changes rapidly and its volume is enormous. Therefore, a dynamic soft sensor model is proposed based on propositional linear temporal logic (PLTL) with a sliding window. First, the sliding window is used to extract dynamic data. Then the extracted data is modeled by FTS to generate an initial forecasting result. Finally, according to the data in the window, a PLTL formula is generated to describe the trend of the data. The generated formula is used as a formal label of the data in the window to weight the initial forecasting result. The proposed method is verified with the TAIEX data set. Analysis of variance is used to test the significance of selected data sets. The experimental results prove that the new method has good regression forecasting performance. Finally, an example of industrial application is introduced. The experimental results demonstrate the effectiveness of the model for industrial time series data.

Loop-Type Sequent Calculi for Temporal Logic

Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic (PLTL) have been invented. In this paper, a sound and complete loop-type sequent calculus $$\mathbf{G} _\text {L}{} \mathbf{T} $$ for PLTL with the temporal operators “next” and “henceforth always” ( $${\mathbf{PLTL}}^{n,a}$$ ) is considered at first. We prove that all rules of $$\mathbf{G} _\text {L}{} \mathbf{T} $$ are invertible and that the structural rules of weakening and contraction, as well as the rule of cut, are admissible in $$\mathbf{G} _\text {L}{} \mathbf{T} $$ . We describe a decision procedure for $${\mathbf{PLTL}}^{n,a}$$ based on the introduced calculus $$\mathbf{G} _\text {L}{} \mathbf{T} $$ . Afterwards, we introduce a sound and complete sequent calculus $$\mathbf{G} _\text {L}{} \mathbf{T} ^\mathcal {U}$$ for PLTL with the temporal operators “next” and “until”.

A derivation-loop method for temporal logic

Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic (PLTL) have been considered in the literature. Cutfree Gentzen-type sequent calculi are convenient tools for backward proof-search search of formulas and sequents. In this paper we present a cut-free Gentzen type sequent calculus for PLTL with the operator

Ontology-mediated query answering over temporal and inconsistent data

Stream-based reasoning systems process data stemming from different sources that are received over time. In this kind of applications, reasoning needs to cope with the temporal dimension and should be resilient against inconsistencies in the data. Motivated by such settings, this paper addresses the problem of handling inconsistent data in a temporal version of ontology-mediated query answering. We consider a recently proposed temporal query language that combines conjunctive queries with operators of propositional linear temporal logic (LTL), and consider these under three inconsistency-tolerant semantics that have been introduced for querying inconsistent description logic knowledge bases. We investigate their complexity for temporal EL ⊥ and DL-LiteR knowledge bases. In particular, we consider two different cases, depending on the presence of negations in the query. Furthermore, we complete the complexity picture for the consistent case. We also provide two approaches toward practical algorithms for inconsistency-tolerant temporal query answering.

Index set expressions can represent temporal logic formulas

In Temporal Logic (TL), a well-formed formula is generally formed by applying rules of its syntax finitely many times. However, under some circumstances, although formulas such as ones expressed by index set expressions, are constructed via applying rules of the syntax infinitely many times, they are possibly still well-formed since their equivalent concise syntax formulas can be found. With this motivation, this paper investigates the relationship between formulas specified by index set expressions and concise syntax by means of fixed-point approach. Firstly, we present two kinds of formulas, namely ⋁i∈N0◯iQ and ⋁i∈N0Qi, and prove they are indeed well-formed by proving they are equivalent to formulas ◇Q and Q⁎ respectively. Further, we generalize ⋁i∈N0◯iQ to ⋁i∈N0P(i)∧◯iQ and explore the least and greatest fixed-points of an abstract equation X≡Q∨P∧◯X. Based on these, some well-formed special instances of ⋁i∈N0P(i)∧◯iQ are obtained. Besides, with the index set expression technique, we equivalently represent ‘U’ (strong until) and ‘W’ (weak until) constructs of propositional Linear Temporal Logic (LTL) within Propositional Projection Temporal Logic (PPTL).

A New Rule for LTL Tableaux

Propositional linear time temporal logic (LTL) is the standard temporal logic for computing applications and many reasoning techniques and tools have been developed for it. Tableaux for deciding satisfiability have existed since the 1980s. However, the tableaux for this logic do not look like traditional tree-shaped tableau systems and their processing is often quite complicated. In this paper, we introduce a novel style of tableau rule which supports a new simple traditional-style tree-shaped tableau for LTL. We prove that it is sound and complete. As well as being simple to understand, to introduce to students and to use, it is also simple to implement and is competitive against state of the art systems. It is particularly suitable for parallel implementations.

Propositional Linear Temporal Logic with Initial Validity Semantics1

Summary In the article [10] a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of “until” operator in a very strict version. The very strict “until” operator enables to express all other temporal operators. In this article we construct a formal system for LTLB with the initial semantics [12]. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [10] and the one introduced in this article. Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in [10] and in this article can be used to carry out such verifications in Mizar [4].

Method of marks for propositional linear temporal logic

It is known that traditional techniques used to ensure termination of a decision procedure in non-classical logics are based on loop-checking, in general. Nowadays, effective loop-check techniques based on histories are used instead of unrestricted loop-check. These techniques are widely and successfully applied also for non-classical logics containing induction-like axioms. These induction-like axioms create new type loops (“good loops”) along with ordinary “bad loops”. In this paper, some loop-check free saturation-like decision procedure based on some technique of marks is proposed. This saturation procedure terminates when special type marked sequents are obtained. This procedure is demonstrated for propositional linear temporal logic (PLTL) with temporal operators “next” and “always”.

Runtime verification using the temporal description logic [formula omitted]-LTL revisited

Formulae of linear temporal logic (LTL) can be used to specify (wanted or unwanted) properties of a dynamical system. In model checking, the system's behaviour is described by a transition system, and one needs to check whether all possible traces of this transition system satisfy the formula. In runtime verification, one observes the actual system behaviour, which at any point in time yields a finite prefix of a trace. The task is then to check whether all continuations of this prefix to a trace satisfy (violate) the formula. More precisely, one wants to construct a monitor, i.e., a finite automaton that receives the finite prefix as input and then gives the right answer based on the state currently reached.In this paper, we extend the known approaches to LTL runtime verification in two directions. First, instead of propositional LTL we use the more expressive temporal logic ALC-LTL, which can use axioms of the Description Logic (DL) ALC instead of propositional variables to describe properties of single states of the system. Second, instead of assuming that the observed system behaviour provides us with complete information about the states of the system, we assume that states are described in an incomplete way by ALC-knowledge bases. We show that also in this setting monitors can effectively be constructed. The (double-exponential) size of the constructed monitors is in fact optimal, and not higher than in the propositional case. As an auxiliary result, we show how to construct Büchi automata for ALC-LTL-formulae, which yields alternative proofs for the known upper bounds of deciding satisfiability in ALC-LTL.

A decidable timeout-based extension of linear temporal logic

We develop a timeout extension of propositional linear temporal logic (which we call ToLTL) to specify timing properties of timeout-based models of real-time systems. A timeout is used to model the execution of an action marking the end of a delay. With a view to expressing such timeout constraints, ToLTL uses a dynamic variable to abstract the timeout behaviour in addition to a variable which captures the global clock and some static timing variables which record time instances when discrete events occur. We propose a corresponding timeout tableau for satisfiability checking of the ToLTL formulas. Further we prove that the validity checking for such an extended logic remains PSPACE-complete. Under discrete time semantics, with bounded timeout increments, the model-checking problem which verifies if a ToLTL-formula holds in a timeout Kripke structure is also shown to be PSPACE complete. We further prove that ToLTL, when interpreted over discrete time, can be embedded in the monadic second-order logic with time and timeout and show an expressive completeness result for ToLTL. However, when ToLTL is interpreted over dense time without non-zenoness, the resulting logic becomes -complete (i.e., highly undecidable).

Sequent systems for PLTL

We consider three sequent calculi for propositional linear temporal logic (PLTL) which allow us to formalize the properties of operator “always”. The main new results presented in the paper are: (1) introduction of the calculus withlooping axioms; (2) the direct proof that the presented calculi are equivalent; (3) the proof of completeness of the calculi with looping axioms and with invariant-like rule based on completeness of the calculus with the infinitary $\omega$-type rule.

Weak Completeness Theorem for Propositional Linear Time Temporal Logic

Summary We prove weak (finite set of premises) completeness theorem for extended propositional linear time temporal logic with irreflexive version of until-operator. We base it on the proof of completeness for basic propositional linear time temporal logic given in [20] which roughly follows the idea of the Henkin-Hasenjaeger method for classical logic. We show that a temporal model exists for every formula which negation is not derivable (Satisfiability Theorem). The contrapositive of that theorem leads to derivability of every valid formula. We build a tree of consistent and complete PNPs which is used to construct the model.

Deductive temporal reasoning with constraints

When modelling realistic systems, physical constraints on the resources available are often required. For example, we might say that at most N processes can access a particular resource at any moment, exactly M participants are needed for an agreement, or an agent can be in exactly one mode at any moment. Such situations are concisely modelled where literals are constrained such that at most N, or exactly M, can hold at any moment in time. In this paper we consider a logic which is a combination of standard propositional linear time temporal logic with cardinality constraints restricting the numbers of literals that can be satisfied at any moment in time. We present the logic and show how to represent a number of case studies using this logic. We propose a tableau-like algorithm for checking the satisfiability of formulae in this logic, provide details of a prototype implementation and present experimental results using the prover.

Cut, invariant rule, and loop-check free sequent calculus for PLTL

In this paper, some loop-check free saturation-like decision procedure is proposed for propositional linear temporal logic (PLTL) with temporal operators “next” and “always”. This saturation procedure terminates when special type sequents are obtained. Properties of PLTL allows us to construct backtracking-free decision procedure without histories and loop-check.

Partial cut elimination for combinations of propositional multi-modal logics with past time

We consider combinations of nine propositional multi-modal logics with propositional discrete linear time temporal logic with past time. For these combinations, we present sound and complete Gentzen-type sequent calculi with a restricted cut rule.

Unification in linear temporal logic LTL

We prove that a propositional Linear Temporal Logic with Until and Next (LTL) has unitary unification. Moreover, for every unifiable in LTL formula A there is a most general projective unifier, corresponding to some projective formula B, such that A is derivable from B in LTL. On the other hand, it can be shown that not every open and unifiable in LTL formula is projective. We also present an algorithm for constructing a most general unifier.

The Axiomatization of Propositional Linear Time Temporal Logic

The Axiomatization of Propositional Linear Time Temporal Logic The article introduces propositional linear time temporal logic as a formal system. Axioms and rules of derivation are defined. Soundness Theorem and Deduction Theorem are proved [9].

Partial cut elimination for propositional discrete linear time temporal logic

We consider propositional discrete linear time temporal logic with future and past operators of time. For each formula ϕ of this logic, we present Gentzen-type sequent calculus Gr(ϕ) with a restricted cut rule. We sketch a proof of the soundness and the completeness of the sequent calculi presented. The completeness is proved via construction of a canonical model.

Non-axiomatizability for the linear temporal logic of knowledge with concrete observability

We show that propositional linear temporal logic with knowledge modalities but without common knowledge has an undecidable satisfiability problem when interpreted in a ‘concrete’ semantics with perfect recall or with perfect recall and synchrony.We then conclude that this concrete semantics is not axiomatizable in the semantics, based on local states.