A Survey on Temporal Logics

Temporal logics can be considered as extensions of classical propositional and first-order logic. In fact, propositional temporal logics are an extension of propositional logic with temporal operators. Similarly, first-order temporal logics are extension of first-order logic with temporal modalities. Temporal logics are also special type of modal logics, where statements are evaluated on ‘worlds’ which represent time instants.

Although various aspects of time and logic have been studied, an up-to date comprehensive analysis of logic of time does not exist in the literature. Some surveys (such as [106,31,105,68]) can be found in the literature but these mainly concentrate on specific formal systems over specific structures of time; therefore, they do not contain a broad analysis. The aim of this paper is to outline main and recent developments in the field in a broad sense by presenting various formal systems dealing with various time structures, and discussing important features, such as (un)decidability results, expressiveness and axiomatization systems.

Temporal formalisms we will analyse include propositional/first-order linear temporal logics, branching temporal logics, partial-order temporal logics and interval temporal logics. We will summarize important results on decidability, axiomatizability, expressiveness, model checking, etc. for each logic analysed. We also provide a comparison of features of the temporal logics discussed.

Note that in some instances we think it is more convenient to refer to the original text for clarification purposes. In the following, we will use quotation marks to use the text from the original resources.

We can classify temporal logics based on several criteria. The common dimensions are ‘propositional versus first-order’, ‘point-based versus interval-based’, ’linear versus branching’, ‘discrete versus continuous’, etc [46,141,14]. Below we discuss the most important criteria to classify temporal logics.

There are two structure types to model time in a temporal logic: points (instants) and intervals. A point structure T can be represented as T, < , where T is a nonempty time points, and < is a ‘precedence’ relation on T . Different temporal relationships can be described using different modal operators. Some logics include modal operators which can express quantification over time. However, a relationship between intervals is difficult to express using a point-based temporal logic [52].

Interval temporal logics are expressive, since these logics can express a relationship between two events, which are represented by intervals. Also, interval logics [128,129,103,85,99,119,74] have a simpler and neater syntax to define a relationship between intervals, which provides a higher level abstraction than a point-based logic when modeling a system. This makes interval logic formulas much simpler and more comprehensive than point-based logic formulas. Some of the known interval operators are meets, before, during [4], which denote the ordering of intervals; chop modality [140], which denotes combining two intervals; and duration, which denotes a length of an interval [31].

Interval structures can be considered in two ways: (i) intervals are ‘primitive’ objects (ii) intervals are composed from points. [139,101,142] consider intervals as primitive objects of time. [139] defines a ‘period structure’ as the tuple I, ⊆, ≺ , where I is a non-empty set of intervals, ⊆ is a sub-interval relation, and ≺ is a precedence relation. One particular problem of this approach is that theoretical analyses are usually very difficult. Also, although it is very easy to define properties linearity, density, discreteness, unboundedness in a point-based logic, it is very difficult to define these properties in an interval logic where intervals are primitive objects. [68,74,140] consider intervals as set of points, where the time flow is assumed as “a strict partial ordering of time points”. Namely, an interval structure is defined as T , I(T ) , where T = T, < is a strict partial ordering and I(T ) is a set of intervals. The properties mentioned above can be defined in an interval logic where intervals are composed of time instants.

We conclude this section with the historical development of interval-based temporal logics. The concept of time intervals was first studied by Walker [143]. Walker considered a nonempty set of intervals, which is partially orderd. However, his work does not cover aspects of temporal logic in a general sense. In [75] philosophical aspects of an interval ontology was analysed. In [79] an interval tense logic was introduced. [43,80,125,28,138,63,131] studied interval logics

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