Expressive Power Of Temporal Logic Research Articles

Arity hierarchy for temporal logics

A major result concerning temporal logics is Kamp’s Theorem which states that the pair of modalities “until” and “since” is expressively complete for the first-order fragment of the monadic logic over the linear-time canonical model of naturals. The paper concerns the expressive power of temporal logics over trees. The main result states that in contrast to Kamp’s Theorem, for every n there is a modality of arity n definable by a monadic logic formula, which is not equivalent over trees to any temporal logic formula which uses modalities of arity less than n . Its proof takes advantage of an instance of Shelah’s composition theorem.This result has interesting corollaries, for instance reproving that C T L ∗ and E C T L + have no finite basis.

The Expressive Power of Temporal Logic of Actions

It is shown that the satisfiability/validity questions in Temporal Logic of Actions can be translated into satisfiability/validity questions in Second Order Temporal Logic and vice versa. The translation from Second Order Temporal Logic into Temporal Logic of Actions is linear and the translation from Temporal Logic of Actions into Second Order Temporal Logic is quadratic.

On the expressive power of temporal logic

We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective algorithm to decide whether a given rational language is expressible. Our main result states a similar condition for the “restricted” temporal logic (RTL), obtained by discarding the “until” operator. A formal language is RTL-expressible if and only if its syntactic semigroup is finite and satisfies a certain simple algebraic condition. This leads to a polynomial time algorithm to check whether the formal language accepted by an n-state deterministic automaton is RTL-expressible.

On the expressive power of temporal logic for infinite words

In this paper, we give an algebraic proof of the equivalence between temporal logic and star-free languages on infinite words. The proof is based on a result of Schützenberger that characterizes the star-free languages of finite words by using particular prefix codes.

Proving entailment between conceptual state specifications

The lack of expressive power of temporal logic as a specification language can be compensated to a certain extent by the introduction of powerful, high-level temporal operators, which are difficult to understand and reason about. A more natural way to increase the expressive power of a temporal specification language is by introducing conceptual state variables, which are auxiliary (unimplemented) variables whose values serve as an abstract representation of the internal state of the process being specified. The kind of specifications resulting from the latter approach are called conceptual state specifications. This paper considers a central problem in reasoning about conceptual state specifications: the problem of proving entailment between specifications. A technique, based on the notion of simulation between machines, is shown to be sound for proving entailment. A kind of completeness result can also be shown if specifications are assumed to satisfy well-formedness conditions. The role played by entailment in proofs of correctness is illustrated by the problem of proving that the concatenation of two FIFO buffers implements a FIFO buffer.