On the expressive power of temporal logic

Abstract

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. * Research on this paper was partially supported by PRC "Mathématiques et Informatique".

Extended Temporal Logic on Finite Words and Wreath Product of Monoids with Distinguished Generators

“…In this paper, we associate a modal operator with each language in a given subclass of regular languages, and use the reverse wreath product (of monoids with distinguished generators) to provide an algebraic characterization of the expressive power of (future) temporal logic on finite words endowed with these modal operators. Our logic is closely related to that proposed in Wolper [14], and our methods are related to those of Cohen, Pin, Perrin [3] and Bazirambawo, McKenzie, Therien [2] andÉsik, Larsen [7]. Moreover, our methods and results extend to ω-words, (countable) ordinal words and, more generally, to all discrete words.…”

“…In [3], Cohen, Perrin and Pin studied the expressive power of the restricted temporal logic RTL whose formulas over an alphabet Σ are constructed from the atomic formulas p σ , σ ∈ Σ by the X and modalities. Let RTL denote the class of languages definable by the formulas in RTL.…”

“…Straubing [11] and Straubing, Therien, Thomas [12], temporal logic and the until hierarchy, Cohen, Pin, Perrin [3] and Therien, Thomas [13], and modular temporal logic, Bazirambawo, McKenzie, Therien [2]. In this paper, we associate a modal operator with each language in a given subclass of regular languages, and use the reverse wreath product (of monoids with distinguished generators) to provide an algebraic characterization of the expressive power of (future) temporal logic on finite words endowed with these modal operators.…”

A Visual Tradeoff Space for Formal Verification and Validation Techniques

“…Coverage is the degree to which a given specification language can actually be used to capture certain properties; a weak formal specification language can only capture simple requirements. For example, the specification language known as Propositional Linear-time Temporal Logic (PLTL) is known to be star-free regular [11] and therefore cannot formally capture requirements that require a stronger formalism, such as requirements that require nontrivial counting. In addition PLTL cannot be used to capture requirements that contain real-time constraints.…”

A normal form for spider diagrams of order

“…Weakly expressive language classes, such as regular languages and star-free regular languages, are used to formalise real-world temporal specifications [5,12]. Due to the real-world application there has been recent interest in incorporating temporal semantics in these diagrammatic logics [1,14].…”