A reified temporal logic

This paper presents a reified temporal logic for representing and reasoning about temporal and non-temporal relationships between non-temporal assertions. A clear syntax and semantics for the logic is formally provided. Three types of predicates, temporal predicates, non-temporal predicates and meta-predicates, are introduced. Terms of the proposed language are partitioned into three types, temporal terms, non-temporal terms and propositional terms. Reified propositions consist of formulae with each predicate being either a temporal predicate or a meta-predicate. Meta-predicates may take both temporal terms and propositional terms together as arguments or take propositional terms alone. A standard formula of the classical first-order language with each predicate being a non-temporal predicate taking only non-temporal terms as arguments is reified as just a propositional term. A general time ontology has been provided which can be specialized to a variety of existing temporal systems. The new logic allows one to predicate and quantify over propositional terms while according a special status of time; for example, assertions such as ‘effects cannot precede their causes’ is ensured in the logic, and some problematic temporal aspects including the delay time between events and their effects can be conveniently expressed. Applications of the logic are presented including the characterization of the negation of properties and their contextual sentences, and the expression of temporal relations between actions and effects.

Guest Editorial. Special Issue: Temporal logic in engineering

Logic-based models are thriving within artificial intelligence. A great number of new logics have been defined, and their theory investigated. Epistemic logics introduce modal operators for knowledge or belief; deontic logics are about norms, and introduce operators of deontic necessity and possibility (i.e., obligation or prohibition). And then we have a much investigated class—temporal logics—to whose application to engineering this special issue is devoted. This kind of formalism deserves increased widespread recognition and application in engineering, a domain where other kinds of temporal models (e.g., Petri nets) are by now a fairly standard part of the modelling toolbox.

A sound and complete reified temporal logic

There are mainly two known approaches to the representation of temporal information in Computer Science: modal logic approaches (including tense logics and hybrid temporal logics) and predicate logic approaches (including temporal argument methods and reified temporal logics). On one hand, while tense logics, hybrid temporal logics and temporal argument methods enjoy formal theoretical foundations, their expressiveness has been criticised as not power enough for representing general temporal knowledge; on the other hand, although current reified temporal logics provide greater expressive power, most of them lack of complete and sound axiomatic theories. In this paper, we propose a new reified temporal logic with a clear syntax and semantics in terms of a sound and complete axiomatic formalism which retains all the expressive power of the approach of temporal reification.

Temporal structure and enablement representation for mutual wills: A Petri net approach

Those temporal formalisms that are sporadically found nowadays in the literature of AI & Law are based on temporal logic. We claim a revived role for another major class of temporal representation: Petri nets. This formalism, popular in computing from the 1970s, had its potential recognized on occasion in the literature of legal computing as well, but apparently the discipline has lost sight of it, and its practitioners on average need be tutored into this kind of representation. Asynchronous, concurrent processes—for which the approach is well‐suited—are found in the legal domain, in disparate contexts. We develop an example for Mutual Wills.

The completeness and soundness of temporal theories

Generally speaking, the term temporal logic refers to any system of rules and symbolism for representing and reasoning about propositions qualified in terms of time. In computer science, particularly in the domain of Artificial Intelligence, there are mainly two known approaches to the representation of temporal information: modal logic approaches including tense logic and hybrid temporal logic, and predicate logic approaches including temporal arguement method and reified temporal logic. On one hand, while tense logic, hybrid temporal logic and temporal argument method enjoy formal theoretical foundations, their expressiveness has been criticised as not power enough for representing general temporal knowledge; on the other hand, although reified temporal logic provides greater expressive power, most of the current systems following the temporal reification lack of complete and sound axiomatic theories. With there observations in mind, a new reified temporal logic with clear syntax and semantics in terms of a sound and complete axiomatic formalism is introduced in this paper, which retains all the expressive power of temporal reification.

Representing temporal knowledge in legal discourse

This paper presents a formalism for representing temporal knowledge in legal discourse that allows an explicit expression of time and event occurrences. The fundamental time structure is characterized as a well‐ordered discrete set of primitive times, i.e. non‐decomposable intervals with positive duration or points with zero duration), from which decomposable intervals can be constructed. The formalism supports a full representation of both absolute and relative temporal knowledge, and a formal mechanism for checking the temporal consistency of a given set of legal statements is provided. The general consistency checking algorithm which addresses both absolute and relative temporal knowledge turns out to be a linear programming problem, while in the special case where only relative temporal relations are involved, it becomes a simple question of searching for cycles in the graphical representation of the corresponding legal text.

Reified temporal logics: an overview

There are three main approaches to the representation of temporal information in AI literature: the so-called method of temporal arguments that simply extends functions and predicates of first-order language to include time as the additional argument; modal temporal logics which are extensions ofthe propositional or predicate calculus with modal temporal operators; and reified temporal logics which reify standard propositions of some initial language (e.g., the classical first-order or modal logic) as objects denoting propositional terms. The objective of this paper is to provide an overview onthe temporal reified approach by looking closely atsome representative existing systems featuring reified propositions, including those of Allen, McDermott, Shoham, Reichgelt, Galton, and Ma and Knight. We shall demonstrate that, although reified logics might be more complicated in expressing assertions about some given objects with respect to different times, they accord a special status to time and therefore have several distinct advantages in talking about some important issues which would be difficult (if not impossible) to express in other approaches.