This paper proposes a logic formalism for reasoning about time and action.

In the first part of the work, a temporal logic (a typed first-order predicate calculus) which is based on the concept of time interval is introduced. A set of primitive, mutually exclusive relations that can hold between time intervals are defined [BEFORE, EQUAL, MEETS, OVERLAPS, DURING, STARTS, FINISHES, and their inverse relations, except for EQUAL which is symmetric], together with the axioms that define their behavior. Properties holding during a time interval and occurrences taking place in a time interval are then introduced for describing both static (e.g., “Cleo owns a car”) and dynamic (e.g., “Cleo runs a race”) aspects of the world. Properties are characterized by the following feature: if a property p holds during a time interval t [HOLDS(p,t)] it holds also during any subinterval of t. Occurrences are formalized by means of the concepts of process and event. An event e is supposed to occur over the smallest time t possible for it to occur [OCCUR(e,t)]; i.e., it does not occur in any subinterval of t (e.g., “Cleo walked to the store”). On the other hand, if a process p is occurring over a time interval t [OCCURRING(p,t)], it is occurring also over at least one subinterval of t (e.g., “Cleo is walking”). So processes fall between properties and events. The concept of event causation is also introduced for representing the fact that event e1 occurring at time t1 causes event e2 occurring at time t2 [ECAUSE(e1,t1,e2,t2)]. Axioms for HOLDS, OCCUR, OCCURRING, and ECAUSE are given.

The paper deals then with the notion of action: an occurrence caused by an animate agent. A basic predicate is introduced for representing agentive causality, that is, the fact that an agent a causes an occurrence o [ACAUSE(a,o)]. The concepts of performance (action that consists of ACAUSEing an event) and activity (action that consists of ACAUSEing a process) are defined. The predicate GENERATES(a1,a2,t) is also introduced for representing the fact that an action a1 produces an action a2 during a given time interval t.

Finally, attention is focused on intentional actions. A basic model of belief and planning is developed using the theoretical framework so far constructed, and the concept of intending is formally defined and illustrated through examples. The expressive power of the proposed approach is shown in a fully worked-out example that deals with the representation of the action of hiding.

The paper does not deal with algorithms and implementation issues. However, an effective algorithm for temporal reasoning based on constraint propagation is presented in [1], and an experimental problem solver implementation built using the framework described above is reported in [2].

The knowledge representation formalism illustrated in the paper is one of the best structured and most comprehensive approaches to the problem of dealing with action and time that has appeared in the literature so far. Also, its expressive power seems sufficiently large to cover a wide class of very subtle cases.

One of the major specific advances offered by the research presented in the paper is the capability of dealing with actions that involve nonactivity, that are not decomposable into subactions, and that occur simultaneously and possibly interact.

Two major theoretical problems have been identified that could deserve further investigation:

The 13 basic relations that are introduced for describing the relationships existing between time intervals, although primitive from an intuitive point of view, seem not to constitute a minimum set of operators from a more formal point of view. E.g., BEFORE can be easily expressed by means of MEETS as:

BEFORE(x,y):3W.3WVz((MEETS(x,z) & MEETS(z,y))

where x,y, and z are time intervals.

The model of time adopted in the paper is based on the concept of time interval, and there is the implicit assumption that intervals are indefinitely decomposable into subintervals. No attention is devoted to the concept of time instant, i.e., a point that can hardly be associated with a temporal duration. Time instants, however, have a clear intuitive meaning (e.g., the point where a time interval begins or ends, an interval that is not decomposable into subintervals, etc.) and play an important role in common sense reasoning.

Moreover, from a practical point of view, it seems worth investigating whether the model proposed can be applied, beyond the simple case studies so far developed in the fields of natural language dialogue, problem solving, and planning, to more complex and challenging domains, such as knowledge-based systems for realtime applications (signal interpretation, process control, vehicle monitoring, etc.).