Actions and Events in Interval Temporal Logic
James F. Allen George Ferguson
University of Rochester
Rochester, NY, 14627-0226
{james,ferguson}@cs.rochester.edu
Abstract
We present a representation of events and action based on interval temporal logic that is
significantly more expressive and more natural than most previous AI approaches. The rep-
resentation is motivated by work in natural language semantics and discourse, temporal logic,
and AI planning and plan recognition. The formal basis of the representation is presented
in detail, from the axiomatization of time periods to the relationship between actions and
events and their effects. The power of the representation is illustrated by applying it to the
axiomatization and solution of several standard problems from the AI literature on action
and change. An approach to the frame problem based on explanation closure is shown to
be both powerful and natu ral when combined with our representational framework. We also
discuss features of the logic th at are beyond the scop e of many traditional representations,
and describ e our app roach to difficult problems such as external events and simultaneous
actions.
1 Introduction
Representing and reasoning about the dynamic aspects of the world—primarily about
actions and events—is a problem of interest to many different disciplines. In AI, we
are interested in such problems for a number of re asons, in particular to model the
reasoning of intelligent agents as they plan to act in the world, and to reason about
causal effects in the world. More specifically, a gener al representation of actions and
events has to support the following somewhat overlapping tasks:
• Prediction: Given a description of a scenario, including actions and events, what
will (or is most likely to) happen?
• Planning: Given an initial description of the world and a desired goal, find a
course of action that will (or is most likely to) achieve that goal.
• Explanation: Given a set of observations about the world, find the b e st expla-
nation of the data. When the observations are another agent’s actions and the
explanation desired is the agent’s plan, and the problem is called plan recogni-
tion.
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Our claim in this paper is that in or der to adequately represent actions and events,
one needs an explicit temporal logic, and that approaches with weaker temporal mod-
els, such a s state spaces (e.g., STRIPS-based approaches) and the situation calculus,
either cannot handle the problems or requir e such dramatic extensions that one in ef-
fect has grafted an explicit temporal logic onto the earlier formalis m. Fur thermore, if
one of these formalisms is extended in this way, the temporal logic part will dominate
and the original formalism plays little role in the solution. We will primarily defend
this position by proposing a specific temporal representation and showing that it ca n
handle a wide range of situations that are often problematic for other formalisms. In
particular, here are some of properties of actions and events that we feel a re essential
to any general representation:
1. Actions and events take time. While some events may be instantaneous, most
occur over an interval of time. During this time, they may have a rich structure.
For instance , the event of driving my car to work involves a wide range of
different actio ns, states of the world and other complications, yet the activity
over that stretch of time can be nicely desc ribed as a single event. Because
events are e xtended in time, different events and actions may overlap in time
and interact. Thus, if while I am driving to work, a rock pierces the gas tank
and the gaso line drains out, I may end up being stranded on the highway rather
than arriving at work on time. We should be able to represent and reason about
such complex interactions.
2. The relationship between actions and events and their effects is complex. Some
effects become true at the end of the event and remain true for some time after
the event. Fo r example, when I put a book on the table, this has the effect that
the book is on the table for at least a short time after the action is completed.
Other effects only hold while the event is in progress. For instance, consider a
flashlight with a button for flashing it. The light is on only when the button is
being press e d down. Finally, other effects might start to hold sometime after
the event has started and stop holding before it finishes. For example, if I am
walking to school alo ng my usual route, at some time during the walk I will be
on the bridge crossing the river. This is an effect of the action even though it
is not true at the end of it.
3. Actions and events may interact in complex ways when they overlap or occur
simultaneously. In some ca ses, they will interfere with certain effects that would
arise if the events were done in isolation. In other cases, the effect of performing
the two actions may be completely different that the effects of each in isolation.
As a radical case of this, co ns ider a wristwatch that is controlled by two buttons
A, and B. Pressing A changes the mode of the display, while B shows the alarm
time se tting. Pressing A and B simultaneously, however, turns the alarm on or
off. The effect of p erforming the actions simultaneously has no causal relation
to the effects of the actions performed in isolation.
4. External changes in the world may occur no matter what actions an agent pla ns
to do, and may interact with the planned actions. Possible external events
should be an important factor when reasoning about wha t effects an action
might have. Certain goals can only be accomplished by depending on external
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events. For example, to sail across a lake, I can put up the sails but I depend
on the wind blowing to bring about the event.
5. Knowledge of the world is necessarily incomplete and unpredictable in detail,
thus prediction can only be done on the basis of certain assumptions. Virtually
no plan is foolproof, and it is imp ortant that a fo rmalism makes the necessary
assumptions e xplicit so that they can be considered in evaluating plans.
Our aim is to develop a general representation of actions and events that supports
a wide range of reasoning tasks, including planning, explanation, and prediction,
but also natural language unders tanding, and commonsens e reasoning in general.
Most of the pr evious work on these problems tends to address only a subset of these
problems. The STRIPS-based planners (e.g., TWEAK [11], SIPE [69], SNLP [41]),
for instance, only address the planning problem, while wor k in the situation calculus
(e.g., [43, 9]) has primarily focussed on the prediction problem or on using it as an
abstract theory for planning (e.g., [25]). Natural language work, on the other hand,
typically only deals with commonsense entailments from statements about actions and
events, sometimes with a focus on plan recognition and explanation (e.g., [4, 30, 57]).
Our representation is intended to serve as a uniform representation to support all
these tasks . As a result, we try to avoid introducing any specific syntactic constructs
that support only one reasoning task. Knowledge should be encoded in a way so that
it is usable no matter what reasoning task is currently being performed.
Section 2 outlines our intuitions about actions and events, and briefly explores
the two predominant models of action: the situation calculus and the state-based
STRIPS-style representations. As typically formulated, neither is powerful enough
for the issues described above. Section 3 then introduces Interva l Temporal Logic,
first defining the temporal structure, then introducing properties, events, and actions.
Section 4 demonstrates how the interval logic can be used to solve a set of simple
problems from the literature in order to facilitate compa rison with other approaches.
The key place where other approaches have difficulty is in repre senting external events
and simultaneous actions. Section 5 explores the implications of external events in
detail, and Section 6 explores intera cting simultaneous actions .
2 Representing Actions and Events
Before starting the formal development, we will attempt to describe the intuitions mo-
tivating the representation. We will then consider why the most commonly accepted
representations of action in AI will not meet our needs.
2.1 Intuitions about Actions and Events
The first issue concerns what an event is. We take the position that events are
primarily linguistic or cognitive in nature. That is , the world do e s not really contain
events. Rather, events are the way by which agents classify certain useful and relevant
patterns of change. As such, there are very few restrictions on what an event might
consist of except that it must involve at least one object over some stretch of time, o r
involve at least one change of state. Thus the very same circumstances in the wor ld
might be described by any number of events. For instance, consider a circumstance
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in which a ball rolled off a table and bounced on the flo or. This already is one
description of what actually happe ned. The very same set of circumstances could
also be des c ribed as the event in which the ball fell off the table, or in which the ball
hit the ground, or in which the ball dropped. Each of these descriptions is a different
way of describing the circumstances, and each is packaged as a description of an event
that occurred. No one description is more corr e c t than the other, although some may
be more informative for certain circumstances, in tha t they help predict some r e quired
information, or suggest a way of reacting to the circumstances.
Of course, the “states” of the wor ld referred to above are also ways of classifying
the world, and are not inher e nt in the world itself either. Thus , the same set of
circumstances described above might be partially described in ter ms of the ball being
red. Given this, wha t can one say about the differences be tween events and states?
Intuitively, one describes change and the other describes asp e c ts that do not change.
In lang uage, we say that events occur, and that states hold. But it is eas y to blur these
distinctions. Thus, while the balling falling from the table to the floor clearly describes
change a nd hence des cribes an event, what about the circumstance where an agent
John holds the door shut for a couple of minutes. While the door remains shut during
this time and thus doesn’t change state, it seems that John holding the door shut
is something that occured and thus is like an event. These issues have been studied
extensively in wor k on the semantics of natural languag e sentences. While there are
many proposals, everyone agrees on a few basic distinctions (e.g., [65, 47, 15]). Of
prime relevance to us here are sentences that describe states of the world, as in “T he
ball is red,” or “John believes that the wor ld is flat,” and sentences that describe
general ongoing activities such as “John ran for an hour,” and events such as “John
climbed the mountain.” Each of these types of sentences has different properties, but
the most impo rtant distinctions occur in their relatio n to time. All these sentences
can be true over an interval of time. But when a state holds over an interval of time
t, one can conclude that the state also held over subintervals of t. Thus, if the ball
is red during the entire day, then it is also red during the morning. This property
is termed homogeneity in the temporal logic literature. Events, on the other hand,
generally have the opposite property and are anti-homogeneous: If an event occurs
over an interval t, then it doesn’t occur over a subinterval of t, as it would not yet be
completed. Thus, if the ball dropped fr om the table to the floor over time t, then over
a subinterval of t is would just be somewhere in the air between the table and floor.
Activities, on the other hand, fa ll in between. They may be homogenous, as in the
holding the door shut example above, but they describe some dynamic as pect of the
world like events. This type of distinction must appreciated by any general purpose
knowledge representation for action and change.
The other general point to make is that intervals of time play an essential role in
any representation of events. Events are defined to occur over intervals o f time, and
cannot be reduced to some set of properties holding at instantaneous points in time.
Of course, semantically, one can define time intervals as a n ordered pair of time points,
but the truth conditions must be defined in terms o f these ordered pairs, and not in
terms of the individual points. Specifically, if an interval were simply a set of points
between two points, and truth over a n interval was defined in terms of truth over all
the points in the interval, then every predicate would have to be ho mogeneous.
Finally, a word on actions. The word “action” is used in many different senses by
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many different people. For us , an action refers to something that a person or robot
might do. It is a way of classifying the different sorts of things than an agent can do
to affect the world, thus it more resembles a sensor y-motor program than a n event.
By performing an action, an a gent causes an event to occur, which in turn may cause
other desired events to also occur. For instance, I have an action of walking that I may
perform in the hope of causing the event of walking to my car. Some theories refer to
the event that was caused as the action, but this is not what we intend here. Rather,
we will draw on an ana logy with the robot situation, and view actions as programs.
Thus, pe rforming an a ction will be described in terms of running a program.
2.2 The Situation Calculus
The situation calculus means different things to different researchers. In its origi-
nal formulation [43], which we will call the general theory o f the situation calculus,
situations are introduced into the ontology as a complete snapshot of the universe
at some instant in time. In effect, the situation calculus is a point-based temporal
logic with a branching time model. In its most common use, which we will call the
constructive situation calculus, it is used in a highly restricted form first proposed by
Green [25], in which the only way situations are intro duced is by constructing them
by action composition from an initial state. This practice has attracted the most
attention precisely b e c ause the formalism is constructive—spe cifically, it can be used
for pla nning. To contruct a plan for a goal G, prove that there exists a situation s in
which G holds. In proving this, the situation is constructed by action composition,
and thus the desired sequence of actions (the plan) can be extracted from the proof.
As others have pointed out (e.g., [55]), most of the criticisms about the expressibility
of the situation calculus concern the constructive form of it rather than the general
theory. Our p osition is that the constructive s ituation calculus is a limited repre-
sentation, especially in dealing with temporally complex a ctions and external events.
The general theory, on the other hand, is much richer and can be extended to a model
much close r to what we are proposing, but at the loss of its constr uctive aspect.
To see some of the difficulties, first consider a very simple example. In a domain
that r e asons about transportation of cargo, we might want to define an action of
a train moving some c argo between two cities. Some of the information needed to
reason about this action is the fo llowing:
1. The train initially starts at the originating city S.
2. The trip typically takes between 4 and 6 hours.
3. The cars must remain attached to the train during the entire trip.
4. The train will be on track segment A1 then it will cross junction J1, a nd be on
track segment A2 for the rest of the trip.
5. The train will end up at the destination city D.
This is all very mundane information, but each fact might be crucial for some planning
task. For instance, knowledge of the track segments tha t the train is on during the
trip mig ht be used to avoid having multiple trains on the same track at the same
time.
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