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\title{Actions and  Movements\thanks{The research reported in this
paper has been made possible by a gift from the System Development
Foundation and was conducted as part of a coordinated research effort
with the Center for the Study of Language and Information, Stanford
University.}}
\author{David Israel \\ John Perry \\  Syun Tutiya}

\begin{document}
\bibliographystyle{plain}
\maketitle



\begin{abstract}

We present an account of action whose main features are that actions
are content properties that agents have in virtue of {\em (i)} the
bodily movements they effect and {\em (ii)} the wider circumstances in
which those movements are effected.  The account includes definitions
of one action being {\em a way of} doing another, and of performing
one action {\em by} performing another.  Although this account is
intended to form part of a theory of intelligent action, including the
deliberate and intentional actions of human agents or of autonomous
robots, in this paper we abstract from the information processing and
cognitive factors involved in such actions.


\end{abstract}


\section{Introduction}


Action is the goal of planning, for planning is reasoning about what
actions to perform, given certain circumstances, in order to achieve a
goal.  Most actions, perhaps all, involve movements of an agent's
body, or more generally of its movable parts.  What is the
relationship between actions and movements?  Are actions movements
that are caused in a special way, for example by way of certain
aspects of the mental states of their agents---their beliefs, desires,
and intentions?  This seems wrong, for our notions of actions involve
much more than mere bodily movements, however caused.  Consider a
simple example: moving a block from one location to another.  First,
we must determine whether we are thinking of one nonrepeatable
particular or of the repeatable kind of action: moving a block from
one location to another.  If the latter, it is clear that different
kinds of bodily movement might be involved.  Moving a block from one
place to another may be a kind of action, but it is not one kind of
movement.  Moreover, whether we have in mind a single nonrepeatable
event or a repeatable kind, it is also clear that moving a block
requires more than a bodily movement, however caused. It also requires
a block, and (roughly) a path through which the block can be moved.
The action of moving a block cannot be characterized solely in terms
of the bodily movements of agents that perform the action; on the
other hand, bodily movements can be characterized independently of the
actions those movements are associated with when performed in given
circumstances.

What is the relationship between actions and movements?  On our
theory, actions are not special kinds of movements, but {\em content
properties of agents} that agents have in virtue of performing those
movements in certain circumstances.  We proceed to explain this idea
and to show how a theory based on it can be used to develop accounts
of relationships between actions that are central to a theory of
intelligent action.

It will be useful to begin by examining a report of an action.  In the
course of this brief examination, we shall introduce much of our
theoretical vocabulary, and some of our theory.

Consider:

\begin{description}

\item[($1$)] John turned on the light.

\end{description}


We take ($1$) to describe an {\em act}: an unrepeatable event in the
past.  {\em Actions}, on the other hand, are temporal properties of
individuals (agents), in the sense of being predicable of an
individual at a time.  Turning on a light is an action, as is turning
on some particular light {\bf l}.  Various people at various times
have had the property of turning on a light; the same is true or could
be true for the property of turning on {\bf l}.  The act described by
($1$) is a movement of John's that has the property of resulting in
{\bf l}'s being turned on.  Thus actions are not kinds of
acts, but properties of individuals at times. 


($1$)  does not mention movements, nor use the term ``result''.  But
there are movements and results involved in John's action, in his
having the property of turning on the light then, and we think these
movements and results are the keys to developing a theory of
movements, actions and the relations among them.  When John turned on
the light a complex movement of his body occurred.  His elbow
straightened somewhat; his upper arm rotated forward and upward at the
shoulder.  His right index finger bent somewhat while the other
fingers of his right hand were bent more into a fist (keeping
them out of the way), and his body remained stable and his feet
stationary.  Given his position in front of the light switch, the
movement resulted in the switch being flipped to the {\em on}
position, the appropriate circuit closing, and the light going on.
John's turning on the light thus consisted in his {\em effecting} a
movement which, given the circumstances of the movement, had certain
results.

Actions are grounded in movements.  Most of the things we do, we
do by producing effects on the objects around us by moving our
bodies.\footnote{We do not here address the issue of purely mental
actions.  In the interests of simplicity, we shall be dealing only
with the kinematics of movements, abstracting completely from
considerations of masses and forces.  Finally, we shall not consider
actions of maintenance and/or prevention.}   We plan what to do in accord
with what we know about relations among actions.  Intentional action
requires {\em executing} various types of movements, with knowledge of
which effects they will have.  Which effects are produced is not
merely a matter of which types of movements are executed, however.
They also depend on the circumstances in which the movement occurs;
not only the immediate circumstances, but ones that are quite remote.
In the example above these would include, for example, the continued
operation of the electrical generating plant that supplied the lights
with power. The type of movement John executed will turn on a light in
circumstances like the ones he was in---which light depending on which
light is connected to the switch he is standing in front of; in other
circumstances it might result in someone being tweaked on the snout,
or someone being insulted, or a circus dog being commanded to do a
somersault.

Reasoning about action, either in planning/practical reasoning or in
plan recognition, must ultimately be grounded in reasoning about
movements.  In designing robots able to act effectively in a wide
variety of environments, we must keep in mind that the things the
robot can directly control are the movements of its own effectors; to
know how to do things in any of a wide variety of circumstances is to
know how to move one's body appropriately in those circumstances.  For
any intelligent agent there will be a repertoire of types of movements
that meet two conditions:

\begin{enumerate}
\item[(A)] The agent can produce movements of these types  in a
very large range of circumstances. 
\item[(B)] The agent knows what effects movements of these
types will have in (at least some of) the types of circumstances it encounters.
\end{enumerate}

\noindent That is, an intelligent agent must grasp at least
part of the causal role of the movements it can produce in
the environments in which it is likely to find itself. It
must associate with the movement types a pattern or relation
between types of environments and the effects movements of
that type will have in these environments.  We call these
relationships the {\em meanings} of types of movements.

The term ``meaning'' suggests the perspective that underlies our
approach.  We regard movements as having propositional contents, and
in this regard there is a structural similarity to utterances.  The
propositional content of an utterance---what is expressed---depends on
the type of sentence used and the context and wider circumstances in
which it occurs.  The propositional content of a movement---what it
results in---depends on the type of movement and the context and wider
circumstances in which it occurs.  The term ``meaning'' has been used
for the relation between circumstances and propositional content
associated with sentences.  Here we extend it to types of movement and
types of result.\footnote{We do not assume, however, that an act has a
unique content.  In \cite{Isrper88,Isrper91}, we develop a notion of
content that allows an event (act, utterance, etc.) to have multiple
contents.}

Our account is based on a good deal of oversimplification and
streamlining, both with respect to agents and the language we use to
describe them, and our goals are strictly limited.  We assume that the
agents in question can be viewed as systems with a set of effectors
related by an architecture; the types of movement of the whole system
are systematically determined by the types of movements available to
the effectors and the architecture.  These agents {\em effect}
movements; movements are concrete, unrepeatable particulars that
belong to various {\em types}.  Acts, then, are movements effected by
agents.  An agent who effects a movement
of a given type is said to {\em execute} that type.  Such {\em
executions}, which play the role of basic actions in our theory,
constitute our first category of actions.

When an agent effects a movement of a given type in certain definite
circumstances, that movement will have various results: propositions
made true by the effects of the movement.  We say that the agent {\em
brings about} these results.  This is our second category of actions,
which we call {\em accomplishments}.\footnote{The result that is
brought about need not be intended; we are using the term
``accomplishment'' in what might be called its wry sense, according to
which one could focus on a quite unintended result of someone's
endeavors, and say, ``That's quite an accomplishment''.} We consider
only these two categories of actions in this paper and focus most of
our attention on accomplishments.

Actions are mainly of interest to people insofar as they are done
purposively, intentionally, vindictively, and the like.  The part of
our theory that we present here does not touch any of these
interesting features of action.  The only reason it doesn't apply to a
tree falling as well as to a man shooting is that the former is not an
act: trees don't {\em effect} movements; they just move. Nothing in
our theory explains the difference between acts and other movements.

Before we can develop an account of the meanings of movements, we need
to develop some ideas about movements themselves.  We turn to this in
the next section, and to the meanings of movements in \S 3.  In \S 4,
we introduce actions and focus especially on those actions
(accomplishments) that can be characterized in terms of the result
brought about.  We also define certain central relations involving
actions and movements.  In \S 5, we briefly discuss related work and
the last section contains some conclusions and a preview of further
research.


\section{Movements} 


We take movements to be concrete particulars.  They belong
to various types, and are effected by agents at particular
times, in particular places, and in specific circumstances.
The results of a movement depend on the type of movement
effected, the agent, time, place and circumstances. Our
example focuses on a a certain coordinated movement of arm,
hand and finger that we call shall  ``flicking''.  As we have noted,
the same type of movement can be used to do different things
by different agents, in different circumstances, at
different times.  Thus it is natural to associate relations
between circumstances and results with types of movements.
We call these relations the {\it meanings} of the 
movement types. 

The movement types with which we associate meanings involve the whole
body and all its effectors.  Consider again John's flicking in the
immediate neighborhood of the light switch.  When we said that that
type of action in that circumstance would have as a result that the
light gets turned on, we didn't really just imagine John flicking.  We
imagined him flicking while standing still.  If he had moved his feet
so as to take a full step backward, while his right arm went forwards
and upwards, he would not have turned the light on, but would merely
have pawed the air ineffectually.  If he had moved his feet so as to
take a full step forward, he would not have turned the light on, but
merely have banged his knuckle against the wall.

When a person flicks, his arm and hand move as a part of an ensemble
of movements and non-movements of other bodily parts.  The movements
that we deal with in this essay are complex movements of the whole
body.  People often think of partial movements (and movement types),
and `flicking' can be thought of as a label for such a partial type.  
In so thinking, they are focusing on a (perhaps) complex partial
movement and ignoring the movements (or non-movements) of the rest of
his body.  These latter form the {\em movement context} for the
salient partial movement.  The flicking is salient because it
constitutes the increment, {\em given} the movement context and the
wider circumstances, necessary for turning on the light.  In a more
complete account, we would need to be able to keep track of these
other movements; to have a theory of movements of persons and other
systems, we need to relate it to a theory of the movements of their
parts.  For our purpose in this paper, however, a very simple
conception will suffice.  


\section{Meanings}

Let us suppose that John executed the flicking movement and turned on
the light intentionally.  We can imagine him realizing that he was
standing right in front of and in easy arm's reach of a switch that he
believed was connected in the appropriate way to the light.  Why does
he effect the movement that he does? 

We might represent what John knows about movements that explains his
doing what he did, as a generalization about (i) movements of the
body, (ii) complex types of movement (iii) circumstances in which
those movements occur, and (iv) results. 


\begin{description}

\item[($2$)]  If I effect a movement of the flicking type, while
otherwise standing still, in circumstances in which I am standing directly in
front of and within easy arm's reach of a light switch of a certain
kind that is in on the {\em off} position and which is correctly
connected up to a functioning light, then a result of such a movement
will be that the light so connected will get turned on.  Moreover, if
I simply stand still, then, in those very same circumstances, the
light will not get turned on. 
\end{description}

The second sentence should not be interpreted as saying that there is
no other way, in those circumstances, for John to bring it about that
the light is turned on.  It simply says that if John's total body
movement is a sufficient condition, in the circumstances, of the light
being turned on, then his flicking movement, in particular, is a
necessary part of that sufficient condition.


Let us now attempt to generalize and abstract, by way of the following
generalization over movements ($m$) involving parameters for total
movement types ($M$), circumstances ($C$, plus auxiliary
parameters for objects and relations involved in $C$), and results
($P$, plus auxiliary parameters).

\begin{description}

\item[(*)]  Any movement $m$ that is of type $M$, that is effected in 
circumstances of type $C(x_{1}, \ldots ,x_{n}, m)$, will have as a
result that $P(x_{i},
\ldots ,x_{l})$ ($1 \leq i \leq l \leq n$).
\end{description}


Given (*), we can associate a relation between (types of)
circumstance, and (types of) results with the movement type $M$.  We
call this its {\em meaning} and denote it as $\Lbrack M
\Rbrack$. Thus we say

\begin{quote}
$\Lbrack M \Rbrack (C, P)$ iff (*).\footnote{Note that $C$ is not
quite a property or type of circumstance, and $P$ is not quite a
proposition.  They are what might be called  parametric
properties and propositions.}
\end{quote}



\section{Actions}

We distinguish two categories of actions, {\em executions} and {\em
accomplishments}. 

\subsection{Executions}

{\em Executions} are actions defined simply by the types of movements
executed.  We use ${\cal E}M$ as short for ``executes $M$''

\begin{quote}

${\cal E}M(\alpha, t, m)$ iff 

\begin{enumerate}
\item $\alpha$ executes $m$ at $t$
\item $m$ is of type $M$
\end{enumerate}
\end{quote}

We assume that the results of a movement of type $M$ occurring are
identical with that of an agent executing a movement of type $M$.
Someone interested in the theory of dance, for example, might be
interested primarily in executions; typically, however, both as agents
and theorists, we are not primarily interested in executions, but in
accomplishments.

\subsection{Accomplishments}

{\em Accomplishments} are actions defined by results; they can be
reported by way of a certain canonical form: $\alpha$ {\em brings it
about that $P$}. We use ${\cal B}P$ as short for ``brings it about
that P''.\footnote{We have in mind cases in which the relevant causal
chain does not involve the beliefs, desires, and intentions of another
agent.  We do ultimately intend to accomodate cases, e.g., in which
one person brings something about by convincing another to perform do
something, but the intuitions we rely on here pertain to the simpler
cases.}  Later, in \S 5.1, we shall introduce a formal language within
which to model some of the logic of accomplishments; in this section,
as in the previous sections, our treatment is informal.

\begin{quote}
${\cal B}P_{x_{i}, \ldots , x_{l}}(\alpha, t)$ iff \\
$\exists M, m, \, \mbox{and} \; C$ such that

\begin{enumerate}
\item ${\cal E}M (\alpha, t, m)$
\item $C(x_{1}, \ldots, x_{n}, m)$
\item $\Lbrack M \Rbrack (C(x_{1}, \ldots, x_{n}, m), P(x_{i}, \ldots , x_{l}))$.
\end{enumerate}
\end{quote}



\subsection{The {\em way of} relation}


Our account allows us to analyze various important relations between
actions.  

Consider the following piece of practical advice or expression of
commonsense know-how:

\begin{description}
\item[($3$)]  Flipping the light switch to the {\em on} position is a
way of turning on the light (to which the switch is  connected in the
appropriate way). 
\end{description}


What do we mean when we say that flipping the switch to the {\em on}
position is {\em a way of} turning on the light?  There is a
relativity to circumstance that is suppressed.  We really mean that
flipping the switch to the {\em on} position is a way of turning on
the light in certain circumstances $C$---when the wiring is installed,
the fuse is not blown, the power is on, etc.  On the view sketched so
far, any case (act) in which an agent flips the switch will involve
that agent's executing a movement type that, in the given
circumstances, has as a result that the switch is flipped.  The same
is true for any case of an agent's turning on the light.  When we say
that accomplishing the first is a way of accomplishing the second, we
are claiming that however, in those fixed circumstances, you bring
about the first, you will have brought about the second. 

It will help to introduce the concept of an execution being a {\em mode
of} an accomplishment in a circumstance:

\begin{quote}
{\bf MO}$(C, {\cal E}M, {\cal B}P)$ iff $\Lbrack M \Rbrack ( C, P)$.
\end{quote}

\noindent  Now let $C$ be fixed as above.  In $C$, bringing it about
that the switch is flipped to the {\em on} position is a way of
bringing it about that the light is on iff any type of (total body)
movement $M$ which is a mode of bringing it about that the switch is
flipped to the {\em on} position is a mode of bringing it about the
light (to which the switch is connected) is turned on.  More
generally, we define a family of two place way of ({\bf WO}) relations
between accomplishments parameterized by $C$:

\begin{quote}

{\bf WO}$_{C}({\cal B}P,{\cal B}Q)$ iff $\forall M$ \, {\bf MO}$(C, {\cal
E}M, {\cal B}P) \Rightarrow$ {\bf MO}$(C, {\cal E}M, {\cal B}Q)$.

\end{quote}

These relations are pre-orders; they are reflexive and transitive.
Reflexivity is a mildly and innocuously counterintuitive property: in
$C$, bringing it about that the switch is flipped is a way of bring it
about that switch is flipped.  Transitivity is central to means-end
reasoning.  What of symmetry and antisymmetry? In the circumstances
$C$, bringing it about that the light is turned on is not a way of
bringing it about the switch is flipped to the {\em on} position, but
nothing in our account rules out cases in which, relative to some
circumstance $C'$, bringing it about that the light is turned on is a
way of bringing it about that switch is flipped to the {\em on}
position and vice-versa.  Plausible examples of symmetry, though, are
hard to come by.  We preempt the search for such cases by declaring in
advance our readiness to accept the anti-symmetry of the {\bf WO}
relations, and thus, our acceptance of the claim that, {\em given a
fixed circumstance $C$}, accomplishments form a partial order. 

\subsection{The {\em VO} and {\em PER} relations}

In the foregoing, we defined two temporal properties of agents, (i)
that of an agent executing a movement of a type at a time, and (ii)
that of an agent bringing it about that $P$ at a time.  We also
introduced relations among action properties and circumstances: (i)
the relation of a movement execution property being a mode of
accomplishing a proposition in a type of circumstance and (ii) the
relation of one accomplishment property being a way of for another
accomplishment property, in a type of circumstance.  We now bring
these together in an analysis of the {\em by} relation.  We do things
by doing other things; that is, we perform some actions by performing
others.  Our analysis of actions involves two categories: executions
and accomplishments.  So, too our analysis of the {\em by} relation
involves two subrelations.  Consider:

\begin{description}
\item[($4$)] John flipped the switch to the {\em on} position 
by flicking.
\item[($5$)] John turned on the light by flipping the switch to the
{\em on} position.
\end{description}

We offer definitions of two relations involving agents and
times, that of an agent bringing it about that $P$ {\em by} executing
a movement of a certain type in a circumstance at a time, and that of
an agent, at a time, bringing it about that $P$ {\em by} bringing it
about that $Q$, in a circumstance.  These two relations together
comprise our analysis of the {\em by} relation.  We use the notation
{\bf VO} (to suggest {\em in virtue of}) for the first and {\bf PER}
for the second.

\begin{quote}

{\bf VO}$(\alpha , t, C, {\cal E}M, {\cal B}P) $ iff

$\exists m$ such that

\begin{enumerate}
\item ${\cal E}M(\alpha, t, m)$ 
\item $C(m)$
\item {\bf MO}$(C, {\cal E}M, {\cal B}P)$
\end{enumerate}

\end{quote}

Thus ($4$) is true iff (roughly):
\begin{description}
\item[($4'$)]   John effected a movement of the flicking
type, in circumstances such that any movement of that type, in that
kind of circumstance, would have as a result that the switch directly
in front of which that movement was effected would be flipped to the
{\em on} position. 
\end{description}


\begin{quote}

{\bf PER}$(\alpha ,t, C, {\cal B}P, {\cal B}Q) $ iff 

$\exists M$ such that

\begin{enumerate}
\item {\bf VO}$(\alpha , t, C, {\cal E}M, {\cal B}P)$ and 
\item {\bf WO}$_{C}(C, {\cal B}P, {\cal B}Q)$.
\end{enumerate}

\end{quote}

Thus ($5$) is true (roughly) iff:
\begin{description}
\item[($5'$)]  John effected a movement of a type and in circumstances
such that any movement of that type in that kind of circumstance would
have as a result that the switch directly in front of which such a
movement was effected would be turned to the {\em on} position and in
that kind of circumstance, any type of movement that had as a result
that a switch so related to a movement of that type was flipped to the
{\em on} position, would also have as a result that the light to which
that switch was appropriately connected would be turned on.
\end{description}


\section{Related work}

There has been a great deal of work in AI on planning, and more
generally on reasoning about action.  McDermott and Allen have
developed general theories of events and actions, with special
attention to their temporal characteristics
\cite{McDermott,Allen84}.   In the situation calculus
\cite{McCarthyHayes}, actions are treated as functions from states (or
situations) to states, where these latter are themselves akin to
instantaneous snapshots of possible worlds.  

Recent work on intention of Cohen and Levesque \cite{CohenLevesque}
presents a more complicated picture.  They add action constructors
from dynamic logic to a multi-sorted first-order modal language, one
sort being of event types, so that standard temporal logic operators
are introduced by definition.  The main focus of their work, though,
is elsewhere: on a theory of the rationally balanced cognitive states
of rational agents, and in particular, a theory of commitment and
intention.  Similar remarks apply to the account of Georgeff and Rao
\cite{GeorgeffRao}.   More closely related to the ideas presented here
is Goldman's theory of action, especially the account of action
generation \cite{Goldman}; see also \cite{Pollack} for an application
to the problem of plan recognition.

\subsection{The Logic of Accomplishment}

In a series of papers, \cite{Segerberg82,Segerberg85} and especially
\cite{Segerberg89}, Krister Segerberg has attempted to exploit the dynamic
logic of programs \cite{Pratt} to provide a formal account of features
of human action.  In dynamic logic, one posits a set $S$ of total
states and the set $S^{\omega}$ of finite or infinite sequences or
{\em histories} of states.  One then associates with each program
$\alpha$ its intension, the set $S^{\omega}_{\alpha}
\subseteq S^{\omega}$ of executions sequences of $\alpha$.  Segerberg
focuses on those (human) actions that can be characterized by way of
their results and for which there exists a program or {\em routine}
for accomplishing those results.  He introduces an operator $\delta$,
to be read as ``bringing it about'' that takes sentences and yields
action (accomplishment) terms.  Where $\Phi$ is a sentence and for any
path $p$, $p(\#)$ the final element of $p$, the
intension of $\Lbrack \delta \Phi \Rbrack = \{p \, \mid \, \exists
\alpha \, (p \in \Lbrack \alpha \Rbrack \, \& \, \forall p' \, (p' \in
\Lbrack \alpha \Rbrack \Rightarrow p'(\#) \in \Lbrack \Phi \Rbrack
))\}$.  These accomplishment terms, rather than program terms (as in
standard dynamic logic), are then used to form the modal operators of
a multi-modal logic.  

Segerberg allows all of the standard operators of regular
propositional dynamic logic \cite{FischerLadner}.  We shall present
here only the sequential composition (`;') fragment of the logic, and
we shall substitute our $\cal B$ for Segerberg's $\delta$.  We had
investigated this fragment independently of Segerberg.  Our intuition
was akin to his: our movement types were the analogues of the programs
of dynamic logic (Segerberg's routines) and, like Segerberg, we had
decided to look at a language in which reference to movements
(programs/routines) is suppressed.

The language $\cal L$ of the logic consists of a language for
classical propositional logic, plus closure under the accomplishment
operators.  These are formed from the special delimiters `[' and `]'
enclosing accomplishment terms.  The sets of {\em sentences} and {\em
terms} are defined by simultaneous inductive definition, as follows:
Where $\cal P$ is a denumerable set of sentence letters,

\begin{enumerate}
\item Each $P \in {\cal P}$ is a sentence.
\item The sentences are closed under a functionally
adequate set of Boolean operators.
\item If $\Phi$ is sentence, then ${\cal B}\Phi$ is a term.
\item If $\tau_{1}$ and $\tau_{2}$ are terms, so is
$\tau_{1};\tau_{2}$.
\item If $\tau$ is a term and $\Phi$, a sentence, then $[\tau]\Phi$ is
a sentence. 
\end{enumerate}

Sentences of the form $[{\cal B} \Phi]\Psi$ are to be read as ``After
bringing it about that $\Phi$, it is the case that
$\Psi$.''\footnote{In any theory in this language, the standard form
encountered would be $\Phi \rightarrow [{\cal B} \Psi]\Theta$.}   The axioms
are as follows.\footnote{These first four axioms cover the sequential
composition fragment of regular propositional dynamic logic.}

\begin{enumerate}
\item any tautology
\item $[\tau](\Phi \wedge \Psi) \equiv ([\tau]\Phi \wedge [\tau]\Psi)$
\item $[\tau_{1};\tau_{2}]\Phi \equiv [\tau_{1}][\tau_{2}]\Phi$
\item $[\tau]${\bf T}
\item $[{\cal B}\Phi]\Phi$
\item $[{\cal B}\Phi]\Psi \rightarrow ([{\cal B}\Psi]\Theta
\rightarrow [{\cal B}\Phi]\Theta)$
\end{enumerate}

The rules are as follows:

\begin{enumerate}
\item Modus Ponens
\item External Extensionality: If $\vdash \Phi \equiv \Psi$, then
$\vdash [\tau]\Phi \equiv [\tau]\Psi$.
\item Internal Extensionality: If $\vdash \Phi \equiv \Psi$, then
$\vdash [{\cal B}\Phi]\Theta \equiv [{\cal B}\Psi]\Theta$.\footnote{We
accepted external extensionality simplicity; but were, and remain,
dubious about internal extensionality.  Still, given Segerberg's
treatment, the logic is congruential, and this is a significant
technical advantage.}
\end{enumerate}

The semantics of $\cal L$ is given in terms of frames of the following
kind: $\langle S, A, {\cal B}, {\cal P} \rangle$, where

\begin{enumerate}
\item $S$ (the domain of possible total states) is a set.
\item $A$ (the set of actions) is a set of binary relations on $S$
which is 
\begin{itemize}
\item closed under relational composition (relative product).
\end{itemize}
\item $\cal B$ (the action operator) is a function from $P$ to
$A$.\footnote{We use the same notation for both the symbol and the
function denoted by it.}
\item $\cal P$ (the set of propositions) is a set of subsets of $S$ which
is
\begin{itemize}
\item closed under set theoretic union, intersection and complement
relative to $S$; and
\item for each $R \in A$, $\cal P$ is closed under the operator $I_{R}$
defined as follows: for all $P  \in {\cal P}$, 
\begin{quote}
$I_{R}(P) = \{s \, \mid \, \forall t (\langle s, t \rangle \in R
\Rightarrow t \in P)\}$.
\end{quote}
\end{itemize}
\end{enumerate}

Segerberg presents soundness and (weak) completeness proofs for a
logic that extends full (regular) propositional dynamic logic.  The
completeness proof also yields a proof of the finite model property
and decidability.

\section{Conclusions and further research}

We have sketched an account of the nature of action whose main
features are that actions are properties that agents have in virtue of
{\em (i)} the bodily movements they effect and {\em (ii)} the wider
circumstances in which those movements are effected.  Though this
account is intended to form part of a theory of intelligent action,
including the deliberate and intentional actions of human agents or of
autonomous robots, we have abstracted quite completely from the
information processing and cognitive factors, including sensory-motor
control factors, involved in such actions.  Some quite preliminary
steps in the direction of a richer theory can be found in
\cite{Israel87,Perry86,Isrper88} and more directly in
\cite{Isrper91}.  Finally, we have borrowed a formal treatment, due to
Krister Segerberg, of the logic of bringing it about.  Further
development in this last direction requires extension to the first
order case.

\section{Acknowledgments}

We would like to thank the members of the {\bf RATAG} (Rational
Agency)  project at CSLI for stimulating discussion and helpful
criticism. 
\bibliography{action-biblio}


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