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\begin{titlepage}
\begin{center}
{\LARGE \bf Representing Change}\\[1cm]
Elisha Sacks\\[.5cm]
MIT Laboratory for Computer Science\\
545 Technology Square, Room 370\\
Cambridge, MA  02139\\
U.S.A.\\
Tel: (617) 253-3447\\
Net: {\tt ELISHA@MIT-ZERMATT.EDU} \\[5mm]
{\Large \bf Abstract}\\[1.2cm]
\parbox{\textwidth}{This paper evaluates knowledge representations for
time-dependent information.  It compares recent work by Moore,
McDermott, and Allen with an earlier proposal by McCarthy and Hayes.
Moore's formalism is faulted for its needless and unmotivated complexity
and a simpler alternative is outlined.  McDermott's formalism is proved
inconsistent and unintuitive.  Allen achieves the most by attempting the
least.  He proposes a simple plausible formalism, which makes few
ontological or computational commitments.  The paper concludes with a
high-level discussion of the merits formal logic as a representation for
empirical knowledge.}

\end{center}
\begin{flushleft}
{\bf Keywords:} knowledge representation, change, time, situation calculus, nonmonotonic logic\\

\vfill
This research was supported (in part) by National Institutes of
Health Grant No. R01 LM04493 from the National Library of Medicine and
National Institutes of Health Grant No. R24 RR01320 from the Division of
Research Resources.
\end{flushleft}

\end{titlepage}

\pagenumbering{roman}
\tableofcontents
\newpage
\pagenumbering{arabic}

\section{Introduction}
How can an artificially intelligent computer make it in the real world?
It must know what the world is like and use that knowledge to achieve
its goals, or in AI terms, it needs a world model on which to base its
inference mechanisms.  A complete model should capture all important
aspects of reality: three-dimensional geometry, topology, objects,
properties, naive physics, elementary human psychology, and the
epistemological kitchen sink.  But that's too much for one paper, so I
focus on representing change.

A robot that ignores changes will run into trouble sooner than Mr.\
Magoo.  The first time it tries to cross a momentarily empty street
without considering oncoming cars, it will probably (definitely, in
Boston) be destroyed.  On the other hand, the robot could get nothing
done if it assumed that anything could change arbitrarily.  It would
never try to cross a totally empty street, since a car might appear from
nowhere, or the asphalt might disappear.  In order to steer between
these two extremes, the robot needs to know how and when things change
in the real world. 

In this paper, I evaluate three recent proposals for representing
change and compare them with an earlier one by McCarthy and Hayes.  All
four use the same strategy, though differing in specifics.  They represent
the world by sentences of some formal logic and reason about the world
within its proof theory.  Initially, I adopt this paradigm on faith
and examine how well the four models implement it.  I discuss
McCarthy and Hayes's situation calculus in section~2 and continue with
the ideas of Moore, McDermott, and Allen in the next three sections.  I
argue that all three (despite some claims to the contrary) are 
simply modified versions of the situation calculus, not radically new
ideas. In section~6, I lose my faith and question the formal paradigm
itself.  According to time-honored tradition, I conclude with a summary.
						
\section{Situation Calculus}
McCarthy and Hayes \cite{mcCarthy1} propose a representation for change,
called situation calculus, that subsumes similar work by McCarthy
\cite{mcCarthy2} and Green \cite{green1}.  They utilize the
{\it possible worlds} model of reality developed by Lewis
\cite{lewis1,lewis2}, Stalnaker \cite{stalnaker1,stalnaker2}, and other
philosophers dating back to Leibniz.  Every conceivable instantaneous
state of the universe corresponds to a possible
world,%
\footnote{Philosophers disagree on just what possible worlds are.  In
fact, many deny their existence altogether.  For more on this issue, see
my paper ``What are possible worlds?'' \cite{sacks1}}
 {\it situation} in McCarthy's terminology.  Successor links connect 
each situation with all other situations that could follow immediately
after it.  Every path through the graph of situations forms a possible
history of the universe, but only one path traces its actual history.

The situation calculus formalizes the possible worlds theory in
first-order predicate calculus.  It represents situation dependent
propositions and functions as predicates and functions on worlds, called 
{\it fluents.}  For example, ``owns a car'' corresponds to the
propositional fluent $owns(p,car,s)$ and ``president'' to the
functional
fluent {\it president(s)}.  The functional fluent $result(p,a,s)\/$
denotes the situation that results from agent {\it p}
performing action {\it a} in situation {\it s},  so ``if you feed a cat
it will be 
quiet'' can be formalized as
\[\forall s \forall c \forall agent
   \left[ cat(c)\wedge (t=result(agent,feed(c),s))\right] 
 \Rightarrow quiet(c,t)\]
where {\it feed} is a function from objects to actions.  Simple actions
are primitive irreducible entities, whereas complex actions, called 
{\it strategies,} are specified by computer programs in which calls to
{\it result} change the value of a global situation variable.  The act
of walking five blocks could be written as
 \begin{quote} 
for i=1 to 5 do s:=result(agent,walk-a-block,s)
\end{quote}
with s denoting the changing world situation.  Unlike the general
possible worlds model, situation calculus is deterministic; every action
has a unique result.  However, nondeterminism could be introduced by
generalizing the {\it result} function to a {\it possible-result\/}
relation between an agent, an action, a prior situation, and a resulting
situation.

McCarthy and Hayes incorporate models of time and knowledge into the
situation calculus, without leaving first-order logic.  Time is
represented by the {\it time} function from situations to their time
of occurrence, a real number, and by the {\it cohistorical} predicate
that
holds between pairs of situations belonging to the same world history.
Intuitively, the {\it time} value of a situation shows when it would
have occurred if its history were the actual one.  Non-cohistorical
situations that have the same {\it time} represent alternate states of
the universe at that instant.  {\it Cohistorical} must be an equivalence
relation in any reasonable theory of time, but other axioms may be added
as desired.  For instance, one could rule out multiple histories by
making all situations cohistorical.  Other axioms could stipulate
continuous, discrete, infinite, finite, or circular models of time.
Once {\it cohistorical} and {\it time} have been introduced, other
useful
concepts can be expressed, such as the $\prec$ precedence relation 
\begin{equation}
s\prec t \stackrel{\rm def}{=}cohistorical(s,t)\wedge time(s)<time(t)
\end{equation}
which says that situation {\it t} lies in the future of {\it s}; {\it t}
occurs in the same history as {\it s}, at a later date.

A useful model of the world must represent what agents know, as well as
what they do, since the two concepts are interdependent.  An ape cannot
write this paper because it does not know English, but I can because I
do.  Conversely, the action of reading this paper affects the readers
knowledge.  Knowledge can be represented by a first-order analog of
Hintikka's \cite{hintikka1} logic of knowledge.  Hintikka uses a special
operator, $know(agent,prop)$ to formalize the proposition ``agent
knows proposition.''  The axioms
\begin{enumerate}
\item $know(a,p)\Rightarrow p$
\item $know(a,p)\Rightarrow know(a,know(a,p))$
\item $know(a,p\Rightarrow q) \Rightarrow \left[ know(a,p)\Rightarrow
      know(a,q)\right]$,
\end{enumerate}
closed under the principle
\begin{enumerate}\setcounter{enumi}{3}
\item if {\it p} is an axiom then $know(a,p)$ is an axiom,
\end{enumerate}
specify the meaning of the {\it know} operator.  Axiom~1 says everything
known is true, axiom~2 says everyone knows what he knows, axiom~3 says
everyone knows Modus Ponens and principle~4  makes all axioms, including
1--3, common knowledge.  Axiom~3 and principle~4 jointly imply that
everyone knows all logical implications of his knowledge, including all
tautologies, since he knows the axioms and how to apply the rule of
inference.  For a fixed agent, these knowledge axioms are equivalent to
the modal axioms of $S4,$ so adding them to predicate calculus produces
a version of modal $S4$ indexed by agents, with a corresponding indexed
version of Kripke's \cite{kripke1} possible worlds semantics.  Instead
of a single accessibility relation, there is one for each agent which
holds between two worlds when the first is compatible with everything he
knows in the second.  The proposition $know(a,p)$ is true in world {\it
w} iff {\it p} is true in all worlds accessible to {\it a} from {\it w}.
This definition, along with the intuitively plausible requirements that
the accessibility relation be reflexive and transitive in its world
arguments, guarantees the validity of Hintikka's axioms. 

Rather than extend situation calculus to include Hintikka's theory of
knowledge, McCarthy and Hayes encode its model theory within the
existing formalism.  The advantage of this approach, discussed at length
by Moore \cite[pp.\ 11--20]{moore1}, is the relative simplicity and
tractability of standard predicate calculus compared with its modal
relatives.  They encode the accessibility relation as a reflexive
transitive predicate $shrug(a,s,t)$ between an agent and two situations.
Two {\it shrug} related situations are {\it epistemologically
equivalent\/} to an agent.  When asked which of the two he is in, he can
only shrug his shoulders.  The modal proposition $know(a,p)$ translates
to the first-order proposition
\begin{equation}
\forall s \: shrug(a,s_0,s)\Rightarrow p(s)
\end{equation}
where $p(s)$ is a fluent and $s_0$ denotes the situation corresponding
to the real world.  This scheme seems to capture the same intuitions as
Hintikka's knowledge axioms, since it translates all of them into
theorems of situation calculus.  For example, axiom~1 translates to 
\begin{equation}
 \left[\forall s\: shrug(a,s_0,s)\Rightarrow p(s)\right]\Rightarrow
    p(s_0)
\end{equation}
which follows from the reflexivity of {\it shrug} after instantiating
{\it s} with $s_0$ and using conditional proof.  Hence, McCarthy and
Hayes feel justified in replacing Hintikka's modal axioms with their
first-order counterparts.

This representation of knowledge seems completely impractical.  First,
it assumes that agents know all logical implications of their premises,
in particular all analytic truths.  The standard answer, that
completeness is an idealization similar to the frictionless point-mass
of mechanics, leaves me unconvinced.  Idealizations should focus
attention on key concepts by suppressing irrelevant details.  For
example, friction distracts a beginner from understanding oscillation or
inertia.  In our case, completeness is not a useful idealization, since the
computational limitations of humans and robots are the concept of
interest, not a tedious detail.  Similarly, frictionless point-masses
are not a useful idealization for modeling torque.  
Moore \cite[p.\ 8]{moore1} recognizes this problem, but offers no
solution beyond vague references to default reasoning and to the work
of Konolige \cite{konolige1}.  Konolige's solution is to represent
knowledge (or other propositional attitudes, such as belief) in an
incomplete deductive system.  An agent only knows the implications
of its belief in its restricted language.  It remains ignorant of
their additional logical consequences.

Second, the user must stipulate what situations {\it shrug} relates,
since the axioms merely constrain it to be reflexive and transitive.
Moore's work, discussed in the next section, provides a partial solution
to this problem by defining {\it shrug} for the results of an action
in
terms of its values on prior situations.  Yet this still leaves a lot to
the user.  He must extend the {\it shrug} relation to each new
situation that arises, except for results of actions. 

In this section, I have described the situation calculus, a first-order
predicate calculus version of the possible worlds model of reality.  It
represents situation-dependent facts by predicates on situations and
change by the {\it result} function, which maps the performance of an
action in a situation to the resulting situation.  Theories of time and
knowledge can be incorporated into this formalism by defining a few
straightforward predicates.  In the next three sections, I discuss
alternative representations for change, all of which claim to improve
upon situation calculus.

\section{Moore's Extension}
The situation calculus represents actions, time, and knowledge, but
fails to tie them together, as McCarthy and Hayes 
\cite[p.\ 497]{mcCarthy1} realize: 
\begin{quote}				
Rather more interesting [than the familiar axioms of time and knowledge]
would be axioms relating `shrug' to `cohistorical' and time;
unfortunately we have been unable to think of any intuitively plausible
ones.  Thus, if two situations are epistemological alternatives (that is,
$shrug(p,s_1,s_2))$ then they may or may not have the same time value
(since we want to allow that {\it p} may not know what the time is), and
they may or may not be cohistorical.
\end{quote}
Moore \cite[sec.\ 1]{moore1} stresses the interdependence of knowledge
and action in planning.  A good planner must reason about knowledge
prerequisites for its actions, as well as physical ones, and fill in
missing knowledge by taking appropriate actions.  For example, a robot
cannot open a combination safe unless it {\it knows} the combination, or
make a phone call unless it knows the number.  However, McCarthy and
Hayes must use the {\it ad hoc} functions {\it idea-of-combination} and
{\it idea-of-phone-number} to formalize these rules.  Moore proposes a
formalism for knowledge and action, in which such examples follow from
general principles, without special rules.  In section~\ref{s-moore}, I
describe a simplified version of his theory that can be expressed
directly in situation calculus.  Before doing so, I argue that the
original gains nothing from its tremendous complexity.

\subsection{The Complicated Original}
Moore bases his theory of knowledge on Hintikka's modal logic, as do
McCarthy and Hayes, but chooses to include both the original object
language $\cal H$ and its model theory in a new first-order object
language $\cal M$, rather than just the model theory.  He represents 
$\cal H$ variables and constants by $\cal M$ constants; and $\cal H$
functions, predicates, and connectives by $\cal M$ functions, so the 
$\cal H$ sentence 
$know(john,\exists x p(x))$
translates to the $\cal M$ term, called an {\it object term,}
$know(john,exist(x,p(x)))$
made up of the {\it know, exist,} and {\it p} functions and {\it john}
and {\it x} constants.  Next, he encodes the model-theoretic relation
``{\it p} is true in world {\it w}'' as an $\cal M$ predicate, $t(w,p)$
on situations and object terms.  Compound object terms can be reduced to
atomic ones by repeated application of appropriate axioms to their main
connective, for instance
\begin{equation}
\forall w\forall p\forall q\: t(w,and(p,q))\Leftrightarrow
 \left[ t(w,p)\wedge t(w,q)\right]
\end{equation}
for conjunction.  An atomic object term $p(t_1,\ldots ,t_n)$ is true in
{\it w} iff the denotations of its arguments in {\it w} satisfy the
corresponding $n+1$ place predicate $:\!p$, according to the axiom
\begin{equation}
\forall w \:t(w,p(t_1\ldots ,t_n))\equiv \,:\!p(w,d(w,t_1)\ldots
    ,d(w,t_n))
\end{equation}
where $d(w,t)\/$ is the denotation of {\it t} in {\it w}.  The {\it d}
function, in turn, can be reduced to rigid designators, but I skip the
details. Finally, {\it t} is defined for {\it know} by the axiom 
\begin{equation}
\forall w \forall p \forall s \: t(w,know(p,s))\equiv
   \forall u\: k(p,w,u)\Rightarrow t(u,s)
\end{equation}
where {\it k} is a synonym for the {\it shrug} accessibility relation
of section~2.

What advantage does $\cal M$ have over the simpler knowledge
representation of situation calculus?  According to Moore,
\begin{quote}
This has the advantage of letting us use either the modal language or
the possible-world language---whichever is more convenient for
a particular purpose---while rigorously defining the connection between
the two. \cite[p.\ 30]{moore1}       
\end{quote}
I reject this claim.  The only connection that Moore defines is
between formulas of the form $t(w,p)$ and other, {\it t} free, formulas.
This connection allows one to interchange propositions about object
terms with equivalent object-term-free propositions, but asserts nothing
about the relation between an $\cal H$ sentence and its corresponding
object term.  To substantiate his claim, Moore must prove some sort of
equivalence between $\cal H$ and $\cal M$, not between object terms and
their definitions.

Although Moore is unaware of his debt to rigor, others have tried
unsuccessfully to discharge it.  As a first step, it seems reasonable to
define the translation of the $\cal H$ formula {\it h} as the $\cal M$
formula $\forall w\: t(w,m)$ where {\it m} is the object term
corresponding to {\it h}.  One might hope that the theorems of $\cal M$
would be exactly the translations of the theorems of $\cal H$.
Unfortunately, this proves false, since $\cal M$ contains theorems, such
as $\forall w\: w=w$ that are not the translations of any $\cal H$
formula, let alone of theorems.  The conjecture 
\begin{conjecture}\label{c-moore}
{\it h} is a theorem of $\cal H$ iff its translation is a theorem of 
$\cal M$ 
\end{conjecture}
avoids this difficulty and seems plausible.  However, I see no way of
proving it, nor have I found one in the philosophical literature.  In
fact, Lewis \cite{lewis2} proposes a translation of modal logic into
first-order predicate calculus that closely resembles $\cal M$, but
makes no equivalence claims.  More recently, Forbes 
\cite[chap.\ 4]{forbes1} discusses the difficulties involved in proving
conjecture~\ref{c-moore}.  In any case, the burden of proof lies with
Moore.  

Theoretical difficulties aside, I think $\cal M$ is a bad idea, because
it confuses data structures with algorithms.  A theory of knowledge,
like any database, should consist of two parts: a representation for
knowledge and algorithms for translating input from external to internal
form and back.  Moore uses the knowledge representation to do both jobs,
thus making his system needlessly complex.  Each knowledge query must be
translated into situation terms, analyzed, and the results translated back
into modal language, as demonstrated by the theorems proved by Moore
\cite[pp.\ 39--40, 61--64]{moore1}.  It is much simpler to keep the
first-order model free of any reference to the modal language and leave
translation to an interface program.  The new formalism cannot express
the (as yet unproved) equivalence between $\cal H$ formulas and their
object terms, but that job belongs to the user interface.  Put another
way, the equivalence should be proved and used in the meta-language%
\footnote{Every paper needs a meta.}
that describes the knowledge representation, not in the object language.

In this section, I criticized  Moore's first-order translation of
Hintikka's logic of knowledge $\cal H$ for being overly complex and
suggested simplifications.  The simplified formalism is equivalent to the
situation calculus theory translation of $\cal H$ described in
section~2.  In the next section, I recast Moore's theory of knowledge
and action in situation calculus, thus demonstrating that the simplified
theory retains sufficient expressive power for his purposes.  His only
justification for the more complicated one, that modal propositions be
interchangeable with their translations, confuses object language and
meta-language.

\subsection{A Simpler Alternative}\label{s-moore}
Moore's primary goal is to formalize the interdependence of knowledge
and action, which eluded McCarthy and Hayes.  He reinterprets the action
argument of the {\it result} function as an {\it event}.  This
distinction has no practical effect on the theory, since the only events
considered consist of actions by agents, represented by the
$do(agent,action)$ function.  However, it reappears in sections~4 and~5,
so keep it in mind.  Just as in situation calculus, complex actions can
be specified by sequences of actions, conditional actions, and iteration
over actions.  Each has an appropriate event associated with it by the
{\it do} function. 

How does action depend on knowledge?  Moore claims that knowing how to
perform an action consists of having a descriptor for that action and
knowing what object it denotes.  Moreover, most routine actions can be
described as applications of known procedures to appropriate arguments.
In these cases, knowing how to perform the action reduces to knowing 
its arguments.  For example, a combination safe can be opened by the
general routine {\it dial(combination(safe),safe)}, so knowing how to
open it reduces to knowing the denotations of {\it safe} and
{\it combination(safe)}, that is knowing which safe to open with what
combination.  Knowing what an expression denotes in world {\it w} is
expressed in situation calculus as
\begin{equation}\label{e-know}
\exists x \forall u\: k(agent,w,u)\Rightarrow x=exp
\end{equation}
where {\it exp} contains fluents with world variable {\it u}.  The
intuitive meaning of~(\ref{e-know}) is that {\it exp} denotes the same
object in every world compatible with what {\it agent} knows in {\it w}.
In particular, a burglar could open a safe if the proposition
\begin{equation}
\exists x \forall u\: k(burglar,\mbox{\it real-world},u)\Rightarrow
           x=combination(safe(u),u)
\end{equation}
were true.  An agent can also know how to perform an action without
initially knowing what object it denotes, if he knows a multiple-step
action whose early steps fulfill the knowledge prerequisites of its
later ones.  Moore claims that these are the only two ways.

An agent's actions affect its knowledge, just as its knowledge affects
its actions.  Moore distinguishes between {\it informative} actions that
tell the agent something about the world and {\it noninformative}
actions that do not.  Reading a thermometer is informative, whereas
dancing is not.  (Of course these are abstractions, since every
real-world action yields some information.)  Let us call the situations
compatible with an agent's prior knowledge {\it antecedents} and those
compatible with his post-action knowledge {\it consequences}.  After
performing a noninformative action, an agent knows that it occurred,
but learns nothing else, so each consequence is the result of some
antecedent.  Equivalently, the consequences are the image under
{\it result} of the antecedents.  This is expressed formally as
\begin{equation}\label{e-inf}
\forall u\forall v\: result(e,u,v)\Rightarrow\left[ \forall w \:k(a,v,w)
   \Leftrightarrow \exists x \:k(a,u,x)\wedge result(e,x,w)\right]
\end{equation}
a state {\it w} is compatible with the result of event {\it e} in state
{\it u} iff {\it w} is the result of {\it e} in some state {\it x}
compatible with {\it u}.  On the other hand, after performing a {\it p}
informative action, an agent knows whether or not {\it p} is true, so
the consequences are those results of antecedents in which {\it p} takes
on the correct truth value.  This is formalized by adding a third
conjunct $p(v)\equiv p(w)$ to~(\ref{e-inf}).  As an example, suppose
$w_3$ and $w_4$ are the results of {\it e} on $w_1$ and $w_2\/$ and that
$p(w_3)$ is true and $p(w_4)$ false.  If {\it e} is noninformative and
$w_1$ and $w_2$ are John's antecedents then $w_3$ and $w_4$ are his
consequences.  If {\it e} informs John that {\it p} is true then $w_3$
is his only consequence.  

\subsection{Summary} 
Moore hypothesizes relations between knowledge and action and formalizes
them in situation calculus.  An agent knows how to perform an action if
it has a descriptor for the action and knows what action the descriptor
denotes.  Performing an action maps an agent's state of knowledge to one
compatible with the results and the agent's prior knowledge.  Moore
improves on McCarthy and Hayes's work by binding together two formerly
independent concepts into a single integrated theory.

\section{McDermott's Confusion}
McDermott \cite{mcdermott1} proposes a temporal logic for representing
causality, continuous change and problem solving.  He prefaces the
theory with a bold claim: 
\begin{quote}
Actually, of course, no one has ever dealt with time correctly in an AI	
program.
\end{quote}
According to him, the underlying logic of existing programs, situation
calculus, cannot adequately model time-varying facts or hypothetical
future events.  In this section, I argue that his theory is logically
confused, philosophically dubious and overly complex.  Nor does his case
against situation calculus seem compelling, since time-varying facts can
be represented by fluents and hypothetical futures by situations.

\subsection{Time, Facts, and Events}
McDermott adopts the situation calculus model of states partially
ordered by the $\prec$ relation and mapped onto the real line by
the {\it time} function.  (In fact, he adopts the nonstrict version of
$\prec$, which he renames $\leq$, and {\it time} as his primitives and
defines {\it cohistorical} in terms of them.)  Whenever one
state strictly precedes another, some third state lies in between.
Next, he defines {\it chronicles,} totally ordered sets of states that
cover all times, symbolically  
\begin{equation}\label{e-chron}
\forall t \exists s \: s\in ch \wedge time(s)=t
\end{equation}
where {\it t} is a time, {\it s} a state, and {\it ch} a chronicle. 
Each chronicle represents a complete possible history of the universe.
An ordered set is {\it convex} if whenever it contains two points, it
contains all points in between.  Figure~\ref{f-convex} provides a
counterexample to McDermott's claim that chronicles are convex by their
definition. 
\begin{figure}

\begin{picture}(50,11)
\thicklines
\put(0,7){\vector(1,0){10}}
\put(10,7){\line(2,1){10}}
\put(20,12){\line(2,-1){10}}
\put(30,7){\vector(1,0){10}}

\thinlines
\put(10,7){\line(2,-1){10}}
\put(20,2){\line(2,1){10}}

\put(9,8){\makebox(2,2){\bf a}}
\put(19,12){\makebox(2,2){\bf b}}
\put(19,0){\makebox(2,2){\bf c}}
\put(30,8){\makebox(2,2){\bf d}}
\end{picture}

\caption{Convexity Counterexample}\label{f-convex}
\end{figure}
State {\it c} is between {\it a} and {\it d}, but outside the {\it abd}
chronicle. 

Fortunately, convexity follows from the restriction that states never
branch into the past 
\begin{equation}\label{e-back}
\forall s \forall t \forall u (\:s\leq u \wedge t \leq u) \Rightarrow
   (s\leq t \vee t\leq s)
\end{equation}
only the future, because there exists only one, perhaps unknown, past,
but many possible futures.  Suppose state {\it s} belongs to chronicle
{\it ch} and {\it t} strictly precedes {\it s}.  There exists $t'\in ch$
such that $time(t')=time(t)$ by~(\ref{e-chron}).  State $t'$ precedes
{\it s} because $time(t')=time(t)<time(s)$ and {\it t} precedes {\it s}
by hypothesis, so either $t'$ precedes {\it t} or {\it t} precedes $t'$
by~(\ref{e-back}).  Either way, the precedence cannot be strict because
$time(t)=time(t')$, so they must be equivalent.

Axiom~(\ref{e-back}) reflects a genuine asymmetry of time: actions can
change the future, but not the past.  It allows an agent to focus on
planning for the future without wasting time constructing an overly
general model of the past.   However, I think this distinction should be
enforced semanticly from outside the formalism, not syntactically from
within.  In some cases, an agent might want to consider branching
pasts---perhaps in reasoning backwards from dinosaur fossils to
situations in which they lived---or nonbranching futures.  This decision
should be made by the agent based on his current needs, not fixed once
and for all by axiom~(\ref{e-back}).  I disapprove of branching futures
too, but postpone discussion to section~5.

Having specified a formalism for time, McDermott turns to facts, events,
and causality.  A {\it fact\/} is a set of states and an {\it event} a
set of intervals: convex sets of states.  A fact is true in each of its
states, while an event occurs exactly once over each of its intervals
with no time left over on either side.  The intervals belonging to an
event are its instances in all possible chronicles.  Facts obey a
generalized continuity principle: for every fact {\it f} and situation
{\it s} in chronicle {\it ch} there are {\it intervals} preceding {\it
s} and following {\it s} which are contained in {\it f} or its
complement.  In other words, a fact cannot change truth-value infinitely
often in the neighborhood of any state. 

These definitions of facts and events are philosophically problematic,
since they reduce intensional concepts to extensional ones.  Surely, a
fact or event is not identical with its times of occurrence.  If that
were true, contemporaneous facts and events would be identical.  For
example, every time Alexander makes a local phone call he is charged one
message unit, but making local phone calls and spending message units
are different events.  Alternately, suppose a bus always leaves from
Boston at the same time a train leaves from Davenport, and arrives in
New York City at the same time the train pulls into Grand Rapids.  Two
distinct events still occur.  Philosophers disagree on whether distinct
events can really have the same instances in all possible situations,
but from a {\it pragmatic\/} point of view the answer is clear.  A real
program can only represent a countable subset, probably only a finite
number, of the uncountably many possible states.  For this reason, it
should never assume that two events that always occur together in its
model are truly cotemporaneous.  The bus and the train may very well
travel together in all represented states, but these trips must
never be reduced to the same event.  McDermott's extensional model of
events and facts encourages such mistakes.

\label{can-so}
McDermott claims that situation calculus cannot represent the duration
and intermediate results of events, since its {\it result} function maps
initial situations directly to the final results of actions.  I find
these charges unconvincing, since situation calculus defines the $\prec$
relation and {\it time} function.  One can represent the duration of an
event by the difference between the times of its final and initial
situations and intermediate events by situations that follow the initial
state and precede the final one, just as McDermott does.  A second
claim, that events that are not fact changes cannot be expressed at all,
deserves more thought.  From a formal point of view, this is false,
since his model is a restricted version of situation calculus: it
contains the same {\it time} and $\prec$ along with additional axioms,
such as~(\ref{e-back}).  Anything that can be expressed in the
restricted system can be expressed in the more general one as well.
However, from a pragmatic perspective, I agree that a useful world model
must provide the user with a richer model of events than does the
situation calculus.  Although McDermott thinks his work is a
revolutionary alternative to situation calculus---as he makes clear in
his introduction---it is really an evolutionary extension, which adds
structure to the existing framework.  

\subsection{Causality}\label{s-causality}
Causality between events is expressed by the $ecause(pre,c,e,d)$
predicate, meaning that effect {\it e} occurs within delay {\it d}
after cause {\it c} if condition {\it pre} holds during that period.
Causation is
a primitive irreducible aspect of reality, so there are no special
inference rules for deriving instances of {\it ecause} from other facts.
If any instance of an event has a cause then all do.  In my opinion,
this latter principle confuses the formalism with reasoning methods, as
does axiom~(\ref{e-back}).  Whether or not a user assumes that all
events have causes should be up to him, not the syntax of his world
model.  In fact, it might be false for some useful and intuitive
theories of causality.   For example, consider two coins, the first
loaded and the second fair, each of which comes down heads 100 times
consecutively.  According to our intuitions, the first event was caused
by the design of the coin, whereas the second was pure chance.

At first glance, it seems natural to represent causality between an
event and a fact analogously to causality between two events.  If an
event causes a fact then the fact remains true for some period after the
event ends.  However, McDermott finds this definition unacceptable
because it says nothing about the fact's duration, or in his words
\begin{quote}
We could do this, but it would be useless.  In this sense, shooting a
bullet past someone would be a way of achieving that it was near him.
\cite[p.\ 119]{mcdermott1}
\end{quote}
I reject this view.  Shooting a bullet past McDermott certainly does
cause it to be near him, albeit for a short while.   An event causes a
fact by leading to a state in which the fact is true.  How long that
state persists is another matter, depending partially, but not solely,
on the causing event.

After rejecting causality between events and facts, McDermott develops
a theory in which an event causes the {\it persistence} of facts for a
given period.  He thinks it perfectly reasonable to reify persistences,
since ``the senses actually tell you about persistences.''  If people
only sensed facts, they would suffer from the frame problem, since what
is true now might well be false in an instant.  They avoid that famous
bugaboo of AI by sensing persistences.  I find this unconvincing: it is
pure unsubstantiated speculation without explanatory power.  If sensing
persistences just means sensing that facts persist, McDermott's theory
merely postulates a solution to the frame problem and names it
persistence, without explaining anything new.  If it means that people
sense entities called persistences, he owes us evidence of their
existence and an account of how they solve the frame problem.  Next,
I evaluate the role of persistences in causality from an artificial
intelligence perspective. 

The $persists(s,p,r)\/$ predicate means that fact {\it p} is true in
state {\it s} and if {\it p} is false for any state {\it t} within {\it
r} time of {\it s} then a $cease(p)\/$ event occurs between {\it s} and
{\it t}. For instance, $persists(now,on(light),1\; hour)\/$ expresses
the proposition that the light will be on for an hour unless something
extinguishes it.  The $cease(p)\/$ event occurs iff it is {\it
inconsistent\/} that it not occur.  With this nonmonotonic definition of
persistence in hand, McDermott defines the {\it pcause} predicate, which
states that an event causes the persistence of a fact for a certain
period.  

McDermott's use of nonmonotonicity raises theoretical and practical
difficulties.  It contradicts his declared motive for proposing a
logic-based model in the first place. 
\begin{quote}
So why do I plan to spend any time at all on logic?\ldots
We want to be assured that our special-purpose modules are not prone to
absurd interactions\ldots One way to guarantee this is to be sure that
the modules' actions are sound with respect to an underlying logic.
\cite[p.\ 103]{mcdermott1}
\end{quote}
That reasoning makes sense for a well-behaved logic like predicate
calculus, where a reasoner can apply rules of inference to axioms and
premises to derive new theorems.  The soundness theorem prevents him
from ever coming up with incorrect results.  However, nonmonotonic
logics have no deduction rules or other local proof methods.  It is
impossible in general to determine whether a formula follows from
premises without generating an entire, necessarily infinite, fixed-point
of the theory.  Worse yet, even the most benign premises may have
several disparate fixed-points or none at all.  A guarantee that all
deductions are correct means little in a logic where none can be made.
As McDermott admits,
\begin{quote}
I try to appeal to nonmonotonic deductions as seldom as possible.  This
is because the logics they are based on are still rather unsatisfactory.
For one thing, even some of the simple deductions in this paper may not
be valid in any existing nonmonotonic system. \cite[p.\ 122]{mcdermott1}
\end{quote}

McDermott believes nonmonotonic logic can be made workable by
logicians, if AI practitioners explain what types of inference they
need.  He treats its shortcomings ``not as problems with this paper, but
as problems with nonmonotonic logic.''  Unfortunately, his optimistic
view runs counter to Moore's \cite[pp.\ 77--79]{moore2} argument that no
sound logic can include default reasoning because it is an inherently
unsound rule of inference.  Any attempt to include defaults in a logic
must fail because default rules sometimes derive incorrect results from
correct premises, but logics never do.  McDermott defeats his original
goal of soundness by introducing nonmonotonicity.    

In my opinion, the whole attempt to introduce nonmonotonicity stems from
the confusion between a model and its use, which I have mentioned
before.  A reasoner may revise his theory of the world nonmonotonically
by deleting old beliefs based on new information.  This metal-level
operation need not be sound in a logical sense---in fact, unsound
revisions probably yield the best results---but the resulting theory
must be logically sound, if its theorems are to be useful.  Hence,
any attempt to express the unsound meta-level objectives in the sound
object language are misguided and futile.

McDermott repeats\footnote{Moore's 1985 paper \cite{moore1} reviews
ideas from his 1980 doctoral thesis, preceding McDermott by several
years.}  Moore's theory of action: the {\it do(agent,action)}
function maps a performance of {\it action} by {\it agent} to an
event.  He composes compound actions, called {\it plans}, from sequences
of simple actions, but eschews more powerful constructs, such as
iteration and local variables.  Unlike Moore and McCarthy, he realizes
that the complexity of program-like actions makes it impractical to
represent them as deduction rules of predicate calculus.   It can be
done, but mechanical, or human, reasoners have little chance of
understanding the results.  Perhaps a different logic, such as Pratt's
\cite{pratt1} dynamic logic, would represent action better than
predicate calculus, but I cannot pursue that topic for lack of time and
space. 

Next, McDermott defines useful types of action, such as allowing,
preventing, forgoing and bringing about.  I only describe preventing
since the rest are similar.  By definition, action {\it a} prevents
event {\it e} if {\it e} depends negatively on the event {\it
do(agent,a)}.  Event $e_2\/$ depends negatively on $e_1\/$ if $e_2$
occurs in some chronicle, does not occur in every chronicle and never
occurs in the same chronicle as $e_1$.  Although this account may be
theoretically adequate, its implementation confuses correlation with
causation.  As discussed earlier, the agent can only represent a tiny
fraction of the possible situations.  It should not assume that event
$e_2\/$ depends negatively on $e_1\/$ just because they happen never to
occur jointly in its data base.  Furthermore, this definition ignores
time of occurrence.  By ringing today, Brook's phone prevents the cord
from being cut yesterday, since the cutting and the ringing could not
share a chronicle.  Yet cutting the cord cannot prevent the phone
ringing, because it might have rung yesterday.  In trying to represent
prevention, a causal relation, as correlation, McDermott has ignored his
own advice, ``there is no way to infer causality merely from correlation
\cite[p.\ 117]{mcdermott1}.''

In a personal communication, Allen charges that real-world actions
never satisfy McDermott's prevention and achievement criteria.  I
cannot really prevent the phone from ringing by destroying it, since
some brilliant ex-employee of Western Electric might reconstruct it.
Nor, to use McDermott's example, can Dudley prevent the train from
crushing Nell by untying her.  She might jump back on the tracks and
commit suicide.  In general, a problem-solver cannot {\it prove} that
any plan achieves or prevents anything without restricting its world
model unduly.  It must choose the plan most likely to succeed and 
ignore improbable obstacles.  McDermott hopes that ``a dose of
non-monotonicity'' will enable his system to prove things based on
default assumptions.  He concedes that it cannot represent probabilistic
information necessary for choosing plans likely to succeed and ignoring
rare events, but claims that this problem transcends his work to
encompass all AI research.  I return to this issue in section~6.

\subsection{Summary}
McDermott rejects situation calculus and proposes a nonmonotonic logic
for time, change, and action.  His model of time resembles situation
calculus, except that time cannot branch into the past, while his model
of action resembles Moore's, except that program-like actions are banned.
He equates facts and events with the time of their truth and occurrence,
leading to philosophical and practical problems.  He reifies
persistences of facts and relates them to events by a nonmonotonic
axiom.  I find this approach unpromising because of the inadequacies of
nonmonotonic logics.  Finally, his definitions of prevention and
achievement are unintuitive and impractical.  In the next section, I
describe a better solution to the problems McDermott addresses.

\section{Allen's Simplification}
Allen \cite{allen1} finds situation calculus inadequate for
representing:
\begin{itemize}
\item nonactivity: standing on the corner for an hour,
\item nondecomposable actions: spending the day hiding from someone, and
\item simultaneous interacting action: walking to the store while
juggling.
\end{itemize}
As I showed on page~\pageref{can-so}, this is a claim about expressive
power, not logical adequacy.  In principle, situation calculus can
represent the same things as Allen's formalism, since they are both
extensions of predicate calculus.  Theoretical equivalence, though, does
not imply practical equivalence.  A formalism that captures explicitly
the important aspects of a domain facilitates reasoning, whereas a
formalism that leaves them implicit causes confusion.  For this reason,
Allen designs his model of time and action to represent explicitly the
domain's key concepts: time, event, process, and action.

\subsection{Properties, Events, and Processes}
Allen uses an interval-based temporal logic, described in \cite{allen2},
general enough to model discrete time, continuous time, and combinations
of the two.  By and large, the precise theory does not affect our topic,
so I will just equate his intervals with the standard concept of a time
interval.  However, there is one exception to this rule: Allen rules out
time points and, consequently, identifies open, half open, and closed
intervals.  He finds points unnecessary, since they can be viewed as
tiny intervals.  I disagree.  Time points are useful and important
concepts.  When does a pendulum reach the end of its swing, if not at a
particular instant?  Of course, one man's points are another
microscope's intervals, but points are useful at any fixed level of
abstraction, regardless of their ultimate ontological status.  Allen
\cite{allen2} himself admits as much when he reinvents the wheel and
constructs points out of intervals. 

Nor do I agree with Allen's argument that introducing points leads to
inconsistency or truth gaps:
\begin{quote}
For example, consider the time of running a race, R, and the time
following after the race, AR.  Let P be the proposition representing the
fact that the race is on; P is true over R and $\sim \rm P$ is true over
AR.  We want AR and R to meet in some sense.  Whether both ends of the
intervals are open or closed, AR and R must either share a time point or
allow time between them.  Thus we have a choice between inconsistency or
truth gaps, i.e., either there is a time when both P and $\sim \rm P$
are true, or there is a time when neither P nor $\sim\rm P$ is true.
\cite[p.\ 128]{allen1}
\end{quote}
All this argument shows is that intervals divide into four similar
subclasses: closed, open, left open and right open.  These distinctions
can often be ignored, but sometimes they must be made.  In the race case,
we may put the boundary point b between R and AR into either R, AR both
or neither.  The predicate $\rm P(b)$ is true iff b is in R.  Allen's
purported truth gap and inconsistency result from a confusion between 
$\neg \rm P\it (t)$, which equals $t\not\in \rm R$, and $t\in AR$. 
These are the same only if b belongs to exactly one of A and AR. 

Adopting the ontology of Mourelatos \cite{mourelatos1}, he divides
time-dependent aspects of the world into {\it properties\/} and 
{\it occurrences}.  Properties represent persistent facts about objects,
such as ``the house was warm all Monday.''  A property holds over an
interval, denoted by {\it holds(p,int)}, iff it holds over every
subinterval.  Thus, the house was warm all Monday iff it was warm over
ever subinterval of Monday.  Although I accept this definition in
principle, I think it fails in Allen's interval-based time, providing
another argument for points.  Some intuitively reasonable properties
hold over an open interval, but not its closure.  This situation
contradicts Allen's condition, so he must impoverish the world
model by ruling out such properties.  For example, consider a ball that
is thrown straight up at time $t_0\/$ and reaches its apex at time
$t_m$.  It has the property of rising on every subinterval of $[t_0,
t_m)$, but not on the closed interval.  Allen must either say it does,
or deny that rising is a property. 

Occurrences model dynamic aspects of the world.  They divide into two
subclasses: {\it events\/} describe discrete activities with definite
endpoints, whereas {\it processes\/} describe continuous activities
without definite outcomes.%
\footnote{McDermott's events lump these events and processes together.
In order to represent Allen-processes, McDermott allows all events to
occur over an interval and its subintervals.  This leads to confusion
and mistakes in the case of Allen-events that {\it cannot} occur over
their subintervals.}
As a general heuristic, one can count events, not processes.  ``He
bought a coat'' refers to an event and ``the ball is falling,'' to a
process.  It makes sense to ask how many times he bought a coat, but not
how many times the ball is falling.  In contrast, ``the ball fell to the
ground'' is an event because it has a clear outcome, so its instances
can be enumerated.  In Allen's terminology events {\it occur} and
processes are {\it occurring}.  In light of the counting principle, if
an event occurs over interval {\it int} then it does not occur over any
subinterval of {\it int}.  Allen does not know exactly what a process's
occurring over an interval implies about its subintervals.  He
stipulates that it must be occurring over {\it some\/} subinterval, for
lack of a stronger condition.  Such a condition would formalize the
intuition that the process is occurring on most subintervals, or on a
large enough percentage to guarantee continuity.  Like Allen, I see no
general way to formalize this.

Allen formalizes causation between events by an {\it ecause} predicate 
analogous to McDermott's.  If event {\it c} occurs over $t_c\/$ and 
$ecause(c,t_c,e,t_e)\/$ is true then event {\it e} occurs over $t_e$.
Backward causations cannot occur, that is $t_c\/$ must be no later that
$t_e$.  Event causation is transitive, anti-reflexive and
anti-symmetric, corresponding to our intuitions: 
\begin{enumerate}
\item if {\it a} causes {\it b} and {\it b} causes {\it c} then {\it
a} causes {\it c},
\item nothing causes itself, and
\item two events cannot cause each other.
\end{enumerate}
It is a primitive
concept irreducible to others, so new causal relations can only be
deduced from existing ones, nothing else.  I think this decision, also
made by McDermott, is a wise one.  Causal relations should be deduced by
examining a world model from outside, not proved from within, since they
are never deductively valid, only inductively reasonable.

\subsection{Actions and Plans}
Actions are a subset of occurrences, as in situation calculus, not a
separate class of entities, as postulated by Moore and McDermott.  The
function $acause(agent,occ)\/$ returns the action of {\it agent} causing
{\it occ}.  This action is an event, called a {\it performance}, if {\it
occ} is an event and a process, called an {\it activity}, if {\it occ}
is a process.  If an agent acauses an event or process over a time
interval, that event or process occurs or is occurring respectively over
the interval.  Many actions consist of agents acausing occurrences.  In
fact, Allen conjectures that all actions can be described in this way
fairly naturally.

Unlike McCarthy, Moore, and McDermott, Allen adopts a single time line,
rather than branching time.  He performs hypothetical reasoning from
outside the formalism by creating and analyzing hypothetical models,
rather than encoding the mechanism within the formalism.  Agents plan by
manipulating three partial world descriptions: the expected world,
desired world, and planned world.  The {\it expected\/} world contains
what the agent believes will happen if certain known events occur, he
does nothing, and everyone else does as little as possible.  In Allen's
words, 
\begin{quote}
The expected world obeys a generalized law of inertia.  Things in
the process of changing continue to change unless prevented, and
everything else remains the same unless disturbed. \cite[p.\ 144]{allen1}
\end{quote}
The {\it desired\/} world consists of the properties, events, and
processes that the agent desires.  In order to achieve its goals, the
agent forms a {\it plan}, a set of decisions to perform actions not in
the expected world and refrain from actions in the expected world.
Applying a plan to the expected world produces a {\it planned\/} world,
a simulation of the plan's effects on the expected world.  Loosely
speaking, an optimal plan for an agent minimizes the differences between
the corresponding planned world and his desired world.  Allen suggests a 
generalized {\sc gps} model of difference reduction as a planning
algorithm, but defers discussion to another paper \cite{allen3}.  My
topic is modeling not planning, so I do the same.

\subsection{Summary} 
Allen prefers an interval-based linear model of time to possible worlds.
He adopts an ontology of properties, events, and processes and represents
them explicitly in his formalism.  This improves on the implicit
representation of McCarthy and Hayes and Moore's partial one, in which
all three categories are lumped together.  Actions are special cases of
occurrences, instances of agents causing occurrences, not a separate
type of entity.  The linear time line simplifies the model, thus
facilitating inference, without harming hypothetical reasoning.  That is
done from outside the formalism, by constructing hypothetical worlds.

\section{Questioning the Paradigm}
So far, I have compared the expressive power of four representations for
time-varying behavior as logical theories.  Now, I turn to their common
strategy, expressed by McCarthy and Hayes \begin{quote} we want a
computer program that decides what to do by inferring in a formal
language that a certain strategy will achieve its assigned goal.
\cite[p.\ 463]{mcCarthy1} \end{quote} They model the world as a logical
theory in order to reduce reasoning about the world to formal deduction,
a mechanizable chore.  Action reduces to theorems about the {\it result}
function, knowledge to {\it shrug} relations, causation to {\it ecause,
pcause,} or {\it acause}, and so on.  In each case, the soundness of the
underlying logic guarantees that all deductions are valid.

I find this approach unpromising.  Syntactic theorem proving methods,
such as resolution or natural deduction, suffer from combinatorial
explosion in toy domains, let alone realistic world models.  Semantic
information must be used to guide the process, as Bledsoe
\cite{bledsoe1} demonstrates for mathematics.  I came to the same
conclusion after taking courses in mathematical logic.  The best way to
prove a theorem is by examining its interpretation in an appropriate
model, such as the integers, rationals, or reals.  An informal proof in
that interpretation often transforms directly into a formal proof, or at
least suggests a line of attack.

Well then, why not use semantic information to guide deduction about
world models?  Therein lies the tale.  Most, if not all, real world
reasoning employs unsound inference schemes, such as induction,
intuition, and defaults.  Its hard to see how these transfer to a sound
logic.  For example, the default rule ``dogs are friendly'' allows me to
walk home without fearing attack.  Yet, my rule cannot be part of a
sound inference system since it has incorrect implications---witness the
scar on my left calf.  Worse yet, human theories of the world contain
inconsistencies.  Our informal inference techniques accommodate minor
flaws that would destroy a formal theory.  I never draw arbitrary
conclusions from the contradictory propositions ``dogs are friendly''
and ``there exists a vicious dog'' contained in my world model, but a
standard theorem prover does.  In section~\ref{s-causality}, I argue
that nonmonotonic reasoning cannot represent unsound default rules, such
as  ``most dogs are friendly,'' because it is sound.  Hence, it cannot
avoid contradictions by replacing general rules with default rules.

The dichotomy between unsound informal inference and sound formal
inference explains most of my previous criticisms of the four
formalisms.  People reason about actions without knowing their exact
effect on the entire universe, but those inferences cannot be captured in
situation calculus.  Some supposedly irrelevant factor might cause them
to fail, whereas infallible inference rules alone can be expressed by the
{\it result} function.  Knowledge is deductively closed in the
formalisms of McCarthy and Moore, but not in people, computers, or any
finite beings.  In all four systems, agents construct plans that
provably achieve their goals, while humans make do with plans that
succeed unless something unusual happens.

Am I rejecting deduction outright?  Not at all, just putting it in
perspective.  As Israel \cite{israel1} explains, formal logic is just
one part of human reasoning. Deduction is great at proving tautologies
and other {\it analytic\/} truths, but poor at deriving 
{\it synthetic\/}, fact-dependent, truths.  Proponents of formal
theories should use it accordingly, as a tool for exploring the logical
connections between facts and assumptions, not an all encompassing model
of reasoning.  Such a model must include the nondeductive inference
methods, such as induction and default reasoning, upon which most real
world reasoning depends.  They cannot be encoded as inference rules of a
formal logic.

\section{Conclusions}
In this paper, I discuss four formalisms for representing events,
actions, and other time-varying information.  The first, proposed by
McCarthy and Hayes, formalizes an ontology of possible worlds, called
situations, in first-order predicate calculus.  Predicates and functions
on situations represent time-dependent facts and actions.  The {\it
result} function maps every situation {\it s} and action {\it a} to the
unique situation resulting from {\it a}'s happening in {\it s}.
McCarthy and Hayes include a theory of time and a first-order
translation of Hintikka's logic of knowledge in their formalism.  Moore
constructs an unnecessarily complicated variant of this theory of
knowledge and uses it to describe the relations between knowledge and
action.  However, his results can be formulated in the simpler situation
calculus formalism just as well.  McDermott adds events and facts to the
situation calculus formalism, increasing its expressive power.
Unfortunately, he represents them by their extensions, the times at
which they hold, creating philosophical and practical problems.  Worse
yet, he defines nonmonotonic persistences of facts, thereby confusing
the formalism and destroying its proof theory.  Allen takes the opposite
approach.  He replaces the ontology of situations with a linear time
order and transforms hypothetical reasoning from an object-level
deduction mechanism to a meta-level, possibly nondeductive, form of
inference.  He improves on McDermott's representation of events by
subdividing them into properties, events, and processes and restoring
the distinction between an occurrence and the time at which it occurs.
He also simplifies Moore's and McDermott's representation of actions by
reducing them to occurrences.

Of the four theories, I find Allen's clearest and most expressive.
However, it does not include a representation for knowledge and belief,
reportedly for lack of space.  In any case, formal logic alone cannot
represent human reasoning, since the former is sound, whereas the latter
need not be.  I predict that attempts to ignore this distinction and
reduce all reasoning to deduction will fail.

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