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\title{Extending Sandewall's Discrete Fluent\\Logic to Continuous Values}

\date{\today\\Rough Draft}

\author{Steven Ketchpel}

\maketitle 

\begin{abstract}
  This would be the abstract.
\end{abstract}
\section{Introduction to Reasoning about Dynamical Systems}
	One of the primary challenges facing AI systems in the real
world is how to represent and reason about the way systems change over
time.  This is a critical ability for autonomous robots as they
attempt to determine the affects that their intended actions will have
as they construct a plan to achieve their goals.  In addition to this
prediction problem, there is a postdiction problem, where an agent
makes certain observations about the state of the world, and tries to
infer the most likely set of occurrences that led to that world state.
	The reasoning process is complicated by the fact that it deals
not only with actions of the agent, but also events, that is, changes
in the environment that are not caused by the agent.
\section{Description of Sandewall's Discrete Fluent Logic}
	In his book {\it Features and Fluents}, Sandewall
proposes a logic based on non-monotonic entailment which is an
extremely expressive system for representing and reasoning about the
facts that an agent might learn about its environment.  Specifically,
each of the relevant aspects of the environment is observed as a
separate feature.  Each feature takes on a value from the domain of
legal possibilities for that feature.  The features are also able to
change values over time, and hence are called fluents.  By appealing
to the notion of inertia, Sandewall is able to come up with a
preferred model (or models) which minimize the number of changes in
the environment.
	Formally, there are three sorts of information:  
\begin{enumerate}
\item{Observations -- at a certain time, the agent observes that a
particular feature has some value from a subset of the legal values.}
\item{Changes -- at a certain time, the agent observes that a
particular feature takes on one of a subset of the legal values.  The
feature did not have that value before.}
\item{Qualification Defaults -- Also time dependent, the qualification
defaults allow the agent to make certain assumptions about information
not directly observed on the basis of prior expectations.}
\end{enumerate}
	There are two distinct preference orders, one over the
occurrence of changes which tries to minimize the number of changes,
and in scenarios with equal number of changes tries to defer them as
late as possible.   One exception to this rule is that certain
features may be unobservable to an agent at certain times.  These
features are explictly marked as ``occluded'' and then there is no
preference as to when in the occluded interval a change takes place.
(This handles Kautz's stolen car example, so that the car is not
always assumed to be stolen from the garage at the last possible
minute.)  The second preference ordering is over qualification
defaults.  Two scenarios are compared to see if one satisfies a
strict superset of the qualifications that the other satisfies.  If
so, the one satisfying more qualification defaults is preferred.
	The preferred models are those which are minimal in the
ordering with respect to changes, satisfy the observations, and then
fulfill the maximal set of qualification defaults.

$$ min(CHG) \cap OBS \models \alpha $$
	
\section{Motivation for Continuous Values}
	Sandewall himself offers compelling reasons for moving to
continuous time, and these same reasons apply to domains other than
those used for temporal elements.  They are
\begin{itemize}
\item{In order to ``interface'' to differential equations and otehr
standard methods for describing continuous change.}
\item{In order to avoid unnecessary complications in foramlizations
due to granularity.}
\item{In order to capture as well as possible the real character of
the actual physical world.}
\end{itemize}
	In fact, many physical phenomenon relate to quantities like
position, volume, brightness, temperature, etc. which may be measured
in real values that change continuously.  Therefore in order to
capture the information at an arbitrarily fine level of detail, real
values must be used.  
\section{Why Sandewall's approach doesn't apply directly}
	In measuring the appropriateness of one model against another, 
\section{An approach that might work}
	Basic idea: for each of the continuous fluents, we add a
rate-of-change feature.  Since the value of a continuous fluent can be
changing at points other than the breakpoint set (because the change
doesn't introduce a discontinutity), the preference ordering which
looks at number of breakpoints doesn't notice continuous changes in
continuous fluents.  But this is a good thing.  Essentially, a
continuous change is likely to be introduced by a single cause (eg, a
billiard ball moving across a table, the temperature in an oven
increasing) and so the preferred models don't need to penalize a
feature which has an infinite number of continuous changes.  But we do
need a mechanism for recording the fact that the value is changing
over that time interval (in order to determine the feature value at a
given time).  So, we introduce a predicate similar to occlusion Xf1,
call it [t1,t2]CCf1, which says that from t1 to t2 (inclusive, with
the square brackets) feature f1 is changing continuously.  But in
order to determine the value, we also need to know the rate at which
its changing.  So, for any interval where CCf1 holds, another real
valued feature f1' exists, which corresponds to the rate of change in
value f1.  But the beauty of it is that f1' is now a first-order
continuous feature, so it, too, could be varying.  So, CCf1' adds
another feature which is f1'', the second derivative of the f1.  It
seems like these would be enough that even a complex example like the
path of a projectile could be handled.  (Though the feature value
determination part looks like it would be a full scale integration
problem.)  I still need to work out the correct order for preferences
among these things (for instance, Sandewall minimizes the extent of
X--should CC also be minimized in extent?  in number?)  [Actually,
Hector's suggestion seems a good one--assume that all features have a
rate of change parameter, with a preference for 0.]
\section{Examples}
\begin{enumerate}
\item{\em{The position of an automobile moving at constant speed} Given only
the facts that:
\begin{enumerate}
\item{pos(1) = 1}
\item{pos(5) = 9}
\item{[1,5] CCpos}
\end{enumerate}
In this case, the minimal model is the one that involves a 
CC-ing feature, the position.  Therefore, a feature which corresponds
to the derivative of the position feature is created.  The value of
this new feature (the velocity of the car) is calculated from the
data.  Here, by assuming that the velocity is constant, the need for
the addition of a second feature (the acceleration) is obviated, so
this model is smaller than any that posits a non-uniform velocity.
  If we did not have Obs3 that [1,5] CCpos, then the same assumptions
would be made as if the position value were discrete, and the minimal
model would describe a single change from 1 to 9, occurring somewhere
in the occluded interval [1,5].
  As a second extension to this example, imagine that we make the
additional observation that at time 3, the position is 5.  This fits
our expectations, so it should not change the model.  In fact, it does
not.  The original model is still the minimal model consistent with
the observed data.
  In a different extension, the car is accelerating over the interval.
We observe the pos(3) = 4, so the car is covering more distance over
the second half of the interval than the first half.  But since we
know that the position is changing continuously over the whole
interval, we are able to deduce the function that describes the
velocity, which is also a continuous function.  So, in our minimal
model to describe these observations, we need to infer the
acceleration with the minimal number of continuously changing
features, and the minimum number of breakpoints.  In this case, we
know that there are no breakpoints, because the observations
explicitly say that the position is changing continuously.
Essentially, we fit a polynomial to the datapoints that we have, and
then differentiate for each degree of the polynomial, calling each one
a continuously changing feature, until we reach a derivative the same
degree of the polynomial which stays constant.}
\item{\em{The moving car, as above, but without the explicit
observation that the position changes continuously over the interval
[1,5].}  When we aren't given the fact that the value is changing
continuously, we have to trade off the complexity of introducing
multiple breakpoints (for each time the feature is known to change)
with the complexity of saying that the feature is changing
continuously and determining the rates of change of the polynomial
that describe its value.  As an added complication, it may be the case
that a simpler model may be found by piecewise combining several
polynomials, but allowing breakpoints (non-continuous changes) between
the intervals described by the various polynomials.
	The real design problem, then, is coming up with preference
ordering that takes these various complications into account in a
meaningful way.  Even in the case where [1,5]CCpos is given, you aren't
that there's only one polynomial.  There might be two that have a point in
common.}
\end{enumerate}
Given only the facts that:
\begin{enumerate}
\item{pos(1) = 100}
\item{pos(5) = 21.6}
\item{[1,5] CCpos}
\end{enumerate}

\section{Interactions between the addition and the old system}
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