\newpage
\subsection{Explicit Minimization}
A common mode for organizing computation is this.  In one phase,
all agents post whatever constraints they want on the variables
of interest.  On quiescence it is desired to determine {\em
minimal} values for the particular variable --- perhaps {\em the}
minimal one (if it is unique), perhaps {\em some} arbitrary one
(if there are many minimal ones; this is sometimes called a
``credulous'' choice), perhaps the logical disjunction of all the
minimal ones (this is sometimes called ``skeptical'' choice),
while keeping the other variables in the store constant or
varying. Various kinds of minimization procedures proposed for
logic programming and for default reasoning (e.g., varieties of
circumscription) fall into this general paradigm.

The \tcc{} model makes possible the specification of ``safe''
versions of such primitive processes: it makes sense to perform
the minimization only on quiescence of the current phase (the
minimization operation is not monotone in its argument, hence
cannot be run while its argument is being accumulated); hence the
new constraint may be added to the store only at the {\em next}
instant; hence structurally in a way that cannot interact with
the actual process of minimization itself.

This leads to the addition of a class of minimization agents to
the language. Because of the variability mentioned above, many
such agents are definable and interesting. Assume $min(X,c)$ is
such a minimization operation --- it returns a constraint minimal
in $X$ given the information $c$ per some chosen definition of
minimization. Then we may define the basic process by:
\begin{eqnarray*}  
\LL{\tt minimize(X)} &=& 
\{\epsilon\} \cup \{c \in \Obs\} \cup
 \{c\cdot d \cdot s \in \Obs \alt d \geq min(X,c)\}
\end{eqnarray*}
Of course once the basic agent is introduced into the language,
it may be combined with all the other combinators in the language
in completely general ways. 

Two concrete examples should illustrate the matter further. 
Consider the {\em histogram} problem studied by researchers in
working on determinate data-flow languages.  The problem is to
efficiently determine the histogram associated with a given input
function $f$. That is, determine, for each point $x$ in the range of
$f$, the number of points in the domain mapped to $x$. 

The following simple algorithm is expressible in \tgentzen{}
augmented with a suitable minimization operation: examine the
tuples {\tt (Index, Value)} in  $f$ in parallel, asserting that
{\tt Index} belongs to a set associated with {\tt Value}. On
quiescence, minimize the extension of these sets, and take their
cardinality. Concretely:
\begin{ccprogram}
\agent histogram(In, Hist) :: (Temp)\Hat
   \[(Index, Value)\$((Index, Value)\ in\ In -> \{Index\ in\ Temp.Value\}),
     minimize(Temp), 
     next\ sizeof(Temp, Hist)\]..
\end{ccprogram}

{\tt minimize(X)} is an example of an ``indexical'' constraint
\cite{cc-FD} --- a functions from constraints to constraints that
is not necessarily monotone. Obviously, several other
constraint-system specific indexicals are possible in this vein.

The second concrete example is taken from work on model
composition in Qualitative Reasoning and is representative of the
kind of default inferencing employed therein. While building
component models one may only be able to specify very incomplete
information about the relationship between some ``global''
variable $X$ and some variable $V$ corresponding to the
component. For example, one may merely know that $X$ varies
monotonically with $V$ (assuming that both vary over some ordered
set such as the reals). This can, of course, be achieved by
asserting the constraint that the partial derivative of $X$ with
respect to $V$ is positive. However, once system structure is
completely determined, and the constraints between $X$ and the
ensemble of component variables $V_1, \ldots, V_n$ established,
another piece of information can be deduced, namely, that $X$ is
a function of {\em just} the inputs $V_1, \ldots, V_n$.
Structurally, this is a safe default inference because no
inference licensed by this deduction should affect the
applicability of the default: once the structure of the system
has been determined, it has been determined. That is, the default
depends only on predications that are going to remain stable for
the rest of the deduction, that are causally independent of the
consequences of the conclusion of the default. 

Such a safe default rule can clearly be directly expressed in
\tgentzen, augmented with minimization predicates. Indeed the
basic notion of {\em staged computation} underlying \tcc{} is in
completely harmony with the ideas of staged model composition
underlying model composition work in Qualitative Reasoning.