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Next: Free category on a Up: Partial constructions Previous: Topological interpretation

   
What about triples?

Dull would he be of soul who could look at the signature of a quadruple and not wonder how the theory of such structures compares with that of the following even simpler structure (A,B,c), where:

This kind of structure represents a binary relation, or directed graph on A. Think of a binary relation as given by the function which assigns to each a : A, as an indexed set $\{\; c(a,b) \; \vert \; b :
B(a) \; \}$, the subset of A related to a. For example, the relation $\leq$ between natural numbers can be represented using the triple:

\begin{displaymath}\begin{array}{lcl}
A & = & \ensuremath{\mathit{Nat}}\\
B &...
...at}} ) \\
c & = & (a : A, b : B(a) \mapsto a + b)
\end{array}\end{displaymath}

From another point of view, a binary relation is a directed graph. Viewed this way, A is the set of nodes, for each node a:A, B(a) is the set of arrows with origin a, and for each such arrow b : B(a) the node which is its destination is c(a,b).) Of course, one might equally well model a directed graph with B(a) being the set of arrows with terminus a, and for each such arrow, c(a,b) the node which is its origin.

What, in this situation, is a simulation? First let us look at things in relational notation. Suppose, given a pair of such systems, we write a >i a' to mean that a'has the form $c_i \; a \; b$ for some $b : B_i\; a$. (Note that > need not be transitive.) Then a simulation of one system (indexed 0) in another (indexed 1) is a relation $R \subseteq A_0 \times A_1$ such that $(R \; ; >_1) \subseteq ((>_0)^+ ; R)$. We insist that for any descending chain in the first, there is a descending chain in the second which has it as a subsequence. (This allows the `implementation' to use `extra steps'.)

In type theory, it is better to define the notion as follows. Again, a simulation is a relation between the states of the two systems together with the following data:

\begin{displaymath}\begin{array}{rcl}
( & \begin{array}{l}
x : R (a, a') \\
...
...a \; (\phi \; x \; l), c_1 \; a' \; l)
\end{array}\end{array}\end{displaymath}

(If the relations are not transitive, then we ask that there is a simulation of the first system in the transitive closure of the second.) Note that `simulating steps' may depend on the proof of the simulation.

A simulation is a down-moving relation (or right-moving?). So if a is accessible, and is simulated by a', then we can derive for each irreducible derivative of asomething that simulates it which is a derivative of a'.

Need to investigate categorical things about simulations. It is a little confusing .. a simulation relation itself is a sort of hom-set structure .. . Inductive definition of the (complement of the) weakest bisimulation on a graph.

Obviously, if there were an equality relation, it would be a simulation. So for our base structure, we are going to start with some kind of self-simulation.

What is the word whose symmetric closure is `similarity'?

We can define a `reflexive' simulation relation on the accessible elements, by recursion.

There are two natural ways to make a triple T=(T.A,T.B,T.c) into a quadruple Q=(Q.A,Q.B,Q.C,Q.d).

\begin{displaymath}\begin{array}{llll}
Q.A = T.A & Q.B(a) = T.B(a) & Q.C(a,b) =...
... = N_1 & Q.C(a,1) = T.B(a) & Q.d(a,1,b) = T.c(a,b)
\end{array}\end{displaymath}

We may follow either of these two mappings with the `star' construction. This gives



 
next up previous
Next: Free category on a Up: Partial constructions Previous: Topological interpretation
Peter Hancock
1998-04-08