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
,
the subset of A related to a. For example, the
relation
between natural numbers can be represented using the
triple:
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
for some
.
(Note that > need
not be transitive.) Then a simulation of one system (indexed 0) in another (indexed 1) is
a relation
such that
.
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:
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).
We may follow either of these two mappings with the `star' construction. This gives