Theoretical and Experimental DNA Computation (Natural ...

4.10 The Translation Process 95
program are dependent only on the number of inputs, this transformation can
always be carried out from just the value of n.
Recall that instk(x), the program instantiated on processor Px by instruction
Ak of the (modified) control program, is a sequence of numbered
instructions drawn from Table 4.4. Furthermore, instk(x) andinstk(y) are
identical except for specific references to memory locations. It follows that in
describing the translation to a combinational circuit, it suffices to describe the
process for a single processor: an identical translation will then apply for all
other processors working in parallel.
The simulation of the program running on a single processor lies at the
core of the combinational circuit translation, and requires some initial transformation
of the given program. To assist with this we employ the concept of
a program control graph.
Definition 2. Let P =< p1,p2,...,pt > be an ordered sequence of numbered
instructions from the set described in Table 4.4. The control graph, G(V,E)
of P is the directed graph with vertex set {1, 2,...,t} and edge set defined as
follows: there is an edge (v, w) directed from v to w if instruction pw immediately
follows instruction pv (note, “immediately follows” means w = v +1
and pv is not an unconditional jump) or pv has the form JUMP w or JEQ
x, w. Notice that each vertex in G has either exactly one edge directed out of it
or, if the vertex corresponds to a conditional jump, exactly two edges directed
out of it. In the latter case, each vertex on a directed path from the source of
the branch corresponds to an instruction which would be executed only if the
condition determining the branch indicates so.
We impose one restriction on programs concerning the structure of their corresponding
control graphs.
Definition 3. Let G(V,E) be the control graph of a program P . G is well
formed if, for any pair of directed cycles C =< c1,...,cr > and D =<
d1,...,ds > in G, it holds that {c1,...,cr} ⊂{d1,...,ds}, {d1,...,ds} ⊂
{c1,...,cr}, or{c1,...,cr}∩{d1,...,ds} = ∅.
It is required of the control graphs resulting from programs that they be
well formed. Notice that the main property of well-formed graphs is that any
two loop structures are either completely disjoint or properly nested. The
next stage is to transform the control graph into a directed acyclic graph
which represents the same computation as the original graph. This can be
accomplished by unwinding loop structures in the same way as was done for
the main control program.
Let HP be the control graph that results after this process has been completed.
Clearly HP is acyclic. Furthermore, if the program corresponding to
HP is executed, the changes made to the memory, M, are identical to those
that would be made by P . Finally, the number of vertices (corresponding to
the number of instructions in the equivalent program) is O(t), where t is the
worst case number of steps executed by P .