Modular Networks and Nondeterministic Sequential Machines

The synthesis of sequential machines by interconnections of copies of a fixed module is considered. A family of modules M
r,p
for positive integers r and p, is defined. Mr,p can be used to synthesize sequential machines with 2
^P
input symbols. A nondeterministic sequential machine (NSM) is said to be r-bounded if it has one initial state and for no state and input are there more than r choices for the next state. It is shown that the problem of finding a network of modules M
r,p
realizing a given event E is equivalent to finding an r-bounded NSM realizing the reverse of E. As a consequence, two upper bounds on the number of copies of the module M
r,p
necessary to realize an event E can be shown. 1) If E is defined by an n-state NSM, then E is defined by a network of at most c
l
n
^2+p
log
r
²
copies of M
r,p
. 2) If E is defined by an n-state deterministic sequential machine, then E is defined by a network of at most c
2
n
^l+v
log
r
²
copies of M
r,p
. c
l
and C
2
are constants, about 4.