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An iterative work is one that is composed of a number of identical cells,
combinational or sequential, interconnected in a regular array. Each cell
will normally have a primary input, which serves as one of the inputs of
the entire network, as well as a primary output, which serves as one of the
outputs of the entire network. In addition, each cell communicates with its
nearest neighbors by transmitting and receiving discrete signals through
synchronous intercell delays. The boundary conditions specify the signals
that are to be applied to the intercell leads entering the edges of the network.
These boundary signals are fixed for all time and are independent of the
number of cells in the network. The set of all (hyper) rectangular networks
composed of cells of a given structure is referred to as the Iterative system
defined by that cell structure and the corresponding boundary conditions.
The total state of a particular cell is specified by giving the values of
all the cell’s inputs and internal state variables; the state of a network is
specified by giving the states of all the cells in the network. If the state
of a network does not change with time, the network is said to be in
equilibrium. If a network experiences a periodic change of states, it is said
to display a cycle. When a network possesses a unique equilibrium state for
every possible pattern of primary input values, it is said to be regular. If, in
addition, the network cannot sustain a cycle, it is said to be stable.
For networks of combinational cells stability implies regularity. A system
is said to be regular (stable) if and only if its networks is regular (stable).
Two regular networks of the same size are equivalent, if in their equilibrium
states, they produce identical primary output patterns whenever they are
presented with identical primary input patterns. Finally, two regular systems
are said to be equivalent if and only if each network of one is equivalent to
the corresponding network of the other.
When analyzing or synthesizing iterative systems, it would be desirable
to have a standard means of testing an arbitrary system for the various
properties described above. Our first task was to develop these tests for
the classes of systems for which they exist, and to offer the proof of the
fact that they do not exist for many other classes. We then investigated
the capabilities of various classes and finally discussed techniques for
network synthesis.
Iterative systems may be classified according to the number of
directions in which intercell signals are transmitted and according to the
ways in which signals flowing in the various directions interact with one
another. Such a classification provides a convenient basis for specifying
when tests for the properties of regularity and equivalence exist. We say
that a class of systems is decidable if there exist general, final tests
(algorithms) for determining whether an arbitrary of the class is regular,
and whether two regular systems of the class are equivalent. If these
tests do not exist, the class is said to be undecidable. The space analog of
a synchronous sequential switching circuit is a one-dimensional array of
combinational cells in which signal flowing is restricted to be unilateral.
Analysis procedures for such systems follow directly from the well-known
procedures for sequential circuits. Systems of this class are inherently
regular; since there is a general procedure for testing the equivalence for
two arbitrary systems, the class is said to be decidable.