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## Approximate Theories of Digital Circuits

Consider first combinational circuits built from logic elements, i.e. without bridges and other sneak paths. The logical elements are treated as boolean functions, defined by their truth tables. The theory defines the behavior of any circuit in terms of composition of the functions associated with the logical elements. The outputs of a circuit are given by the theory for any combination of boolean inputs. Fan-in and fan-out restrictions are outside the logical theory, as are timing considerations.

Now consider sequential circuits including flipflops. Now what happens is defined only for some combinations of inputs. For example, the behavior of a D flipflop is not defined when its 0 and 1 inputs are given the same value, whether that value be 0 or 1. The behavior is only defined when the inputs are opposite.

The manufacturer does not say what will happen when these and other restrictions are not fulfilled, does not warrant that two of his flipflops will behave the same or that a flipflop will retain whatever behavior it has in these forbidden cases.

This makes the concept of D flipflop itself approximate, perhaps not in the same sense as some other approximate theories.

Thus the theory of sequential circuits is an approximate theory, and it is not an approximation to a definite less approximate theory of purely digital circuits. This is in spite of the fact that there is (or can be) an electronic theory of these digital circuits which describes their behavior. In that theory one D flipflop is different from another and changes its behavior as it ages.

John McCarthy
Wed Feb 2 15:59:04 PST 2000