Finite Digital Synchronous Circuits Are Characterized by 2-Algebraic Truth Tables

Abstract

A digital function maps sequences of binary inputs, into sequences of binary outputs. It is causal when the output at cycle N is a boolean function of the input, from cycles 0 through N. A causal digital function f is characterized byit s truth table, an infinite sequence of bits (F N) which gathers all outputs for all inputs. It is identi fied to the power series ∑F N z N, with coefficients in the two elements field F 2. Theorem 1. A digital function can be computed by a finite digital synchronous circuit, if and only if it is causal, and its truth table is an algebraic number over F 2[z], the field of polynomial fractions (mod 2). A data structure, f, is introduced to provide a canonical representation, for each finite causal function f. It can be mapped, through finite algorithms, into a circuit SDD(f), an automaton SBA(f), and a polynomial poly(f); each is characteristic of f. One can thus automaticallysy nthesize a canonical circuit, or software code, for computing any finite causal function f, presented in some effective form. Through recursive sampling, one can verify, in finite time, the validityof anyh ardware circuit or software program for computing f.