On circuits and numbers

We establish new, yet intimate relationships between the 2-adic integers ₂Z from arithmetics and digital circuits, both finite and infinite, from electronics. 1) Rational numbers with an odd denominator correspond to output only synchronous circuits. 2) Bit-wise 2-adic mappings correspond to combinational circuits. 3) Online functions ∀n∈N,x∈₂Z:f(x)=f(xmodd2ⁿ)mod2 ⁿ), correspond to synchronous circuits. 3) Continuous functions, ₂Z→₂Z, correspond to circuits with output enable. The proof is obtained by constructing synchronous decision diagrams SDDs. They generalize to sequential circuits as classical BDD constructs do for combinational circuits. From simple identities over ₂Z, we derive both classical and new bit-serial circuits for computing: {+,-,×,1/(1-2x), (1+8x)}. The correctness of each circuit directly follows from the 2-adic definition of the corresponding operator. All but the adders (+,-) above are infinite. Yet the use of reset signals reduces all previously infinite operators to finite circuits. The present work lays out the semantic basis of a new language for describing synchronous circuits. Language 2Z incorporates arithmetic synthesis for some of the above bit-serial operators, and for periodic binary constants (logic from chronograms). It also provides for the powerful deeply binding synchronous enable and reset operators, whose meaning is discussed