Digital Algebra and Circuits

Abstract

Digital numbers D are the world’s most popular data representation: nearly all texts, sounds and images are coded somewhere in time and space by binary sequences. The mathematical construction of the fixed-point D ≃ Z ₂ and floating-point D′ ≃ Q ₂ digital numbers is a dual to the classical constructions of the real numbers R.

The domain D′ contains the binary integers N and Z, as well as Q. The arithmetic operations in D′ are the usual ones when restricted to integers or rational numbers. Similarly, the polynomial operations in D′ are the usual ones when applied to finite binary polynomials F ₂[z] or their quotients F ₂ (z). Finally, the set operations in D are the usual ones over finite or infinite subsets of N.

The resulting algebraic structure is rich, and we identify over a dozen rings, fields and Boolean algebras in D and D′. Each structure is well-known in its own right. The unique nature of digital numbers is to combine all into a single algebraic structure, where operations of different nature happily mix. The two’s complement formula −x = 1 + ¬x is an example. Digital algebra is concerned with the relations between a dozen operators. Digital synchronous circuits are built from a simple subset of these operators: three Boolean gates and the unit-delay z.

Digital analysis is simpler and more intuitive than analysis in R. The computable digital functions D ₦ D are continuous: each output bit depends upon finitely many input bits. Infinite circuits compute causal functions: present output depends upon past inputs. Sequential functions are equivalently computed by FSM and by finite circuits.

The v-transform is an infinite binary truth-table for causal functions. The v-transform provides a natural one-to-one correspondence between algebraic digital numbers and sequential functions. Questions about sequential functions are transformed by v into questions about algebraic digital numbers, where the whole of digital algebra applies.

An algebraic digital number is finitely represented by a unique minimal regular binary tree RBT. The inverse transform of the RBT is the minimal deterministic FSM for computing the (reversed) sequential function. An algebraic digital number is finitely represented by a unique minimal up-polynomial MUP of which it is root. The MUP is smaller than the RBT. It is exponentially smaller than the minimal deterministic FSM for a shift-register circuit.

The net-list of a circuit is transformed by v into the isomorphic truth-list: a system of equations over algebraic numbers. Circuit examples show how the truth-list is cast to normal form — either RBT or MUP — through a sequence of simple identities within digital algebra. This contribution is dedicated to Zohar Manna on his 64-th birthday.