Derivation of a logarithmic time carry lookahead addition circuit

Abstract

Using Haskell as a digital circuit description language, we transform a ripple carry adder that requires O(n) time to add two n-bit words into a parallel carry lookahead adder that requires O(log n) time. The ripple carry adder uses a scan function to calculate carry bits, but this scan cannot be parallelized directly since it is applied to a non-associative function. Several techniques are applied in order to introduce parallelism, including partial evaluation and symbolic function representation. The derivation given here constitutes a semi-formal correctness proof, and it also brings out explicitly each of the ideas underlying the algorithm.