EXOR Projected Sum of Products

Abstract

In this paper we introduce a new algebraic form for Boolean function representation, called EXOR-Projected Sum of Products (EP-SOP), resulting in a four level network that can be easily implemented in practice. We prove that deriving an optimal EP-SOP from an optimal Sum of Products (SOP) form is a hard problem (NPNP-hard); nevertheless we propose a very efficient approximation algorithm, which returns in polynomial time an EP-SOP form whose cost is guaranteed to be near the optimum. Experimental evidence shows that for about 35% of the classical synthesis benchmarks the EP-SOP networks have a smaller area and delay with respect to the optimal SOPs (sometimes gaining even 40-50% of the area). Since the computational times required are extremely short, we recommend the use of the proposed approach as a post-processing step after SOP minimization

An approximation algorithm for fully testable kEP-SOP networks

“…In this section we recall some basic definitions from [1]. Let f : {0, 1} n → {0, 1} be a Boolean function depending on n variables x 1, x2, .…”

“…We can represent the function f as the sum (union) of the two projections of φ, φ⊕ and φ ⊕ , onto these two spaces: ξij = (xi ⊕ xj)φ⊕ + (xi ⊕ xj)φ ⊕ . The expression ξij is called the (i, j)-EP-SOP of f [1].…”

“…These projections reduce the Hamming distances among the cubes appearing in each subspace, so that further merges can be performed using standard SOP heuristics. Due to the high complexity of their synthesis, we minimize kEP-SOP forms with a polynomial time approximation algorithm, whose approximation ratio improves the one derived in [1] for EP-SOP forms. Thus our overall method yields a polynomial time approximation algorithm that guarantees near-optimum solutions.…”

“…However, no constraint is given on the number of levels, and therefore the delay of the networks is not guaranteed to be low. On the contrary, bounded multi-level forms (e.g., EXSOP [6], OR-AND-OR [7], SPP [11], ESPP [10], EP-SOP [1]), while still keeping a very compact area, result into circuits of constant depth, usually three or four levels. Bounded multi-level forms are therefore quite promising for the high quality of the corresponding networks, but the main problem of these models is the huge minimization time required for their synthesis.…”

“…The main idea is to manipulate an already minimal SOP expression in order to derive, in polynomial time, a more compact but still bounded network. Generalizing the strategy described in [1], we project the minimal SOP onto subspaces of the Boolean space {0, 1} n . These projections reduce the Hamming distances among the cubes appearing in each subspace, so that further merges can be performed using standard SOP heuristics.…”