A unified approach to canonical form-based Boolean matching

Abstract

In this paper, we face the problem of P-equivalence Boolean matching. We outline a formal framework that unifies some of the canonical form-based approaches to the problem.As a first major contribution, we show how these approaches are particular cases of a single generic algorithm, parametric with respect to a given linear transformation of the input function.As a second major contribution, we identify a linear transformation that can be used to significantly speed up Boolean matching with respect to the state of the art. Experimental results show that, on average, our approach is five times faster than the main competitor on 20-variables input functions, and scales better, allowing to match even larger components.

Generalized Boolean symmetries through nested partition refinement

“…As future research, we plan to extend our algorithm to detect symmetry under permutation and negation of I/Os. Applications of such an algorithm include solving the general Boolean matching problem [2], and reducing samples for logic simulation [16].…”

“…9 shows the routine that checks for a functional conflict in an OPP. This routine searches the database of simulation pairs for two consistent input vectors (lines [1][2][3][4][5][6][7][8][9][10][11][12][13]. Suppose that it finds input vectors P and Q consistent with regard to the OPP, and suppose that P, R and Q, U are the simulation pairs that correspond to P and Q.…”

“…Also, ι denotes the identity. 2 Group theory is a branch of abstract algebra that studies the algebraic structures known as groups. A group comprises a non-empty set of elements with a binary operation that is associative, admits an identity element, and is invertible.…”

“…MUX: z = a 0 s 1 s 0 + a 1 s 1 s 0 + a 2 s 1 s 0 + a 3 s 1 s 0 γ 1 : ι γ 2 : (a 0 , a 2 )(a 0 , a 2 )(a 1 , a 3 )(a 1 , a 3 )(s 1 , s 1 ) γ 3 : (a 1 , a 2 )(a 1 , a 2 )(s 0 , s 1 )(s 0 , s 1 ) γ 4 : (a 0 , a 1 , a 3 , a 2 )(a 0 , a 1 , a 3 , a 2 )(s 0 , s 1 , s 0 , s 1 ) γ 5 : (a 0 , a 1 )(a 0 , a 1 )(a 2 , a 3 )(a 2 , a 3 )(s 0 , s 0 ) γ 6 : (a 0 , a 3 )(a 0 , a 3 )(a 1 , a 2 )(a 1 , a 2 )(s 0 , s 0 )(s 1 , s 1 ) γ 7 : (a 0 , a 2 , a 3 , a 1 )(a 0 , a 2 , a 3 , a 1 )(s 0 , s 1 , s 0 , s 1 ) γ 8 : (a 0 , a 3 )(a 0 , a 3 )(s 0 , s 1 )(s 0 , s 1 ) γ 9 : (a 0 , a 0 )(a 1 , a 1 )(a 2 , a 2 )(a 3 , a 3 )(z, z ) γ 10 : (a 0 , a 2 )(a 0 , a 2 )(a 1 , a 3 )(a 1 , a 3 )(s 1 , s 1 )(z, z ) γ 11 : (a 0 , a 0 )(a 1 , a 2 )(a 1 , a 2 )(a 3 , a 3 )(s 0 , s 1 )(s 0 , s 1 )(z, z ) γ 12 : (a 0 , a 1 , a 3 , a 2 )(a 0 , a 1 , a 3 , a 2 )(s 0 , s 1 , s 0 , s 1 )(z, z ) γ 13 : (a 0 , a 1 )(a 0 , a 1 )(a 2 , a 3 )(a 2 , a 3 )(s 0 , s 0 )(z, z ) γ 14 : (a 0 , a 3 )(a 0 , a 3 )(a 1 , a 2 )(a 1 , a 2 )(s 0 , s 0 )(s 1 , s 1 )(z, z ) Generating Set: g = {γ 3 , γ 5 , γ 9 } inputs and outputs is investigated [1], [2]. Other applications include BDD minimization [3] and circuit power optimization [4].…”

Large-Scale Boolean Matching

“…The goal of signature-based matching is to prune the Boolean matching space by filtering out impossible I/O correspondences [4,1]. On the other hand, in matching based on canonical forms, first canonical representations of two Boolean functions are computed and then compared against each other to find valid I/O matches [3,2]. Here, our PP-equivalence checking method first prunes the search space using graph algorithms and simulation signatures, then it invokes SAT-solving until exact I/O matches are found ( Figure 2).…”