Large-Scale Boolean Matching

Generalized Boolean symmetries through nested partition refinement

“…As part of our study, we encoded several instances of Boolean matching as symmetry detection, and used a slightly modified version of our algorithm to solve them (command bm2 in ABC). Table II demonstrates the results, and compares the runtimes of our matcher to that proposed in [1] (we chose [1] since it solves large-scale matching and its source code is publicly available). All the results in Table II are averaged over 10 re-runs, where each re-run 1) randomly permuted I/Os of the circuits, and 2) reconstructed the circuits using ABC's synthesis commands to ensure structural difference.…”

“…1. 1 The set of all symmetries of an object forms a group 2 under functional composition. This group is referred to as the symmetry group of the object.…”

“…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.…”

“…One common application is in Boolean matching, where functional equivalence of Boolean functions under permutation (and negation) of 1 A rotational symmetry of the form (a 1 , a 2 , a 3 , ..., an) maps a 1 to a 2 , a 2 to a 3 , ..., and an to a 1 . Also, ι denotes the identity.…”

“…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].…”