Towards Parallel Boolean Functional Synthesis

Abstract

Given a relational specification ϕ(X, Y ), where X and Y are sequences of input and output variables, we wish to synthesize each output as a function of the inputs such that the specification holds. This is called the Boolean functional synthesis problem and has applications in several areas. In this paper, we present the first parallel approach for solving this problem, using compositional and CEGAR-style reasoning as key building blocks. We show by means of extensive experiments that our approach outperforms existing tools on a large class of benchmarks.

Knowledge Compilation for Boolean Functional Synthesis

“…Specifically, the correctness of line 3 is trivial. The correctness of lines 6 and 8 use Propositions 5(2) and 5 (2). The correctness of lines 12 and 17 use Lemma 6(1).…”

“…. It has been argued in [14], [11], [2], [12] that the BFnS problem for F (X, Y) can be solved by first ordering the outputs, say as x 1 ≺ x 2 · · · ≺ x n , and then synthesizing a function…”

“…∨ β k being the subformula represented by D, β r ∧ β s is a contradiction for all distinct indices r, s. 6) Positive form of input specification: Given a specification F (X, Y) in NNF, we denote by F (X, X, Y) the formula obtained by replacing every occurrence of ¬x i (x i ∈ X) in F with a fresh variable x i . This is also called the positive form of the specification and has been used earlier in [2]. Observe that for any F in NNF, F is positive unate (or monotone) in all variables in X and X.…”

“…While variants of the problem have been studied since long [17], [3], there has been significant recent interest in designing practically efficient algorithms for Boolean functional synthesis. The resulting breed of algorithms [14], [23], [22], [11], [25], [18], [13], [2], [1], [15], [7], [24] have been empirically shown to work well on large collections of benchmarks. Nevertheless, there are not-so-large examples that are currently not solvable within reasonable resources by any known algorithm.…”

“…However, if we are interested in only the result of existentially quantifying a given set of variables, these forms can be unnecessarily restrictive and exponentially larger. Recent work on Boolean functional synthesis [13], [14], [18], [24], [11], [2], [1], [8] has focused more on algorithms to directly synthesize outputs as functions of inputs. Some of these algorithms (viz.…”