Functional Synthesis via Input-Output Separation

Abstract

Boolean functional synthesis is the process of constructing a Boolean function from a Boolean specification that relates input and output variables. Despite significant recent developments in synthesis algorithms, Boolean functional synthesis remains a challenging problem even when state-of-the-art methods are used for decomposing the specification. In this work we bring a fresh decomposition approach, orthogonal to existing methods, that explores the decomposition of the specification into separate input and output components. We make use of an input-output decomposition of a given specification described as a CNF formula, by alternatingly analyzing the separate input and output components. We exploit well-defined properties of these components to ultimately synthesize a solution for the entire specification. We first provide a theoretical result that, for input components with specific structures, synthesis for CNF formulas via this framework can be performed more efficiently than in the general case. We then show by experimental evaluations that our algorithm performs well also in practice on instances which are challenging for existing state-of-the-art tools, serving as a good complement to modern synthesis techniques.

Sequential Relational Decomposition

“…On one hand the problem of solving TDP/PDP becomes uninteresting. On the other, however, The challenge is to find a TDP/PDP solution (R 1 , R 2 ) that allows good synthesis for a given relation R in the sense that the composition of a separate synthesis to R 2 and R 2 can be a synthesis solution to R. A recent work inspired by this approach can be found in [9].…”

“…Note that a variant of PDP, in which the required relations in the solution are functions can be viewed as an instance of the problem of Boolean functional synthesis, e.g. [16,9].…”

Knowledge Compilation for Boolean Functional Synthesis

“…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.…”

“…As we show later, wDNNF can itself be generalized to SynNNF. In another line of investigation, it was shown [8] that if a CNF specification is decomposed into an input-part and an output-part, then synthesis can be achieved in time linear in the size of the CNF specification and k, where k is the smaller of the count of maximal falsifiable subsets (MFS) of the input-part and the count of maximal satisfiable subsets (MSS) of the output-part. However, this does not yield an algorithm whose running time is polynomial in the size of the representation of F (X, Y).…”