What’s Hard About Boolean Functional Synthesis?

Abstract

Given a relational specification between Boolean inputs and outputs, the goal of Boolean functional synthesis is to synthesize each output as a function of the inputs such that the specification is met. In this paper, we first show that unless some hard conjectures in complexity theory are falsified, Boolean functional synthesis must generate large Skolem functions in the worst-case. Given this inherent hardness, what does one do to solve the problem? We present a two-phase algorithm, where the first phase is efficient both in terms of time and size of synthesized functions, and solves a large fraction of benchmarks. To explain this surprisingly good performance, we provide a sufficient condition under which the first phase must produce correct answers. When this condition fails, the second phase builds upon the result of the first phase, possibly requiring exponential time and generating exponential-sized functions in the worst-case. Detailed experimental evaluation shows our algorithm to perform better than other techniques for a large number of benchmarks.

Knowledge Compilation for Boolean Functional Synthesis

“…It was recently shown [1] that unless some long-standing complexity theoretic conjectures are falsified, Boolean functional synthesis must necessarily require super-polynomial (or even exponential) space and time. In the same work [1], it was also shown that if a specification is represented in weak decomposable negation normal form wDNNF, synthesis can be accomplished in time polynomial in the size of the specification.…”

“…It was recently shown [1] that unless some long-standing complexity theoretic conjectures are falsified, Boolean functional synthesis must necessarily require super-polynomial (or even exponential) space and time. In the same work [1], it was also shown that if a specification is represented in weak decomposable negation normal form wDNNF, synthesis can be accomplished in time polynomial in the size of the specification. While this was a first step towards identifying a normal form with the explicit objective of polynomial-time synthesis, experimental results in [1] indicate that wDNNF doesn't really characterize specifications that admit efficient synthesis.…”

“…In the same work [1], it was also shown that if a specification is represented in weak decomposable negation normal form wDNNF, synthesis can be accomplished in time polynomial in the size of the specification. While this was a first step towards identifying a normal form with the explicit objective of polynomial-time synthesis, experimental results in [1] indicate that wDNNF doesn't really characterize specifications that admit efficient synthesis. Specifically, experiments in [1] showed that a polynomial-time algorithm intended for synthesis from wDNNF specifications ends up solving the synthesis problem for a large class of specifications not in wDNNF.…”

“…While this was a first step towards identifying a normal form with the explicit objective of polynomial-time synthesis, experimental results in [1] indicate that wDNNF doesn't really characterize specifications that admit efficient synthesis. Specifically, experiments in [1] showed that a polynomial-time algorithm intended for synthesis from wDNNF specifications ends up solving the synthesis problem for a large class of specifications not in wDNNF. This motivates us to ask if there exists a weaker (than wDNNF) sub-class of Boolean relational specifications that admit polynomial-time synthesis.…”

“…wDNNF, dDNNF, DNNF, FBDD, ROBDD), unless some longstanding complexity theoretic conjectures are falsified. • We show that by suitably weakening SynNNF, we can precisely characterize the class of Boolean specifications that admit polynomial-time synthesis by a simple algorithm originally proposed in [1]. • We define a natural notion of refinement of specifications w.r.t synthesis and show that every specification that admits polynomial-time synthesis necessarily has a polynomial-sized refinement that is in SynNNF.…”