A Normal Form Characterization for Efficient Boolean Skolem Function Synthesis

TL;DR

This paper introduces SAUNF, a normal form that exactly characterizes when Boolean Skolem function synthesis can be performed efficiently, bridging the gap between theoretical hardness and practical algorithms.

Contribution

The paper proposes SAUNF, a new normal form that precisely characterizes tractable Boolean functional synthesis and is more succinct than existing forms like BDDs and DNNFs.

Findings

SAUNF can be compiled from specifications in polynomial time if and only if the synthesis is polynomial-time.

Specifications with polynomial-sized solutions have polynomial-sized SAUNF representations.

SAUNF is exponentially more succinct than BDDs and DNNFs, and subsumes SynNNF.

Abstract

Boolean Skolem function synthesis concerns synthesizing outputs as Boolean functions of inputs such that a relational specification between inputs and outputs is satisfied. This problem, also known as Boolean functional synthesis, has several applications, including design of safe controllers for autonomous systems, certified QBF solving, cryptanalysis etc. Recently, complexity theoretic hardness results have been shown for the problem, although several algorithms proposed in the literature are known to work well in practice. This dichotomy between theoretical hardness and practical efficacy has motivated the research into normal forms or representations of input specifications that permit efficient synthesis, thus explaining perhaps the efficacy of these algorithms. In this paper we go one step beyond this and ask if there exists a normal form representation that can in fact precisely characterize "efficient" synthesis. We present a normal form called SAUNF that precisely characterizes tractable synthesis in the following sense: a specification is polynomial time synthesizable iff it can be compiled to SAUNF in polynomial time. Additionally, a specification admits a polynomial-sized functional solution iff there exists a semantically equivalent polynomial-sized SAUNF representation. SAUNF is exponentially more succinct than well-established normal forms like BDDs and DNNFs, used in the context of AI problems, and strictly subsumes other more recently proposed forms like SynNNF. It enjoys compositional properties that are similar to those of DNNF. Thus, SAUNF provides the right trade-off in knowledge representation for Boolean functional synthesis.