A Normal Form Characterization for Efficient Boolean Skolem Function Synthesis

Abstract

Boolean Skolem function synthesis concerns syn¬thesizing 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 research on normal forms of specification representation that guarantee efficient synthesis, thus partially explaining the efficacy of some of these algorithms.In this paper we go one step further and ask if there exists a normal form representation of the specification that precisely characterizes "efficient" synthesis. We present a normal form called SAUNF that answers this question affirmatively. Specifically, 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.