Knowledge Compilation for Boolean Functional Synthesis

Given a Boolean formula $F(\mathrm{X},\ \mathrm{Y})$, where X is a vector of outputs and Y is a vector of inputs, the Boolean functional synthesis problem requires us to compute a Skolem function vector $\Psi(\mathrm{Y})$ such that $F(\Psi(\mathrm{Y}),\ \mathrm{Y})$ holds whenever $\exists \mathrm{X}F(\mathrm{X},\ \mathrm{Y})$ holds. In this paper, we investigate the relation between the representation of the specification $F(\mathrm{X},\ \mathrm{Y})$ and the complexity of synthesis. We introduce a new normal form for Boolean formulas, called SynNNF, that guarantees polynomial-time synthesis and also polynomial-time existential quantification for some order of quantification of variables. We show that several normal forms studied in the knowledge compilation literature are subsumed by SynNNF, although SynNNF can be super-polynomially more succinct than them. Motivated by these results, we propose an algorithm to convert a specification in CNF to SynNNF, with the intent of solving the Boolean functional synthesis problem. Experiments with a prototype implementation show that this approach solves several benchmarks beyond the reach of state-of-the-art tools.