Incremental Determinization

Abstract

We present a novel approach to solve quantified boolean formulas with one quantifier alternation (2QBF). The algorithm incrementally adds new constraints to the formula until the constraints describe a unique Skolem function -or until the absence of a Skolem function is detected. Backtracking is required if the absence of Skolem functions depends on the newly introduced constraints. We present the algorithm in analogy to search algorithms for SAT and explain how propagation, decisions, and conflicts are lifted from values to Skolem functions. The algorithm improves over the state of the art in terms of the number of solved instances, solving time, and the size of the certificates.

Knowledge Compilation for Boolean Functional Synthesis

“…Therefore, any algorithm that compiles a CNF specification to an ROBDD can be viewed as an alternative to C2Syn for compiling a CNF specification to SynNNF (albeit without refinement). We compare the performance of C2Syn with that of a BDD compiler and two state-of-the-art boolean function synthesis tools, namely, (i) the AIG-NNF pipeline of BFSS [1] with ABC's MiniSat as the SAT solver and (ii) CADET [22], [24]. For the BDD Compiler, the .qdimacs input was converted to an AIG using simple Tseitin variable detection; this AIG was then simplified and ROBDDs built using dynamic variable ordering (of all input and output variables) -this is part of the BDD pipeline of BFSS [1], henceforth called BDD BFSS .…”

“…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.…”

Understanding and Extending Incremental Determinization for 2QBF

“…Recently, Incremental Determinization (ID) has been suggested to overcome this problem [8]. ID represents a departure from the CEGIS approach in that it is structured around identifying which variables have unique Skolem functions.…”

“…In this paper, we first formalize and generalize ID [8] (Section 3). This helps us to disentangle the working principles of the algorithm from implementation-level design choices.…”

Challenges for Fast Synthesis Procedures in SMT

“…Recent work by Preiner et al [19] uses a syntax-guided approach for constructing instantiations for quantified bit-vector formulas. An approach is used for constructing Skolem functions for QBF formulas by Rabe et al [20] that is also similar in spirit.…”