Synthesis for regular specifications over unbounded domains

Synthesis from declarative specifications is an ambitious automated method for obtaining systems that are correct by construction. Previous work includes synthesis of reactive finite-state systems from linear temporal logic and its fragments. Further recent work focuses on a different application area by doing functional synthesis over unbounded domains, using a modified Presburger arithmetic quantifier elimination algorithm. We present new algorithms for functional synthesis over unbounded domains based on automata-theoretic methods, with advantages in the expressive power and in the efficiency of synthesized code. Our approach synthesizes functions that meet given regular specifications defined over unbounded sequences of input and output bits. Thanks to the translation from weak monadic second-order logic to automata, this approach supports full Presburger arithmetic as well as bitwise operations on arbitrary length integers. The presence of quantifiers enables finding solutions that optimize a given criterion. Unlike synthesis of reactive systems, our notion of realizability allows functions that require examining the entire input to compute the output. Regardless of the complexity of the specification, our algorithm synthesizes linear-time functions that read the input and directly produce the output. We also describe a technique to synthesize functions with bounded lookahead when possible, which is appropriate for streaming implementations. We implemented our synthesis algorithm and show that it synthesizes efficient functions on a number of examples.