An Efficient Decision Procedure for the Theory of Fixed-Sized Bit-Vectors

In this paper we describe a decision procedure for the core theory of xed-sized bit-vectors with extraction and composition than can readily be integrated into Shostak's procedure for deciding combinations of theories. Inputs to the solver are unquantiied bit-vector equations t = u and the algorithm returns true if t = u is valid in the bit-vector theory, false if t = u is unsatissable, and a system of solved equations otherwise. The time complexity of the solver is O(j t j log n + n 2), where t is the length of the bit-vector term t and n denotes the number of bits on either side of the equation. Then, the solver for the core bit-vector theory is extended to handle other bit-vector operations like bitwise logical operations, shifting, and arithmetic interpretations of bit-vectors. We develop a BDD-like data-structure called bit-vector BDDs to represent bit-vectors, various operations on bit-vectors, and a solver on bit-vector BDDs. The overall procedure has been integrated with the decision procedures of the PVS prover. The implementation has been tested with typical lemmas from the domain of microprocessor veriication. The implementation has also been applied to proofs found in the veriication of a commercial microprocessor. By using our decision procedure for bit-vectors we have simpliied a number of proofs by eliminating manual proof steps that were previously necessary for reasoning about bit-vectors.