On the Complexity of Fixed-Size Bit-Vector Logics with Binary Encoded Bit-Width

Bit-precise reasoning is important for many practical applications of Satisfiability Modulo Theories (SMT). In recent years efficient approaches for solving fixed-size bit-vector formulas have been developed. From the theoretical point of view, only few results on the complexity of fixed-size bit-vector logics have been published. In this paper we show that some of these results only hold if unary encoding on the bit-width of bit-vectors is used. We then consider fixed-size bit-vector logics with binary encoded bit-width and establish new complexity results. Our proofs show that binary encoding adds more expressiveness to bit-vector logics, e.g. it makes fixed-size bit-vector logic even without uninterpreted functions nor quantification NExpTime-complete. We also show that under certain restrictions the increase of complexity when using binary encoding can be avoided.