Synthesis for Unbounded Bit-Vector Arithmetic

Abstract

Abstract. We propose to describe computations using QFPAbit, a language of quantifier-free linear arithmetic on unbounded integers with bitvector operations. We describe an algorithm that, given a QFPAbit formula with input and output variables denoting integers, generates an efficient function from a sequence of inputs to a sequence of outputs, whenever such function on integers exists. The starting point for our method is a polynomial-time translation mapping a QFPAbit formula into the sequential circuit that checks the correctness of the input/output relation. From such a circuit, our synthesis algorithm produces solved circuits from inputs to outputs that are no more than singly exponential in size of the original formula. In addition to the general synthesis algorithm, we present techniques that ensure that, for example, multiplication and division with large constants do not lead to an exponential blowup, addressing a practical problem with a previous approach that used the MONA tool to generate the specification automata.

More on the Complexity of Quantifier-Free Fixed-Size Bit-Vector Logics with Binary Encoding

“…In [10,15], the authors give a translation from quantifier-free Presburger arithmetic with bitwise operations (QFPAbit) to Sequential Circuits. We can adopt their approach in order to construct a translation for QF BV2 31 .…”

“…Let x rns , y rns denote bit-vector variables, c rns a bit-vector constant, and t 1 rns , t 2 rns bit-vector terms only containing bitvector variables and bitwise operations. Following [10,15] we further assume w.l.o.g that Φ only consists of three types of expressions: t 1 rns t 2 rns , x rns c rns , and x rns y rns 3 1 rns , since any QF BV2 31 formula can be written like this with only a linear growth in the number of original variables.…”

“…We establish two new complexity results for restricted bit-vector logics and bring together our previous results in [1] with work on linear arithmetic on non-fixed-size bit-vectors [10,15] and work on the reduction of bit-widths [16,17]. The formula in the given example only contains bitwise operations, equality, and addition.…”

“…Working with non-fixed-size bit-vectors has been considered for instance in [4,9], and more recently in [10], but is not the focus of this paper. Most industrial applications (and examples in the SMT-LIB) have fixed bit-width.…”

Reductions for Synthesis Procedures

“…Previous work described synthesis procedures for linear arithmetic and sets [8,9] as well as extensions to unbounded bitvector constraints [4,18]. In this paper we make further steps towards systematic derivation of synthesis procedures by showing how inference rules that describe decision procedure steps (possibly for a combination of theories) can be generalized to synthesis procedures.…”