(theory Fixed_Size_BitVectors :written_by {Silvio Ranise, Cesare Tinelli, and Clark Barrett} :date {May 7, 2007} :notes "Against the requirements of the current SMT-LIB standard this theory does not provide a value for the formal attributes :sorts, :funs, and :preds. The reason is that the theory has an infinite number of sort, function, and predicate symbols, and so they cannot be specified formally in the current SMT-LIB language. While extending SMT-LIB's type system with dependent types would allow a finitary formal specification of all the symbols in this theory's signature, such an extension does not seem to be worth the trouble at the moment. As a temporary ad-hoc solution, this theory declaration specifies the signature, in English, in the user-defined attributes :sorts_description, :funs_description, and :preds_description. " :sorts_description { All sort symbols of the form BitVec[m], where m is a numeral greater than 0. } :funs_description { Constant symbols bit0 and bit1 of sort BitVec[1] } :funs_description { All function symbols with arity of the form (concat BitVec[i] BitVec[j] BitVec[m]) where - i,j,m are numerals - i,j > 0 - i + j = m } :funs_description { All function symbols with arity of the form (extract[i:j] BitVec[m] BitVec[n]) where - i,j,m,n are numerals - m > i >= j >= 0, - n = i-j+1. } :funs_description { All function symbols with arity of the form (op1 BitVec[m] BitVec[m]) or (op2 BitVec[m] BitVec[m] BitVec[m]) where - op1 is from {bvnot, bvneg} - op2 is from {bvand, bvor, bvadd, bvmul, bvudiv, bvurem, bvshl, bvlshr} - m is a numeral greater than 0 } :preds_description { All predicate symbols with arity of the form (pred BitVec[m] BitVec[m]) where - pred is from {bvult} - m is a numeral greater than 0 } :definition "This is a core theory for fixed-size bitvectors where the operations of concatenation and extraction of bitvectors as well as the usual logical and arithmetic operations are overloaded. The theory is defined semantically as follows. The sort BitVec[m] (for m > 0) is the set of finite functions whose domain is the initial segment of the naturals [0...m), meaning that 0 is included and m is excluded, and the co-domain is {0,1}. The semantic interpretation [[_]] of well-sorted BitVec-terms is inductively defined as follows. - Variables If v is a variable of sort BitVec[m] with 0 < m, then [[v]] is some element of [{0,...,m-1} -> {0,1}], the set of total functions from {0,...,m-1} to {0,1}. - Constant symbols bit0 and bit1 of sort BitVec[1] [[bit0]] := \lambda x : [0,1). 0 [[bit1]] := \lambda x : [0,1). 1 - Function symbols for concatenation [[(concat s t)]] := \lambda x : [0...n+m). if (x= 0 and y > 0 returns the integer part of x divided by y (i.e., truncated integer division). o (x rem y) where x and y are integers with x >= 0 and y > 0 returns the remainder when x is divided by y. Note that we always have the following equivalence (for y > 0): (x div y) * y + (x rem y) = x. o bv2nat which takes a bitvector b: [0...m) --> {0,1} with 0 < m, and returns an integer in the range [0...2^m), and is defined as follows: bv2nat(b) := b(m-1)*2^{m-1} + b(m-2)*2^{m-2} + ... + b(0)*2^0 o nat2bv[m], with 0 < m, which takes a non-negative integer n and returns the (unique) bitvector b: [0,...,m) -> {0,1} such that b(m-1)*2^{m-1} + ... + b(0)*2^0 = n rem 2^m Now, we can define the following operations. Suppose s and t are both terms of sort BitVec[m], m > 0. [[(bvneg s)]] := nat2bv[m](2^m - bv2nat([[s]])) [[(bvadd s t)]] := nat2bv[m](bv2nat([[s]]) + bv2nat([[t]])) [[(bvmul s t)]] := nat2bv[m](bv2nat([[s]]) * bv2nat([[t]])) [[(bvudiv s t)]] := if bv2nat([[t]]) != 0 then nat2bv[m](bv2nat([[s]]) div bv2nat([[t]])) [[(bvurem s t)]] := if bv2nat([[t]]) != 0 then nat2bv[m](bv2nat([[s]]) rem bv2nat([[t]])) - Shift operations Suppose s and t are both terms of sort BitVec[m], m > 0. We make use of the definitions given for the arithmetic operations, above. [[(bvshl s t)]] := nat2bv[m](bv2nat([[s]]) * 2^(bv2nat([[t]]))) [[(bvlshr s t)]] := nat2bv[m](bv2nat([[s]]) div 2^(bv2nat([[t]]))) Finally, we can define the binary predicate bvult: (bvult s t) is interpreted to be true iff bv2nat([[s]]) < bv2nat([[t]]) Note that the semantic interpretation above is underspecified because it does not specify the meaning of (bvudiv s t) or (bvurem s t) in case bv2nat([[t]]) is 0. Since the semantics of SMT-LIB's underlying logic associates *total* functions to function symbols, we then consider as models of this theory *any* interpretation conforming to the specifications above (and defining bvudiv and bvurem arbitrarily when the second argument evaluates to 0). Benchmarks using this theory should only include a :status sat or :status unsat attribute if the status is independent of the particular choice of model for the theory. " )