Determinism versus Nondeterminism with Arithmetic Tests and Computation
نویسنده
چکیده
For each natural number d we consider a finite structure Md whose universe is the set of all 0, 1-sequence of length n = 2d, each representing a natural number in the set {0, 1, ..., 2n − 1} in binary form. The operations included in the structure are the constants 0, 1, 2n − 1, n, multiplication and addition modulo 2n, the unary function min{2x, 2n − 1}, the binary functions bx/yc, max(x, y), min(x, y), and the boolean vector operations ∧,∨,¬ defined on 0, 1 sequences of length n by performing the operations on all components simultaneously. These are essentially the arithmetic operations that can be performed on a RAM by a single instruction. We show that there exists a term (that is, an algebraic expression) F (x, y) built up from the mentioned operations, with the only free variables x, y, such that for all terms G(y), which is also built up from the mentioned operations, the following holds. For infinitely many positive integers d, there exists an a ∈Md such that the following two statements are not equivalent: (i) Md |= ∃x, F (x, a), (ii) Md |= G(a) = 0. In other words, the question whether an existential statement, depending on the parameter a ∈Md is true or not, cannot be decided by evaluating an algebraic expression at a. We also show that this theorem remains true if we include the operation min{xy, 2n − 1} into the structure Md. A general theorem is proved as well which describes sufficient conditions for a set of operations on a sequence of structures Kd, d = 1, 2, ... which guarantees that the analogue of the mentioned theorem holds for the structure Kd too.
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