On the degree of restrictions of q-valued logic vector functions to linear manifolds

Abstract

Abstract We obtain estimates for the probability that for a randomly selected k-dimensional n-place q-valued logic vector function there exists a linear manifold of fixed dimension such that the degree of the restriction of the function to this manifold is not larger than the given value. The asymptotics of the number of manifolds on which the restrictions are affine is obtained. It is shown that if n → ∞ and k ≤ n/q, then for almost all k-dimensional n-place vector functions the maximum dimension of a manifold on which the restriction is affine lies in the interval [ ⌊ log q n k + log q log q n k ⌋ , ⌈ log q n k + log q log q n k ⌉ ] $ [\lfloor \log_q \frac{n}{k}+\log_q \log_q \frac{n}{k} \rfloor, \lceil \log_q \frac{n}{k}+\log_q \log_q \frac{n}{k} \rceil] $ , while the analogous parameter for the case of fixed variables lies in the range [ ⌊ log q n k ⌋ , ⌈ log q n k ⌉ ] $ [\lfloor \log_q \frac{n}{k} \rfloor, \lceil \log_q \frac{n}{k} \rceil] $ .