Soft nonlinearity constraints and their lower-arity decomposition

In this paper we express nonlinearity requirements in terms of soft global n-ary constraints. We describe a method to project global nonlinearity constraints into redundant lowerarity hard constraints. The nonlinearity constraints apply to the inputs and outputs of discrete functions f : ℤ_2n → ℤ_2m mapping n-bit inputs to m-bit outputs, n > m. No output bit (or linear function on a subset of output bits) of the function f should be too close to a linear function of (a subset of) its input bits. For example, if we select any output bit position and any subset of the six input bit positions, the fraction of inputs for which this output bit equals the exclusive-OR of these input bits should not be close to 0 or 1, but rather should be near 1/2. We analyze this constraint and find that the obtained redundant constraints increase the efficiency of an arc consistency maintenance solver by several orders of magnitude.