We address the problem of providing a logical formalization of arithmetic in declarative modelling languages for NP search problems. The challenge is to simultaneously allow quantification over an infinite domain such as the natural numbers, provide natural modelling facilities, and control expressive power of the language. To address the problem, we introduce an extension of the model expansion (MX) based framework to finite structures embedded in an infinite secondary structure, together with "double-guarded" logics for representing MX specifications for these structures. The logics also contain multi-set functions (aggregate operations). Our main result is that these logics capture the complexity class NP on "small-cost" arithmetical structures.

Eugenia Ternovska, David G. Mitchell