On the Expressiveness of Essence

Development of languages for specifying or modelling problems is an important direction in constraint modelling. To provide greater abstraction and modelling convenience, these languages are becoming more syntactically rich, leading to a variety of questions about their expressive power. In this paper, we consider the expressiveness of Essence, a specification language with a rich variety of syntactic features. We identify natural fragments of Essence that capture the complexity classes P, NP, all levels Σ p i of the polynomial hierarchy, and all levels k-NEXP of the nondeterministic exponential-time hierarchy. The union of these classes is the very large complexity class ELEMENTARY. One goal is to begin to understand which features play a role in the high expressive power of the language and which are purely features of convenience. We also discuss the expressive limit of Essence, a notion of capturing NP-search which is slightly different than that of capturing NP decision problems, and the formalization of arithmetic in Essence and related languages. Our study of arithmetic raises some, perhaps surprising, issues. Our study is an application of descriptive complexity theory, and illustrates the value of taking a logic-based view of modelling and specification languages.