# Algorithmically Efficient Syntactic Characterization of Possibility Domains

We call domain any arbitrary subset of a Cartesian power of the set {0,1} when we think of it as reflecting abstract rationality restrictions on vectors of two-valued judgments on a number of issues. In Computational Social Choice Theory, and in particular in the theory of judgment aggregation, a domain is called a possibility domain if it admits a non-dictatorial aggregator, i.e. if for some k there exists a unanimous (idempotent) function F:D^k → D which is not a projection function. We prove that a domain is a possibility domain if and only if there is a propositional formula of a certain syntactic form whose set of satisfying truth assignments, or models, comprise the domain. A formula whose set of satisfying truth assignments is equal to a given arbitrary domain D is sometimes referred to as an integrity constraint for D. So we call possibility integrity constraints the formulas of the specific syntactic type we define. Given a possibility domain D, we show how to construct a possibility integrity constraint for D efficiently, i.e, in polynomial time in the size of the domain. We also show how to distinguish formulas that are possibility integrity constraints in linear time in the size of the input formula. Our result falls in the realm of classical results that give syntactic characterizations of logical relations that have certain closure properties, like e.g. the result that logical relations component-wise closed under logical AND are precisely the models of Horn formulas. However, our techniques draw from results in judgment aggregation theory as well from results about propositional formulas and logical relations.

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