eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-07-04
50:1
50:13
10.4230/LIPIcs.ICALP.2019.50
article
Algorithmically Efficient Syntactic Characterization of Possibility Domains
Díaz, Josep
1
Kirousis, Lefteris
2
1
https://orcid.org/0000-0002-4912-8959
Kokonezi, Sofia
2
https://orcid.org/0000-0002-4580-6150
Livieratos, John
2
https://orcid.org/0000-0001-6409-4286
Computer Science Department, Universitat Politècnica de Catalunya, Barcelona
Department of Mathematics, National and Kapodistrian University of Athens
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, sometimes called an integrity constraint, whose set of satisfying truth assignments, or models, comprise the domain. 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. Finally, we prove the analogous results for local possibility domains, i.e. domains that admit an aggregator which is not a projection function, even when restricted to any given issue. 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol132-icalp2019/LIPIcs.ICALP.2019.50/LIPIcs.ICALP.2019.50.pdf
collective decision making
computational social choice
judgment aggregation
logical relations
algorithm complexity