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, 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.
collective decision making
computational social choice
judgment aggregation
logical relations
algorithm complexity
Theory of computation~Theory and algorithms for application domains
50:1-50:13
Track A: Algorithms, Complexity and Games
A full version of the paper is available at [Díaz et al., 2019], https://arxiv.org/abs/1901.00138.
We are grateful to Bruno Zanuttini for his comments that improved the presentation and simplified several proofs. Lefteris Kirousis is grateful to Phokion Kolaitis for initiating him to the area of Computational Social Choice Theory. We thank Eirini Georgoulaki for her valuable help in the final stages of writing this paper.
Josep
Díaz
Josep Díaz
Computer Science Department, Universitat Politècnica de Catalunya, Barcelona
Research partially supported by TIN2017-86727-C2-1-R, GRAMM.
Lefteris
Kirousis
Lefteris Kirousis
Department of Mathematics, National and Kapodistrian University of Athens
Computer Science Department, Universitat Politècnica de Catalunya, Barcelona
https://orcid.org/0000-0002-4912-8959
Research carried out while visiting the Computer Science Department of the Universitat Politècnica de Catalunya and supported by TIN2017-86727-C2-1-R, GRAMM.
Sofia
Kokonezi
Sofia Kokonezi
Department of Mathematics, National and Kapodistrian University of Athens
https://orcid.org/0000-0002-4580-6150
John
Livieratos
John Livieratos
Department of Mathematics, National and Kapodistrian University of Athens
https://orcid.org/0000-0001-6409-4286
10.4230/LIPIcs.ICALP.2019.50
Rina Dechter and Judea Pearl. Structure identification in relational data. Artificial Intelligence, 58(1-3):237-270, 1992.
Alvaro del Val. On 2-SAT and renamable Horn. In Proceedings of the National Conference on Artificial Intelligence, pages 279-284. Menlo Park, CA; Cambridge, MA; London; AAAI Press; MIT Press; 1999, 2000.
Josep Díaz, Lefteris Kirousis, Sofia Kokonezi, and John Livieratos. Algorithmically Efficient Syntactic Characterization of Possibility Domains. arXiv preprint, 2019. URL: http://arxiv.org/abs/1901.00138.
http://arxiv.org/abs/1901.00138
Franz Dietrich. A generalised model of judgment aggregation. Social Choice and Welfare, 28(4):529-565, 2007.
Elad Dokow and Ron Holzman. Aggregation of binary evaluations for truth-functional agendas. Social Choice and Welfare, 32(2):221-241, 2009.
Elad Dokow and Ron Holzman. Aggregation of binary evaluations. Journal of Economic Theory, 145(2):495-511, 2010.
Ramez Elmasri and Sham Navathe. Fundamentals of database systems. Pearson London, 2016.
Herbert Enderton and Herbert B Enderton. A mathematical introduction to logic. Elsevier, 2001.
Ulle Endriss and Ronald de Haan. Complexity of the winner determination problem in judgment aggregation: Kemeny, Slater, Tideman, Young. In Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems, pages 117-125. International Foundation for Autonomous Agents and Multiagent Systems, 2015.
Umberto Grandi and Ulle Endriss. Binary aggregation with integrity constraints. In IJCAI Proceedings-International Joint Conference on Artificial Intelligence, volume 22, page 204, 2011.
Umberto Grandi and Ulle Endriss. Lifting integrity constraints in binary aggregation. Artificial Intelligence, 199:45-66, 2013.
Lefteris Kirousis, Phokion G Kolaitis, and John Livieratos. Aggregation of votes with multiple positions on each issue. In Proceedings 16th International Conference on Relational and Algebraic Methods in Computer Science, pages 209-225. Springer, 2017. Expanded version to appear in ACM Transactions on Economics and Computation.
Harry R Lewis. Renaming a set of clauses as a Horn set. Journal of the ACM (JACM), 25(1):134-135, 1978.
Christian List. The theory of judgment aggregation: An introductory review. Synthese, 187(1):179-207, 2012.
Klaus Nehring and Clemens Puppe. Abstract arrowian aggregation. Journal of Economic Theory, 145(2):467-494, 2010.
Gabriella Pigozzi. Belief merging and the discursive dilemma: an argument-based account to paradoxes of judgment aggregation. Synthese, 152(2):285-298, 2006.
Willard V Quine. On cores and prime implicants of truth functions. The American Mathematical Monthly, 66(9):755-760, 1959.
Thomas J. Schaefer. The complexity of satisfiability problems. In Proc. of the 10th Annual ACM Symp. on Theory of Computing, pages 216-226, 1978.
Robert Wilson. On the theory of aggregation. Journal of Economic Theory, 10(1):89-99, 1975.
Susumu Yamasaki and Shuji Doshita. The satisfiabilty problem for a class consisting of Horn sentences and some non-Horn sentences in proportional logic. Information and Control, 59(1-3):1-12, 1983.
Bruno Zanuttini and Jean-Jacques Hébrard. A unified framework for structure identification. Information Processing Letters, 81(6):335-339, 2002.
Joseph Díaz, Lefteris Kirousis, Sofia Kokonezi, and John Livieratos
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