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

APPROACHES TO THE THEORY OF AGGREGATION

Παρουσιάζουμε διάφορες επίσημες προσεγγίσεις στη θεωρία συμψηφισμού απόψεων, όπου οι όχι/ναι θέσεις μιας ομάδας ατόμων πάνω σε μια σειρά από m θέματα πρέπει να συναθροιστούν σε μια συλλογική απόφαση, και δείχνουμε ότι αυτές οι προσεγγίσεις είναι κατά μία έννοια «ισοδύναμες». Στη συνέχεια, εστιάζουμε σε δύο από αυτές τις προσεγγίσεις: το αφαιρετικό πλαίσιο (abstract framework) όπου το πεδίο της διαδικασίας συμψηφισμού είναι ένα υποσύνολο του {0,1}^m, το οποίο θεωρείται ότι αντιπροσωπεύει τις «λογικές» ψήφους και το πλαίσιο που βασίζεται στους περιοριστές της ακαιρεότητας (integrity constraint based framework), όπου ένας τύπος της προτασιακής λογικής, που ονομάζεται περιοριστής της ακαιρεότητας (integrity constraint), καθορίζει ποια διανύσματα θεωρούνται «λογικά», υπό την έννοια ότι το πεδίο της διαδικασίας συμψηφισμού είναι το σύνολο των απονομών αληθοτιμών που τον ικανοποιούν. Ενδιαφερόμαστε μόνο για διαδικασίες συμψηφισμού που διατηρούν αυτή την έννοια λογικότητας, χωρίς όμως να δίνουν όλη τη δύναμη απόφασης σε έναν μόνο ψηφοφόρο. Αυτές οι διαδικασίες ονομάζονται μη-δικτατορικοί συμψηφιστές. Παρέχουμε ικανές και αναγκαίες συνθήκες, που αφορούν στη συντακτική μορφή ενός περιοριστή ακεραιότητας, έτσι ώστε το πεδίο που περιγράφει να δέχεται έναν μη-δικτατορικό συμψηφιστή. Ονομάζουμε αυτό το είδος τύπων δυνητικούς περιοριστές ακεραιότητας (possibility integrity constraints). Δείχνουμε ότι οι δυνητικοί περιοριστές ακεραιότητας είναι εύκολα αναγνωρίσιμοι και παρέχουμε αλγόριθμους οι οποίοι, δοθέντος ενός πεδίου D \subseteq {0,1}^m, ελέγχουν σε χρόνο πολυωνυμικό στο μέγεθός του εάν δέχεται έναν μη-δικτατορικό συμψηφιστή, και παράγουν έναν δυνητικό περιοριστή ακεραιότητας που το περιγράφει, σε περίπτωση που αυτό συμβαίνει. Μελετάμε επίσης διάφορες υποκατηγορίες μη-δικτατορικών συμψηφιστών, συγκεκριμένα τοπικά μη-δικτατορικούς συμψηφιστές (locally non-dictatorial aggregators), συμψηφιστές που δεν είναι γενικευμένες δικτατορίες (not generalized dictatorships), ανωνυμικούς (anonymous), μονοτονικούς (monotone), ισχυρά δημοκρατικούς (StrongDem) και συστηματικούς συμψηφιστές (systematic aggregators). Χαρακτηρίζουμε συντακτικώς τους αντίστοιχους περιοριστές ακεραιότητας και αποδεικνύουμε ότι κάθε ένα από αυτά τα είδη περιοριστών ακεραιότητας μπορεί να αναγνωριστεί αποτελεσματικά. Τέλος, δείχνουμε ότι δοθέντος ενός πεδίου D, μπορούμε αμφότερα να ελέγξουμε αποτελεσματικά αν περιγράφεται από έναν τέτοιο τύπο και, σε περίπτωση που αυτό συμβαίνει, να τον κατασκευάσουμε.We present various formal approaches to the theory of judgement aggregation, where no/yes positions of a group of individuals over a set of m issues need to be aggregated into a collective one, and show that these approaches are in a sense "equivalent". Then, we focus on two of these approaches: the abstract framework where the domain of the aggregation process is a subset of {0,1}^m, thought to represent the "rational" judgements and the integrity constraint framework, where a formula of propositional logic, called the integrity constraint defines which ballots are considered "rational", in the sense that the domain of the aggregation process is the set of its satisfying truth assignments. We are only interested in aggregation procedures that preserve this notion of rationality, without giving all decision power to a single voter. These procedures are called non-dictatorial aggregators. We provide necessary and sufficient conditions, regarding the syntactic type of an integrity constraint, so that the domain it describes admits a non-dictatorial aggregator. We call this type of formulas possibility integrity constraints. We show that possibility integrity constraints are easily recognisable and provide algorithms that, given a domain D \subseteq {0,1}^m, check in time polynomial in its size whether it admits a non-dictatorial aggregator, and actually produce a possibility integrity constraint that describes it in case it does. We also study various sub-classes of non-dictatorial aggregators, namely locally non-dictatorial aggregators, aggregators that are not generalized dictatorships, anonymous, monotone, StrongDem and systematic aggregators. We syntactically characterize the corresponding integrity constraints and show that each of these types of integrity constraints can be recognized efficiently. Finally, we show that given a domain, we can both efficiently check if it is described by such a formula and, in case it is, construct it

Algorithmically efficient syntactic characterization of possibility domains

In the field of Judgment Aggregation, a domain, that is a subset of a Cartesian power of {0, 1}, is considered to reflect abstract rationality restrictions on vectors of two-valued judgments on a number of issues. We are interested in the ways we can aggregate the positions of a set of individuals, whose positions over each issue form vectors of the domain, by means of unanimous (idempotent) functions, whose output is again an element of the domain. Such functions are called non-dictatorial, when their output is not simply the positions of a single individual. Here, we consider domains admitting various kinds of non-dictatorial aggregators, which reflect various properties of majority aggregation: (locally) non-dictatorial, generalized dictatorships, anonymous, monotone, StrongDem and systematic. We show that interesting and, in some sense, democratic voting schemes are always provided by domains that can be described by propositional formulas of specific syntactic types we define. Furthermore, we show that we can efficiently recognize such formulas and that, given a domain, we can both efficiently check if it is described by such a formula and, in case it is, construct it. Our results fall in the realm of classical results concerning the syntactic characterization of domains with specific closure properties, like domains closed under logical AND which are the models of Horn formulas. The techniques we use to obtain our results draw from judgment aggregation as well as propositional logic and universal algebra.The first two authors’ research was partially supported by TIN2017-86727-C2-1-R, GRAMM. The research of the second author was carried out while visiting the Computer Science Department of the Universitat Politècnica de Catalunya.Peer ReviewedPostprint (published version

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