Jérôme Lang

A Short Introduction to Computational Social Choice

Computational social choice is an interdisciplinary field of study at the interface of social choice theory and computer science, promoting an exchange of ideas in both directions. On the one hand, it is concerned with the application of techniques developed in computer science, such as complexity analysis or algorithm design, to the study of social choice mechanisms, such as voting procedures or fair division algorithms. On the other hand, computational social choice is concerned with importing concepts from social choice theory into computing. For instance, the study of preference aggregation mechanisms is also very relevant to multiagent systems. In this short paper we give a general introduction to computational social choice, by proposing a taxonomy of the issues addressed by this discipline, together with some illustrative examples and an (incomplete) bibliography.

Fuzzy sets in approximate reasoning, part 2: logical approaches

Abstract This second part of an overview of fuzzy set-based methods for approximate reasoning is devoted to deductive approaches dealing with knowledge bases made of a collection of symbolic expressions (e.g. logical formulas) to which numerical weigths are attached. Then such a knowledge base can be viewed as a fuzzy set of formulae (especially if the weights belong to the interval [0, 1]). An important distinction is made, in the interpretation of the weights, between degrees of truth and degrees of uncertainty; the former can be assumed to behave in a fully compositional way while the latter cannot. Rule-based inference systems where fuzzy set operations are used for propagating and combining ‘certainty factors’, are briefly discussed. Then an extensive survey of truth-functional fuzzy logic is provided, and it is followed by a brief presentation of possibilistic logic, a possibility theory-based logic of uncertainty (which is not compositional with respect to every connective). In both cases, the automated deduction and logic programming aspects are emphasized while purely logical considerations which are not directly relevant for approximate reasoning, are not developed.

Automated reasoning using possibilistic logic: semantics, belief revision, and variable certainty weights

An approach to automated deduction under uncertainty, based on possibilistic logic, is described; for that purpose we deal with clauses weighted by a degree that is a lower bound of a necessity or a possibility measure, according to the nature of the uncertainty. Two resolution rules are used for coping with the different situations, and the classical refutation method can be generalized with these rules. Also, the lower bounds are allowed to be functions of variables involved in the clauses, which results in hypothetical reasoning capabilities. In cases where only lower bounds of necessity measures are involved, a semantics is proposed in which the completeness of the extended resolution principle is proved. The relation between our approach and the idea of minimizing abnormality is briefly discussed. Moreover, deduction from a partially inconsistent knowledge base can be managed in this approach and captures a form of nonmonotonicity. >

Uncertainty in Constraint Satisfaction Problems: a Probalistic Approach

We propose a framework for dealing with probabilistic uncertainty in constraint satisfaction problems, associating with each constraint the probability that it is a part of the real problem (the latter being only partially known). The probability degrees on the relevance of the constraints enable us to define, for each instanciation, the probability that it is a solution of the real problem. We briefly give a methodology for the search of the best solution (maximizing this probability).

Logical Preference Representation and Combinatorial Vote

We introduce the notion of combinatorial vote, where a group of agents (or voters) is supposed to express preferences and come to a common decision concerning a set of non-independent variables to assign. We study two key issues pertaining to combinatorial vote, namely preference representation and the automated choice of an optimal decision. For each of these issues, we briefly review the state of the art, we try to define the main problems to be solved and identify their computational complexity.

Efficiency and envy-freeness in fair division of indivisible goods: logical representation and complexity

We consider the problem of allocating fairly a set of indivisible goods among agents from the point of view of compact representation and computational complexity. We start by assuming that agents have dichotomous preferences expressed by propositional formulae. We express efficiency and envy-freeness in a logical setting, which reveals unexpected connections to nonmonotonic reasoning. Then we identify the complexity of determining whether there exists an efficient and envy-free allocation, for several notions of efficiency, when preferences are represented in a succinct way (as well as restrictions of this problem). We first study the problem under the assumption that preferences are dichotomous, and then in the general case.

The computational complexity of dominance and consistency in CP-Nets

We investigate the computational complexity of testing dominance and consistency in CP-nets. Up until now, the complexity of dominance has been determined only for restricted classes in which the dependency graph of the CP-net is acyclic. However, there are preferences of interest that define cyclic dependency graphs; these are modeled with general CP-nets. We show here that both dominance and consistency testing for general CP-nets are PSPACE-complete. The reductions used in the proofs are from STRIPS planning, and thus establish strong connections between both areas.

Utilitarian Desires

Autonomous agents reason frequently about preferences such as desires and goals. In this paper we propose a logic of desires with a utilitarian semantics, in which we study nonmonotonic reasoning about desires and preferences based on the idea that desires can be understood in terms of utility losses (penalties for violations) and utility gains (rewards for fulfillments). Our logic allows for a systematic study and classification of desires, for example by distinguishing subtly different ways to add up these utility losses and gains. We propose an explicit construction of the agents preference relation from a set of desires together with different kinds of knowledge. A set of desires extended with knowledge induces a set of ‘distinguished’ utility functions by adding up the utility losses and gains of the individual desires, and these distinguished utility functions induce the preference relation.

How many candidates are needed to make elections hard to manipulate

In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. The complexity of manipulating realistic elections where the number of candidates is a small constant was recently studied [4], but the emphasis was on the question of whether or not a protocol becomes hard to manipulate for some constant number of candidates. That work, in many cases, left open the question: How many candidates are needed to make elections hard to manipulate? This is a crucial question when comparing the relative manipulability of different voting protocols. In this paper we answer that question for the voting protocols of the earlier study: plurality, Borda, STV, Copeland, maximin, regular cup, and randomized cup. We also answer that question for two voting protocols for which no results on the complexity of manipulation have been derived before: veto and plurality with runoff. It turns out that the voting protocols under study become hard to manipulate at 3 candidates, 4 candidates, 7 candidates, or never.

Towards mathematical morpho-logics

In this paper, we suggest a new way to process information represented in a logical framework based on mathematical morphology. We show how the basic morphological operations can be expressed in a logical setting. We give some properties, show some links with revision and fusion, and ideas illustrate possible use of morpho-logics to approximation, reasoning and decision.