Complexity of manipulation and bribery in premise-based judgment aggregation with simple formulas

Judgment aggregation is a mathematical theory of collective decision-making. It concerns the methods whereby individual opinions about logically interconnected issues of interest can, or cannot, be aggregated into one collective stance. Aggregation problems have traditionally been of interest for disciplines like economics and the political sciences, as well as philosophy, where judgment aggregation itself originates from, but have recently captured the attention of disciplines like computer science, artificial intelligence and multi-agent systems. Judgment aggregation has emerged in the last decade as a unifying paradigm for the formalization and understanding of aggregation problems. Still, no comprehensive presentation of the theory is available to date. This Synthesis Lecture aims at filling this gap presenting the key motivations, results, abstractions and techniques underpinning it.

Preference aggregation and judgment aggregation are two basic research models of group decision making. And preference aggregation has been deeply studied in social choice theory. However, researches of social choice theory gradually focus on judgment aggregation which appears recently. Judgment aggregation focuses on how to aggregate many consistent logical formulas into one, from the perspective of logic. We try to start with judgment aggregation model based on logic and then explore different solutions to problem of judgment aggregation.

In logic programming, a program is stipulated as a set of axioms; execution of such a program involves posing a goal statement (i.e., theorem) and having the system construct a proof. Mechanical reasoning programs are not able to operate with logical sentences expressed in a broad form instead they use a simpler and more restricted form known as Horn Clauses. Note that Horn clauses have the additional restriction of including at most one non‐negative disjunctive term. The origins of logic programming can be found in the work of the French mathematician Jacques Herbrand. He developed a theorem, published in his doctoral dissertation and which now bears his name, that provides a method for mechanically determining whether a set of clauses is satisfiable. Robinson published a paper in 1965 describing the unification algorithm and the resolution principle. Kowalski built upon this work and demonstrated that Horn clause statements can be interpreted as recursive procedures.

There are two different fundamental conceptions of the task of social aggregation: interest aggregation and judgment aggregation. While the former aims at the fair and efficient trade-off of personal interests, the latter aims the best resolution of disagreement. Both are possible interpretations of the standard preference aggregation and preference-based voting problems. This note makes two broad points. First, the judgment aggregation conception is not yet well understood and remains undertheorized. Second, the difference between the two conceptions matters; it is a difference that makes a difference. To substantiate the first point, we argue that judgment aggregation as resolution of disagreement cannot be subsumed under statistical aggregation in the manner of Condorcet’s jury theorem, and sketch a more specific articulation as “reflexive consensus under arguable disagreement”. To substantiate the second point, we illustrate the difference in the context of two important voting rules, the Kemeny rule and Majority Judgment.

Judgment aggregation is a field in which individuals are required to vote for or against a certain decision (the conclusion) while providing reasons for their choice. The reasons and the conclusion are logically connected propositions. The problem is how a collective judgment on logically interconnected propositions can be defined from individual judgments on the same propositions. It turns out that, despite the fact that the individuals are logically consistent, the aggregation of their judgments may lead to an inconsistent group outcome, where the reasons do not support the conclusion. However, in this paper we claim that collective irrationality should not be the only worry of judgment aggregation. For example, judgment aggregation would not reject a consistent combination of reasons and conclusion that no member voted for. In our view this may not be a desirable solution. This motivates our research about when a social outcome is `compatible' with the individuals' judgments. The key notion that we want to capture is that any individual member has to be able to defend the collective decision. This is guaranteed when the group outcome is compatible with its members views. Judgment aggregation problems are usually studied using classical propositional logic. However, for our analysis we use an argumentation approach to judgment aggregation problems. Indeed the question of how individual evaluations can be combined into a collective one can also be addressed in abstract argumentation. We introduce three aggregation operators that satisfy the condition above, and we offer two definitions of compatibility. Not only does our proposal satisfy a good number of standard judgment aggregation postulates, but it also avoids the problem of individual members of a group having to become committed to a group judgment that is in conflict with their own individual positions.

We present a method for specialising the constraints in constrained Horn clauses with respect to a goal. We use abstract interpretation to compute a model of a query-answer transformation of a given set of clauses and a goal. The effect is to propagate the constraints from the goal top-down and propagate answer constraints bottom-up. Our approach does not unfold the clauses at all; we use the constraints from the model to compute a specialised version of each clause in the program. The approach is independent of the abstract domain and the constraints theory underlying the clauses. Experimental results on verification problems show that this is an effective transformation, both in our own verification tools (convex polyhedra analyser) and as a pre-processor to other Horn clause verification tools.

We propose in this paper a slight modification of the Knuth and Bendix algorithm for solving the equality problem in non-equational theories defined by a set of Horn clauses. We prove that the completeness property of the algorithm is then preserved, provided that a weak axiomatization of boolean calculus and equality has been given to the algorithm. In particular, we need only the reflexivity axiom for equality.This algorithm can also be interpreted as the extension to Horn clauses of the resolution/narrowing algorithm proposed by Lankford [LA], which applies only when the equality predicate does not occur positively in non-unit clauses. We give some examples of theorems proved by this method, which show its efficiency in comparison with other paramodulation-based methods.Another application of the Knuth and Bendix algorithm is the proof by induction (Musser [MU], Huet-Hullot [HH], and others). We show that our version of the completion algorithm can be used for proving universally quantified formulas (not only equations) in the initial model defined by a set of Horn clauses.KeywordsNormal FormInference RuleEquational TheoryCritical PairValid ConsequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Logic programming is a language paradigm based on logic, more specifically on resolution theorem proving in the predicate calculus as proposed in Robinson [1965]. Robinson had the foresight to distinguish the importance of two components in automatic theorem proving: a single inference rule called resolution and the testing for equality of trees called unification. Resolution is an inference step used to prove the validity of predicate calculus formulas expressed as clauses. In its simplest version: P ~ Q and -P ~ R imply Q ~ R, which is called the resolvant. Unification is the matching of terms used in a resolution step. It consists of testing the satisfiability of the equality of terms (i.e., labeled trees) whose leaves may contain variables. For example, the unification of the terms p(X, q(Z, a)) and p(b, q(a, Y)) succeeds, yielding the bindings X 5 b, Z 5 a, and Y 5 a. Prolog, the main representative of LP, consists of a sequence of Horn clauses. A Horn clause is one containing (at most) one positive literal. The term definite clause is used to denote a clause with exactly one positive literal. Prolog programs can be viewed as a set of definite clauses in which the positive literal is the head of the rule and the negative literals constitute the body or tail of the rule. From a procedural point of view, a head corresponds to the definition of a Boolean function whose body consists of conjunctions of calls to the Boolean functions representing the tail [Kowalski 1979]. A quintessential example of a Prolog program is that of append. It consists of two Horn clauses specifying that list L3 is the concatenation of two lists, L1 and L2:

International audience

Judgment aggregation is a recent theory that combines aggregation problems previously studied by social choice theory with logic. Social choice theory is a vast subject, including not only the study of preference aggregation and voting theory but also topics like social welfare and justice. Given the tight links between judgment aggregation and preference aggregation, in this first chapter we give a concise survey on some historical aspects of preference aggregation and then introduce and motivate the more recent field of judgment aggregation. The chapter builds on [Sen99, Sen86, Bla58] for the historical overview on social choice theory, and on [KS93, Kor92] for the informal introduction to judgment aggregation.

The power system is constantly changing. New elements are being added. The network structure is being changed. Existing network infrastructure is being modernised. This results in the need to perform various types of computational analyses. These analyses are designed to assess the impact of new and existing power system components on its operation. These analyses can be complicated, complex and difficult. These analyses are influenced by the size of the actual grid, technical conditions, and the specific requirements of grid operators. Therefore, there is a constant search for methods that simplify them without significantly compromising the accuracy of the obtained results. In real networks, computation time is also a crucial parameter, which can be excessive when analysing multiple operating variants. Linearisation of the computational model significantly contributes to reducing computational time but also affects their accuracy. Particular attention should be paid to two frequently used and compared linearization methods: the DC method and the first-order Taylor expansion. The DC (Direct Current) method offers relatively simple formulas. These relationships are based on simplifying assumptions. They are well suited for error analysis, at the expense of neglecting reactive power and voltage changes. Taylor linearization preserves the full AC structure around a selected operating point. It takes reactive power and voltage changes into account. Its accuracy is local and depends on the base point. It may require re-linearization for large changes in the power system operating states. This article presents a detailed literature review. It concerns the linearization of the power flow problem in the context of solving various computational problems in the power system. Selected works are grouped and categorised by topic. Linearisation methods used in the literature are presented, and the most frequently used ones are also indicated. Research gaps that may be addressed in future work are highlighted.

We devise a resolution calculus that tests the satisfiability of infinite families of clause sets, called clause set schemata. For schemata of propositional clause sets, we prove that this calculus is sound, refutationally complete, and terminating. The calculus is extended to first-order clauses, for which termination is lost, since the satisfiability problem is not semi-decidable for nonpropositional schemata. The expressive power of the considered logic is strictly greater than the one considered in our previous work.

We consider constrained Horn clause solving from the more general point of view of solving formula equations. Constrained Horn clauses correspond to the subclass of Horn formula equations. We state and prove a fixed-point theorem for Horn formula equations which is based on expressing the fixed-point computation of a minimal model of a set of Horn clauses on the object level as a formula in first-order logic with a least fixed point operator. We describe several corollaries of this fixed-point theorem, in particular concerning the logical foundations of program verification, and sketch how to generalise it to incorporate abstract interpretations.

Automatic program repair modifies a faulty program to make it correct with respect to a specification. Previous approaches have typically been restricted to specific programming languages and a fixed set of syntactical mutation techniques — e.g., changing the conditions of if statements. We present a more general technique based on repairing sets of unsolvable Horn clauses. Working with Horn clauses enables repairing programs from many different source languages, but also introduces challenges, such as navigating the large space of possible repairs. We propose a conservative semantic repair technique that only removes incorrect behaviors and does not introduce new behaviors. Our proposed framework allows the user to request the best repairs — it constructs an optimization lattice representing the space of possible repairs, and uses a novel local search technique that exploits heuristics to avoid searching through sub-lattices with no feasible repairs. To illustrate the applicability of our approach, we apply it to problems in software-defined networking (SDN), and illustrate how it is able to help network operators fix buggy configurations by properly filtering undesired traffic. We show that interval and Boolean lattices are effective choices of optimization lattices in this domain, and we enable optimization objectives such as modifying the minimal number of switches. We have implemented a prototype repair tool, and present preliminary experimental results on several benchmarks using real topologies and realistic repair scenarios in data centers and congested networks.

The abstraction and over-approximation of protocols and web services by a set of Horn clauses is a very successful method in practice. It has however limitations for protocols and web services that are based on databases of keys, contracts, or even access rights, where revocation is possible, so that the set of true facts does not monotonically grow with state transitions. We extend the scope of these over-approximation methods by defining a new way of abstraction that can handle such databases, and we formally prove that the abstraction is sound. We realize a translator from a convenient specification language to standard Horn clauses and use the verifier ProVerif and the theorem prover SPASS to solve them. We show by a number of examples that this approach is practically feasible for wide variety of verification problems of security protocols and web services