Strict Partial Order Research Articles

On continuity of incomplete preferences

A weak (strict) preference relation is continuous if it has a closed (open) graph; it is hemicontinuous if its upper and lower contour sets are closed (open). If preferences are complete these four conditions are equivalent. Without completeness continuity in each case is stronger than hemicontinuity. This paper provides general characterizations of continuity in terms of hemicontinuity for weak preferences that are modeled as (possibly incomplete) preorders and for strict preferences that are modeled as strict partial orders. Some behavioral implications associated with the two approaches are also discussed.

Modelling of a’fortiori reasoning

The paper presents the model of two variants of a’fortiori reasoning applicable in the case of statutory law as well the example of the genuine law case, which has been modeled with use of established methodology. The model of reasoning assumes the existence of “less–more” relation between the analyzed actions, which has been expressed by means of strict partial order and some additional assumptions. The paper also contains the implementation of the analyzed example.

A strict partial order on payoff configurations and its some properties

This paper proposes a method for comparison of payoff configurations in the framework of characteristic function form games with non-transferable utility. The proposed method in this paper is a relation derived from objections and counter-objections. Some examples which show how the proposed method works are given. This paper presents propositions which show that the proposed method satisfies the properties, called strict partial order and independence from linear transformation and parallel shift. An example verifies that the proposed method does not satisfy the property called monotonicity.

Consistent union and prioritized consistent union: new operations for preference aggregation

We discuss some new approaches to preference aggregation, keeping the natural property of transitivity of strict preferences in mind. In a previous paper, we discussed various ways in which to construct and process strict partial order relations in the context of ranking objects on the basis of multiple criteria. We now broaden the scope to include more general expressions of preferences as inputs and introduce the concept of a NIP-triple, composed of a relation of necessary couples, a relation of impossible couples and a relation of possible couples. The use of NIP-triples allows for a more straightforward characterization of the consistent and prioritized consistent union as well as a smooth formulation of algorithmic implementations. We also introduce a NIP-triple closing operation, which can be combined with the consistent union operations for increased flexibility. Some properties of the proposed operations are examined. The consistent union operation is commutative, as is its composition with the closing operation. Both the consistent and prioritized consistent union are associative, but not when they are composed with the closing operation. Nevertheless, the composed operations surely have their use, which is also discussed.

Contracting preference relations for database applications

The binary relation framework has been shown to be applicable to many real-life preference handling scenarios. Here we study preference contraction: the problem of discarding selected preferences. We argue that the property of minimality and the preservation of strict partial orders are crucial for contractions. Contractions can be further constrained by specifying which preferences should be protected. We consider preference relations that are finite or finitely representable using preference formulas. We present algorithms for computing minimal and preference-protecting minimal contractions for finite as well as finitely representable preference relations. We study relationships between preference change in the binary relation framework and belief change in the belief revision theory. We evaluate the proposed algorithms experimentally and present the results.

Convex cone-based partial order for multiple criteria alternatives

In this paper, we consider the problem of finding a preference-based strict partial order for a finite set of multiple criteria alternatives. We develop an approach based on information provided by the decision maker in the form of pairwise comparisons. We assume that the decision maker's value function is not explicitly known, but it has a quasi-concave form. Based on this assumption, we construct convex cones providing additional preference information to partially order the set of alternatives. We also extend the information obtained from the quasi-concavity of the value function to derive heuristic information that enriches the strict partial order. This approach can as such be used to partially rank multiple criteria alternatives and as a supplementary method to incorporate preference information in, e.g. Data Envelopment Analysis and Evolutionary Multi-Objective Optimization.

The r-Dominance: A New Dominance Relation for Interactive Evolutionary Multicriteria Decision Making

Evolutionary multiobjective optimization (EMO) methodologies have gained popularity in finding a representative set of Pareto optimal solutions in the past decade and beyond. Several techniques have been proposed in the specialized literature to ensure good convergence and diversity of the obtained solutions. However, in real world applications, the decision maker is not interested in the overall Pareto optimal front since the final decision is a unique solution. Recently, there has been an increased emphasis in addressing the decision-making task in searching for the most preferred alternatives. In this paper, we introduce a new variant of the Pareto dominance relation, called r-dominance, which has the ability to create a strict partial order among Pareto-equivalent solutions. This fact makes such a relation able to guide the search toward the interesting parts of the Pareto optimal region based on the decision maker's preferences expressed as a set of aspiration levels. After integrating the new dominance relation in the NSGA-II methodology, the efficacy and the usefulness of the modified procedure are assessed through two to ten-objective test problems a priori and interactively. Moreover, the proposed approach provides competitive and better results when compared to other recently proposed preference-based EMO approaches.

Symbolic Implementation of Connectors in BIP

BIP is a component framework for constructing systems by superposing three layers of modeling: Behavior, Interaction, and Priority. Behavior is represented by labeled transition systems communicating through ports. Interactions are sets of ports. A synchronization between components is possible through the interactions specified by a set of connectors. When several interactions are possible, priorities allow to restrict the non-determinism by choosing an interaction, which is maximal according to some given strict partial order. The BIP component framework has been implemented in a language and a tool-set. The execution of a BIP program is driven by a dedicated engine, which has access to the set of connectors and priority model of the program. A key performance issue is the computation of the set of possible interactions of the BIP program from a given state. Currently, the choice of the interaction to be executed involves a costly exploration of enumerative representations for connectors. This leads to a considerable overhead in execution times. In this paper, we propose a symbolic implementation of the execution model of BIP, which drastically reduces this overhead. The symbolic implementation is based on computing boolean representation for components, connectors, and priorities with an existing BDD package.

An Order-Theoretical Extension of the Guttman Scale to Less Simple Orders

A perfect Guttman scale is rarely found in real data. Pairwise dominance relations between items to be scaled, however, often meet the conditions for less simple orders, such as strict partial orders, interval orders, and semiorders. Examples are thus provided for an extension of the Guttman scale to less simple orders in the framework of ordinal theory, or more specifically, the theory of representations with thresholds. The study is methodologically based on ordering theory. Three illustrative constructions of less simple orders demonstrate that they much more strongly account for real data than do Guttman scales, and that some uniqueness in scale values and thresholds is found in semiorders and interval orders.

Stochastic applications of media theory: Random walks on weak orders or partial orders

This paper presents the axioms of a real time random walk on the set of states of a medium and some of their consequences, such as the asymptotic probabilities of the states. The states of the random walk coincide with those of the medium, and the transitions of the random walk are governed by a probability distribution on the set of token-events, together with a Poisson process regulating the arrivals of such events. We examine two special cases. The first is the medium on strict weak orders on a set of three elements, the second the medium of strict partial orders on the same set. Thus, in each of these cases, a state of the medium is a binary relation. We also consider tune in-and-out extensions of these two special cases. We review applications of these models to opinion poll data pertaining to the 1992 United States presidential election. Each strict weak order or strict partial order is interpreted as being the implicit or explicit opinion of some individual regarding the three major candidates in that election, namely, Bush, Clinton and Perot. In particular, the strict partial order applications illustrate our notion of a response function that provides the link between theory and data in situations where, in contrast to previous papers, the permissible responses do not span the entire set of permissible states of the medium.

Database querying under changing preferences

We present here a formal foundation for an iterative and incremental approach to constructing and evaluating preference queries. Our main focus is query modification: a query transformation approach which works by revising the preference relation in the query. We provide a detailed analysis of the cases where the order-theoretic properties of the preference relation are preserved in the revision. We consider a number of different revision operators: union, prioritized and Pareto composition. We also formulate algebraic laws that enable incremental evaluation of preference queries. Finally, we consider two variations of the basic framework: finite restrictions of preference relations and weak-order extensions of strict partial order preference relations.

A characterization theorem for injective model classes axiomatized by general rules

We continue the work in Zhu et al. [Normal conditions for inference relations and injective models, Theoret. Comput. Sci. 309 (2003) 287–311]. A class Ω of strict partial order structures ( posets, for short) is said to be axiomatizable if the class of all injective preferential models from Ω may be characterized in terms of general rules. This paper aims to obtain some characteristics of axiomatizable classes. To do this, a monadic second-order frame language is presented. The relationship between ℵ 0 -axiomatizability and second-order definability is explored. Then a notion of an admissible set is introduced. Based on this notion, we show that any preferential model, which does not contain any four-node substructure, must be a reduct of some injective model. Furthermore, we furnish a necessary and sufficient condition for the axiomatizability of classes of injective preferential models using general rules. Finally, we show that, in some sense, the class of all posets without any four-node substructure is the largest among axiomatizable classes.

Precedence–inclusion patterns and relational learning

In this paper we introduce precedence–inclusion patterns, which are sets with a strictly partially ordered set of strict partial orders, along with some additional structure. The definition of these structures reflects how multiple partial orders interact in a number of situations such as in text, images, and video. In particular, precedence–inclusion patterns generalize constituent structure trees familiar to computational linguists. Our interest in these objects was initially sparked by their connection with problems of relational learning. We develop the general mathematical theory of precedence–inclusion patterns, we show that each finite set of finite precedence–inclusion relations has a minimal most specific generalization that is unique up to isomorphism, and we explain how this result relates to relational learning.

The k-Track Assignment Problem on Partial Orders

The k-track assignment problem is a scheduling problem in which a collection of jobs, represented by intervals, are to be processed by k machines. Two different jobs can be processed by the same machine only if the corresponding intervals do not overlap. We give a compact formulation of the problem and state some polyhedral results for the associated polytope, working in the more general context where job compatibility stems not necessarily from intervals but rather from an arbitrary strict partial order.

User preference mining techniques for personalized applications

Advanced personalized e-applications require comprehensive knowledge about their users’ likes and dislikes in order to provide individual product recommendations, personal customer advice, and custom-tailored product offers. In our approach we model such preferences as strict partial orders with “A is better than B” semantics, which has been proven to be very suitable in various e-applications. In this paper we present preference mining techniques for detecting strict partial order preferences in user log data. Real-life e-applications like online shops or financial services usually have large log data sets containing the transactions of their customers. Since the preference miner uses sophisticated SQL operations to execute all data intensive operations on database layer, our algorithms scale well even for such large log data sets. With preference mining personalized e-applications can gain valuable knowledge about their customers’ preferences, which can be applied for personalized product recommendations, individual customer service, or one-to-one marketing.

情報源の明らかな選好関係を信念とした融合

Belief fusion, instead of AGM belief revision, was first proposed to solve the problem of inconsistency, that arised from repetitive application of the operation when agents' knowledge were amalgamated. In the preceding work of Maynard-Reid II and Shoham, the fusion operator is applied to belief states, which is total preorders over possible worlds which is based on the semantics of belief revision. Moreover, they introduced the pedigreed belief state, which represented multiple sources of belief states, ordered by a credibility ranking. However in the theory, all the sources must be totally ordered and thus applicable area is quite restrictive. In this paper, we realize the fusion operator of multiple agents for partially ordered sources. When we consider such a partial ranking over sources, there is no need to restrict that each agent has total preorders over possible worlds. The preferential model, based on the semantics on nonmonotonic reasoning, allows each agent to have strict partial orders over possible worlds. Especially, such an order is called a preferential relation, that prescribes a world is more plausible than the other. Therefore, we introduce an operation which combines multiple preferential relations of agents. In addition, we show that our operation can properly include the ordinary belief fusion.

Compatible extensions of fuzzy relations

In 1930 Szpilrajn proved that any strict partial order can be embedded in a strict linear order. This theorem was later refined by Dushnik and Miller (1941), Hansson (1968), Suzumura (1976), Donaldson and Weymark (1998), Bossert (1999). Particularly Suzumura introduced the important concept of compatible extension of a (crisp) relation. These extension theorems have an important role in welfare economics. In particular Szpilrajn theorem is the main tool for proving a known theorem of Richter that establishes the equivalence between rational and congruous consumers. In 1999 Duggan proved a general extension theorem that contains all these results. In this paper we introduce the notion of compatible extension of a fuzzy relation and we prove an extension theorem for fuzzy relations. Our result generalizes to fuzzy set theory the main part of Duggan’s theorem. As applications we obtain fuzzy versions of the theorems of Szpilrajn, Hansson and Suzumura. We also prove that an asymmetric and transitive fuzzy relation has a compatible extension that is total, asymmetric and transitive. Our results can be useful in the theory of fuzzy consumers. We can prove that any rational fuzzy consumer is congruous, extending to a fuzzy context a part of Richter’s theorem. To prove that a congruous fuzzy consumer is rational remains an open problem. A proof of this result can somehow use a fuzzy version of Szpilrajn theorem.

Fusion of Pedigreed Preferential Relations

Belief fusion, instead of AGM belief revision, was first proposed to solve the problem of inconsistency, that arise from repetitive application of the operation when agents' knowledge were amalgamated. However in the theory, all the sources must be totally ordered and thus applicable area is quite restrictive. In this paper, we realize the belief fusion of multiple agents for partially ordered sources. When we consider such a partial ranking over sources, there is no need to restrict that each agent has total preorders over possible worlds. The preferential model allows each agent to have strict partial orders over possible worlds. Especially, such an order is called a preferential relation, that prescribes a world is more plausible than the other. We introduce an operation which combines multiple preferential relations of agents. In addition, we show that our operation can properly include the ordinary belief fusion.

Media theory

This paper reviews and extends previous results of the first author concerning a particular semigroup of transformations on a finite set of states. The noun ‘medium’ given to this semigroup stems from an important application in which the transformations formalize the effects, on an individual, of ‘tokens’ of information delivered by the environment—i.e. the ‘medium’—thereby modifying his or her opinions. The axioms containing the semigroup actually capture a wide variety of examples ranging from convex analysis to combinatorics. A common characteristic is that any transformation of a state—if it is effective—leaves a trace which is a partially defining feature of the state produced. For instance, the family of all strict partial orders on a finite set, equipped with the set of transformations consisting in adding (or removing) an ordered pair to (or from) a partial order to form another partial order is an instance of a medium. As suggested by this example, while these transformations are never one-to-one functions, each transformation has a unique ‘reverse’ transformation undoing its actions. We introduce the concepts of ‘orientation’ and ‘closure’ for a medium and derive some consequences. A recently published application of media theory to the analysis of opinion polls data is briefly discussed.

A New Approach to Preferential Structures

This paper deals with some fundamental concepts and questions of preferential structures. Traditionally, a model for preferential reasoning is a strict partial order on the set of classical models of the language; in this article it will be a total order on the classical models. Instead of representing non-monotonic inference relations by individual partial orders, we represent them by sets of total orders. We thus stay close to the way completeness proofs are done in classical logic. Our new approach will also justify multiple copies (or labelling functions) present in most work on preferential structures. A representation result for the finite case is proven; for the infinite case it remains an open question.