Preference inference based on lexicographic models

With personalisation becoming more prevalent, it can often be useful to be able to infer additional preferences from input user preferences. Preference inference techniques assume a set of possible user preference models, and derive inferences that hold in all models satisfying the inputs; the more restrictive one makes the set of possible user preference models, the more inferences one gets. Sometimes it can be useful to have an adventurous form of preference inference when the input information is relatively weak, for example, in a conversational recommender system context, to give some justification for showing some options before others. This paper considers an adventurous inference based on assuming that the user preferences are lexicographic, and also an inference based on an even more restrictive preference model. We show how preference inference can be efficiently computed for these cases, based on a relatively general language of preference inputs

Preference inference based on lexicographic and Pareto models

Preferences play a crucial part in decision making. When supporting a user in making a decision, it is important to analyse the user’s preference information to compute good recommendations or solutions. However, often it is impractical or impossible to obtain complete knowledge on preferences. Preference inference aims to exploit given preference information and deduce more preferences. More specifically, the Deduction Problem asks whether another preference statement can be deduced from a given set of preference statements. The closely related Consistency Problem asks whether a given set of user preferences is consistent, i.e., the statements are not contradicting each other. We present approaches for preference inference based on qualitative preference models that are based on lexicographic and Pareto orders. We consider user preference statements that are given in the form of comparisons of alternatives or alternative sets. For some model types and preference statements we formulate efficient algorithms; for others we show NP-completeness and coNP-completeness results. In particular, we find that the Deduction and Consistency problem are polynomial time solvable for comparative preference statements for lexicographic and simple Pareto preference models by a detailed analysis of the problem structures. The computational efficiency for these models makes them particularly appealing for practical uses. The Deduction and Consistency Problem are coNP-complete and NP-complete, respectively, for hierarchical and generalised Pareto models, which make these models less practical even for simple preference languages. However, we still formulate a quite efficient algorithmic approach to solve the Consistency Problem (and implicitly the Deduction) for hierarchical models. We also analyse deduction and consistency for preference statements that are (strongly) compositional under some set of preference models. (Strong) compositionality is a property of preference statements in connection with a set of preference models. It demands inference of preference statements for certain combinations of preference models. We find many interesting results for this case, which ultimately leads to a general greedy algorithm to solve the Consistency Problem for strongly compositional preference statements. This indicates that strong compositionality is an important property that can deliver immediate algorithmic approaches when present. We find many types of preference statements, e.g., conjunctions of strongly compositional statements, are strongly compositional. The considered comparative preferences statements are also strongly compositional for many of the discussed preference models - different lexicographic and hierarchical models. We can make use of the Deduction Problem to find a set of optimal alternatives, e.g., to be recommended to a user that are undominated with respect to different notions of optimality. We analyse this connection for general lexicographic models and find computationally efficient solutions

Efficient inference and computation of optimal alternatives for preference languages based on lexicographic models

We analyse preference inference, through consistency, for general preference languages based on lexicographic models. We identify a property, which we call strong compositionality , that applies for many natural kinds of preference statement, and that allows a greedy algorithm for determining consistency of a set of preference statements. We also consider different natural definitions of optimality, and their relations to each other, for general preference languages based on lexicographic models. Based on our framework, we show that testing consistency, and thus inference, is polynomial for a specific preference language L′ pqT , which allows strict and non-strict statements, comparisons between outcomes and between partial tuples, both ceteris paribus and strong statements, and their combination. Computing different kinds of optimal sets is also shown to be polynomial; this is backed up by our experimental results

Preference inference based on hierarchical and simple lexicographic models

Preference Inference involves inferring additional user preferences from elicited or observed preferences, based on assumptions regarding the form of the user’s preference relation. In this paper we consider a situation in which alternatives have an associated vector of costs, each component corresponding to a different criterion, and are compared using a kind of lexicographic order, similarly to the way alternatives are compared in a Hierarchical Constraint Logic Programming model. It is assumed that the user has some (unknown) importance ordering on criteria, and that to compare two alternatives, firstly, the combined cost of each alternative with respect to the most important criteria are compared; only if these combined costs are equal, are the next most important criteria considered. The preference inference problem then consists of determining whether a preference statement can be inferred from a set of input preferences. We show that this problem is coNP-complete, even if one restricts the cardinality of the equal-importance sets to have at most two elements, and one only considers non- strict preferences. However, it is polynomial if it is assumed that the user’s ordering of criteria is a total ordering (which we call a simple lexicographic model); it is also polynomial if the sets of equally important criteria are all equivalence classes of a given fixed equivalence relation. We give an efficient polynomial algorithm for these cases, which also throws light on the structure of the inference. We give a complete proof theory for the simple lexicographic model case, and analyse variations of preference inference

A reconstruction of the multipreference closure

The paper describes a preferential approach for dealing with exceptions in KLM preferential logics, based on the rational closure. It is well known that the rational closure does not allow an independent handling of the inheritance of different defeasible properties of concepts. Several solutions have been proposed to face this problem and the lexicographic closure is the most notable one. In this work, we consider an alternative closure construction, called the Multi Preference closure (MP-closure), that has been first considered for reasoning with exceptions in DLs. Here, we reconstruct the notion of MP-closure in the propositional case and we show that it is a natural variant of Lehmann's lexicographic closure. Abandoning Maximal Entropy (an alternative route already considered but not explored by Lehmann) leads to a construction which exploits a different lexicographic ordering w.r.t. the lexicographic closure, and determines a preferential consequence relation rather than a rational consequence relation. We show that, building on the MP-closure semantics, rationality can be recovered, at least from the semantic point of view, resulting in a rational consequence relation which is stronger than the rational closure, but incomparable with the lexicographic closure. We also show that the MP-closure is stronger than the Relevant Closure.Comment: 57 page

Probabilistic Default Reasoning with Conditional Constraints

We propose a combination of probabilistic reasoning from conditional constraints with approaches to default reasoning from conditional knowledge bases. In detail, we generalize the notions of Pearl's entailment in system Z, Lehmann's lexicographic entailment, and Geffner's conditional entailment to conditional constraints. We give some examples that show that the new notions of z-, lexicographic, and conditional entailment have similar properties like their classical counterparts. Moreover, we show that the new notions of z-, lexicographic, and conditional entailment are proper generalizations of both their classical counterparts and the classical notion of logical entailment for conditional constraints.Comment: 8 pages; to appear in Proceedings of the Eighth International Workshop on Nonmonotonic Reasoning, Special Session on Uncertainty Frameworks in Nonmonotonic Reasoning, Breckenridge, Colorado, USA, 9-11 April 200

A strengthening of rational closure in DLs: reasoning about multiple aspects

We propose a logical analysis of the concept of typicality, central in human cognition (Rosch,1978). We start from a previously proposed extension of the basic Description Logic ALC (a computationally tractable fragment of First Order Logic, used to represent concept inclusions and ontologies) with a typicality operator T that allows to consistently represent the attribution to classes of individuals of properties with exceptions (as in the classic example (i) typical birds fly, (ii) penguins are birds but (iii) typical penguins don't fly). We then strengthen this extension in order to separately reason about the typicality with respect to different aspects (e.g., flying, having nice feather: in the previous example, penguins may not inherit the property of flying, for which they are exceptional, but can nonetheless inherit other properties, such as having nice feather)