We introduce lexicographic logic, an extension of propositional logic that can represent a variety of preferences, most notably lexicographic ones. The proposed logic supports a simple new connective whose semantics can be defined in terms of finite lists of truth values. We demonstrate that, despite the well-known theoretical limitations that pose barriers to the quantitative representation of lexicographic preferences, there exists a subset of the rational numbers over which the proposed new connective can be naturally defined. Lexicographic logic can be used to define in a simple way some well-known preferential operators, like "A and if possible B", and "A or failing that B". We argue that the new logic is an effective formalism for ranking query results according to the satisfaction level of user preferences.

Logic Programs with Ordered Disjunction (LPODs) extend classical logic programs with the capability of expressing alternatives with decreasing degrees of preference in the heads of program rules. Despite the fact that the operational meaning of ordered disjunction is clear, there exists an important open issue regarding its semantics. In particular, there does not exist a purely model-theoretic approach for determining the most preferred models of an LPOD. At present, the selection of the most preferred models is performed using a technique that is not based exclusively on the models of the program and in certain cases produces counterintuitive results. We provide a novel, model-theoretic semantics for LPODs, which uses an additional truth value in order to identify the most preferred models of a program. We demonstrate that the proposed approach overcomes the shortcomings of the traditional semantics of LPODs. Moreover, the new approach can be used to define the semantics of a natural class of logic programs that can have both ordered and classical disjunctions in the heads of clauses. This allows programs that can express not only strict levels of preferences but also alternatives that are equally preferred. This work is under consideration for acceptance in TPLP.

We consider the problem of concisely representing and handling preferences in logic programming and relational databases. Our starting point is a well-known proposal [8] which advocates the embedding of first-order preference formulas into relational algebra through a single winnow operator that is parameterized by a database relation and a preference formula. We argue that despite its elegance, the framework of [8] has a number of shortcomings: only intrinsic preference formulas are supported, the preference relations and preference queries are expressed in two different languages, and there is no direct way to define alternative operators beyond winnow. We propose the use of higher-order logic programming as a logical framework that remedies all the above deficiencies. In particular, the proposed framework supports both intrinsic and extrinsic preference formulas, it can represent both preference relations as-well-as queries, and it can be used to define a variety of interesting alternative operators beyond winnow. We demonstrate the feasibility of our approach by presenting an implementation of all the proposed concepts in the higher-order logic programming language Hilog.

We propose the new logic programming language PrefLog, which is based on an infinite-valued logic in order to support operators for expressing preferences. We demonstrate that if the operators used are continuous over the infinite-valued underlying domain, then the resulting logic programming language retains the well-known properties of classical logic programming (and most notably the existence of a least Herbrand model). We argue that one can define simple and natural new continuous operators by using a small set of operators that are easily shown to be continuous. Finally, we demonstrate that despite the fact that the underlying truth domain and the set of possible interpretations of a PrefLog program are infinite, we can define a terminating bottom-up proof procedure for implementing a significant and useful fragment of the language.