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We recall general metalogical axioms for a reflective logic based on the notion of a universal theory, that is, a theory that can simulate the deductions of all other theories in a class of theories of interest, including itself. We then show that conditional rewriting logic is reflective, generalizing in two stages: first to the unsorted conditional case, and then to the many-sorted conditional case, the already known result for unconditional and unsorted rewriting logic (Reflection in Rewriting Logic: Metalogical Foundations and Metaprogramming Applications. CSLI Publications, 2000). This work should be seen as providing foundations for many useful applications of rewriting logic reflection. The results presented here have greatly influenced the design of the Maude language, which implements rewriting logic and supports its reflective capabilities, and have been used as a theoretical foundation for applications such as internal rewrite strategies, reflective design of theorem proving tools, module algebra and metaprogramming, and metareasoning in metalogical frameworks. [Copyright &y& Elsevier]

Maude is a high-level language and a high-performance system supporting executable specification and declarative programming in rewriting logic. Since rewriting logic contains equational logic, Maude also supports equational specification and programming in its sublanguage of functional modules and theories. The underlying equational logic chosen for Maude is membership equational logic, that has sorts, subsorts, operator overloading, and partiality definable by membership and equality conditions. Rewriting logic is reflective, in the sense of being able to express its own metalevel at the object level. Reflection is systematically exploited in Maude endowing the language with powerful metaprogramming capabilities, including both user-definable module operations and declarative strategies to guide the deduction process. This paper explains and illustrates with examples the main concepts of Maude''s language design, including its underlying logic, functional, system and object-oriented modules, as well as parameterized modules, theories, and views. We also explain how Maude supports reflection, metaprogramming and internal strategies. The paper outlines the principles underlying the Maude system implementation, including its semicompilation techniques. We conclude with some remarks about applications, work on a formal environment for Maude, and a mobile language extension of Maude. [Copyright &y& Elsevier]

ELAN implements computational systems, a concept that combines two first class entities: rewrite rules and rewriting strategies. ELAN can be used either as a logical framework or to describe and execute deterministic as well as non-deterministic rule-based processes. With the general goal to make precise a rewriting logic-based semantics of ELAN, this paper has three contributions: a presentation of the concepts of rules and strategies available in ELAN, an expression of rewrite rules with matching conditions in conditional rewriting logic, and finally an enrichment mechanism of a rewrite theory into a strategy theory in conditional rewriting logic. [Copyright &y& Elsevier]

The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and many-valued logic, lattice theory, game theory, and topology. One source of this richness is the inherent non-monotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on fixpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known fixpoint treatments developed for other programming methodologies. [Copyright &y& Elsevier]

A main task in document transformation and information retrieval is locating subtrees satisfying some pattern. Therefore, unary queries, i.e., queries that map a tree to a set of its nodes, play an important role in the context of structured document databases. The motivation of this work is to understand how the natural and well-studied computation model of tree automata can be used to compute such queries. We define a query automaton (QA) as a deterministic two-way finite automaton over trees that has the ability to select nodes depending on the state and the label at those nodes. We study QAs over ranked as well as over unranked trees. Unranked trees differ from ranked ones in that there is no bound on the number of children of nodes. We characterize the expressiveness of the different formalisms as the unary queries definable in monadic second-order logic (MSO). In contrast to the ranked case, special stay transitions had to be added to QAs over unranked trees to capture MSO. We establish the complexity of the non-emptiness, containment, and equivalence of QAs to be complete for EXPTIME. [Copyright &y& Elsevier]

This paper presents a complete axiomatization of two decidable propositional real-time linear temporal logics: Event Clock Logic (EventClockTL) and Metric Interval Temporal Logic with past (MetricIntervalTL). The completeness proof consists of an effective proof building procedure for EventClockTL. From this result we obtain a complete axiomatization of MetricIntervalTL by providing axioms translating MetricIntervalTL formulae into EventClockTL formulae, the two logics being equally expressive. Our proof is structured to yield axiomatizations also for interesting fragments of these logics, such as the linear temporal logic of the real numbers (TLR). [Copyright &y& Elsevier]