Although sorts and unary predicates are semantically identical in order-sorted logic, they are classified as different kinds of properties in formal ontology (e.g. sortal and non-sortal). This ontological analysis is an essential notion to deal with properties (or sorts) of objects in knowledge representation and reasoning. In this paper, we propose an extension of an order-sorted logic with the ontological property classification. This logic contains types (rigid sorts), non-rigid sorts and unary predicates to distinguishably express the properties: substantial sorts, non-substantial sorts and non-sortal properties. We define a sorted Horn-clause calculus for such property expressions in a knowledge base. Based on the calculus, we develop a reasoning algorithm for many separated knowledge bases where each knowledge base can extract rigid property information from other knowledge bases (called rigid property derivation).

Abstract. In this paper we elaborate on a specific application in the context of hybrid description logic programs (hybrid DLPs), namely description logic Semantic Web type systems (DL-types) which are used for term typing of LP rules based on a polymorphic, order-sorted, hybrid DL-typed unification as procedural semantics of hybrid DLPs. Using Semantic Web ontologies as type systems facilitates interchange of domainindependent rules over domain boundaries via dynamically typing and mapping of explicitly defined type ontologies.

Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in order-sorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework where equational, membership and existence formulas can be expressed. A complete deduction calculus is provided to incorporate the interaction between all these formulas. The notion of decorated terms is proposed to memorize local sort information, dynamically changed by a rewriting process. A completion procedure for equational presentations with ordered sorts computes a set of rewrite rules with which not only equational theorems of the form (t = t 0 ), but also typing theorems of the for...

Integrating ontologies and rules on the Semantic Web enables software agents to interoperate between them; however, this leads to two problems. First, reasoning services in SWRL (a combination of OWL and RuleML) are not decidable. Second, no studies have focused on distributed reasoning services for integrating ontologies and rules in multiple knowledge bases. In order to address these problems, we consider distributed reasoning services for ontologies and rules with decidable and effective computation. In this paper, we describe multiple order-sorted logic programming that transfers rigid properties from knowledge bases. Our order-sorted logic contains types (rigid sorts), non-rigid sorts, and unary predicates that distinctly express essential sorts, non-essential sorts, and non-sortal properties. We formalize the order-sorted Horn-clause calculus for such expressions in a single knowledge base. This calculus is extended by embedding rigid-property derivation for multiple knowledge bases, each of which can transfer rigidproperty information from other knowledge bases. In order to enable the reasoning to be effective and decidable, we design a query-answering system that combines order-sorted linear resolution and rigid-property resolution as top-down algorithms.

In an algebraic framework, where equational, membership and existence formulas can be expressed, decorated terms and rewriting provide operational semantics and decision procedures for these formulas. We focus in this work on testing sort inheritance, an undecidable property of specifications, needed for unification in this context. A test and three specific processes, based on completion of a set of rewrite rules, are proposed to check sort inheritance. They depend on the kinds of membership formulas (t : A) allowed in the specifications: flat and linear, shallow and general terms t are studied.

An order-sorted logic can be regarded as a generalized first-order predicate logic that includes many and ordered sorts (i.e. a sort-hierarchy). In the fields of knowledge representation and AI, this logic with sort-hierarchy has been used to design a logic-based language appropriate for representing taxonomic knowledge. By incorporating the sort-hierarchy, order-sorted resolution and sorted logic programming have been formalized that provide efficient reasoning mechanisms with structural representation. In this work, Beierle et al. developed an order-sorted logic to couple separated taxonomic knowledge and assertional knowledge. Namely, its language allows us to make use of sorts to denote not only the types of terms but also unary predicates (called sort predicates). In this paper, we propose a sorted logic programming language with sort predicates in order to improve the practicability of the logic proposed by Beierle et al. The linear resolution is obtained by adding inference relative to sort predicates and subsort relations. In the semantics, the terms and formulas that follow the sorted signature extended with sort predicates are interpreted over its corresponding Σ +-structures. Finally, we build the Herbrand models of programs containing sort predicates, and thus prove the soundness and completeness of this logic programming. order-sorted logic, sort predicate, logic programming, knowledge base system 1

We develop an order-sorted higher-order calculus suitable for automatic theorem proving applications by extending the extensional simply typed lambda calculus with a higher-order ordered sort concept and constant overloading. Huet's well-known techniques for unifying simply typed lambda terms are generalized to arrive at a complete transformation-based unification algorithm for this sorted calculus. Consideration of an order-sorted logic with functional base sorts and arbitrary term declarations was originally proposed by the second author in a 1991 paper; we give here a corrected calculus which supports constant rather than arbitrary term declarations, as well as a corrected unification algorithm, and prove in this setting results corresponding to those claimed there.

We investigate the application of automated deduction techniques to retrieve software components based on their formal specifications. The application profile has major impacts on the problem solving process and requires an open system architecture in which different deductive engines work in combination because the proof problems are too difficult for a single monolithic system. We describe our system architecture, a pipeline of filters of increasing deductive strength, and concentrate on the final filter, in which theorem provers are applied. Here, we use the Ilf-system as a control and integration shell to combine different provers. We support two different combination styles, competition and cooperation. Experiments confirm our approach. With moderate timeouts we already achieve an overall recall of approximately 80%.

This paper develops a sound and complete transformation-based algorithm for unification in an extensional order-sorted combinatory logic supporting constant overloading and a higher-order sort concept. Appropriate notions of order-sorted weak equality and extensionality --- reflecting order-sorted fij-equality in the corresponding lambda calculus given by Johann and Kohlhase --- are defined, and the typed combinator-based higher-order unification techniques of Dougherty are modified to accommodate unification with respect to the theory they generate. The algorithm presented here can thus be viewed as a combinatory logic counterpart to that of Johann and Kohlhase, as well as a refinement of that of Dougherty, and provides evidence that combinatory logic is well-suited to serve as a framework for incorporating order-sorted higher-order reasoning into deduction systems aiming to capitalize on both the expressiveness of extensional higher-order logic and the efficiency of order-sorted calc...

Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.4 Special Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Semantics of Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.1 Semantics of Ada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.2 Action Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.7 Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.1 Early Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.2 Recent Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . 55 4.7.3 The Common Framework Initiative. . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Methodology 57 5.1 Development Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.1 Applica...