Undecidability of the unification and admissibility problems for modal and description logics

We show that the unification problem `is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logic with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ)

Hybrid Unification in the Description Logic EL

Unification in Description Logics (DLs) has been proposed as an inference service that can, for example, be used to detect redundancies in ontologies. For the DL EL, which is used to define several large biomedical ontologies, unification is NP-complete. However, the unification algorithms for EL developed until recently could not deal with ontologies containing general concept inclusions (GCIs). In a series of recent papers we have made some progress towards addressing this problem, but the ontologies the developed unification algorithms can deal with need to satisfy a certain cycle restriction. In the present paper, we follow a different approach. Instead of restricting the input ontologies, we generalize the notion of unifiers to so-called hybrid unifiers. Whereas classical unifiers can be viewed as acyclic TBoxes, hybrid unifiers are cyclic TBoxes, which are interpreted together with the ontology of the input using a hybrid semantics that combines fixpoint and descriptive semantics. We show that hybrid unification in EL is NP-complete and introduce a goal-oriented algorithm for computing hybrid unifiers

The generation of concurrent code for declarative languages

PhD ThesisThis thesis presents an approach to the implementation of declarative languages on a simple, general purpose concurrent architecture. The safe exploitation of the available concurrency is managed by relatively sophisticated code generation techniques to transform programs into an intermediate concurrent machine code. Compilation techniques are discussed for 1'-HYBRID, a strongly typed applicative language, and for 'c-HYBRID, a concurrent, nondeterministic logic language. An approach is presented for 1'- HYBRID whereby the style of programming influences the concurrency utilised when a program executes. Code transformation techniques are presented which generalise tail-recursion optimisation, allowing many recursive functions to be modelled by perpetual processes. A scheme is also presented to allow parallelism to be increased by the use of local declarations, and constrained by the use of special forms of identity function. In order to preserve determinism in the language, a novel fault handling mechanism is used, whereby exceptions generated at run-time are treated as a special class of values within the language. A description is given of ,C-HYBRID, a dialect of the nondeterministic logic language Concurrent Prolog. The language is embedded within the applicative language 1'-HYBRID, yielding a combined applicative and logic programming language. Various cross-calling techniques are described, including the use of applicative scoping rules to allow local logical assertions. A description is given of a polymorphic typechecking algorithm for logic programs, which allows different instances of clauses to unify objects of different types. The concept of a method is derived to allow unification Information to be passed as an implicit argument to clauses which require it. In addition, the typechecking algorithm permits higher-order objects such as functions to be passed within arguments to clauses. Using Concurrent Prolog's model of concurrency, techniques are described which permit compilation of 'c-HYBRID programs to abstract machine code derived from that used for the applicative language. The use of methods allows polymorphic logic programs to execute without the need for run-time type information in data structures.The Science and Engineering Research Council

Knowledge Representation Concepts for Automated SLA Management

Outsourcing of complex IT infrastructure to IT service providers has increased substantially during the past years. IT service providers must be able to fulfil their service-quality commitments based upon predefined Service Level Agreements (SLAs) with the service customer. They need to manage, execute and maintain thousands of SLAs for different customers and different types of services, which needs new levels of flexibility and automation not available with the current technology. The complexity of contractual logic in SLAs requires new forms of knowledge representation to automatically draw inferences and execute contractual agreements. A logic-based approach provides several advantages including automated rule chaining allowing for compact knowledge representation as well as flexibility to adapt to rapidly changing business requirements. We suggest adequate logical formalisms for representation and enforcement of SLA rules and describe a proof-of-concept implementation. The article describes selected formalisms of the ContractLog KR and their adequacy for automated SLA management and presents results of experiments to demonstrate flexibility and scalability of the approach.Comment: Paschke, A. and Bichler, M.: Knowledge Representation Concepts for Automated SLA Management, Int. Journal of Decision Support Systems (DSS), submitted 19th March 200

Improving PARMA Trailing

Taylor introduced a variable binding scheme for logic variables in his PARMA system, that uses cycles of bindings rather than the linear chains of bindings used in the standard WAM representation. Both the HAL and dProlog languages make use of the PARMA representation in their Herbrand constraint solvers. Unfortunately, PARMA's trailing scheme is considerably more expensive in both time and space consumption. The aim of this paper is to present several techniques that lower the cost. First, we introduce a trailing analysis for HAL using the classic PARMA trailing scheme that detects and eliminates unnecessary trailings. The analysis, whose accuracy comes from HAL's determinism and mode declarations, has been integrated in the HAL compiler and is shown to produce space improvements as well as speed improvements. Second, we explain how to modify the classic PARMA trailing scheme to halve its trailing cost. This technique is illustrated and evaluated both in the context of dProlog and HAL. Finally, we explain the modifications needed by the trailing analysis in order to be combined with our modified PARMA trailing scheme. Empirical evidence shows that the combination is more effective than any of the techniques when used in isolation. To appear in Theory and Practice of Logic Programming.Comment: 36 pages, 7 figures, 8 table

Hybrid Rules with Well-Founded Semantics

A general framework is proposed for integration of rules and external first order theories. It is based on the well-founded semantics of normal logic programs and inspired by ideas of Constraint Logic Programming (CLP) and constructive negation for logic programs. Hybrid rules are normal clauses extended with constraints in the bodies; constraints are certain formulae in the language of the external theory. A hybrid program is a pair of a set of hybrid rules and an external theory. Instances of the framework are obtained by specifying the class of external theories, and the class of constraints. An example instance is integration of (non-disjunctive) Datalog with ontologies formalized as description logics. The paper defines a declarative semantics of hybrid programs and a goal-driven formal operational semantics. The latter can be seen as a generalization of SLS-resolution. It provides a basis for hybrid implementations combining Prolog with constraint solvers. Soundness of the operational semantics is proven. Sufficient conditions for decidability of the declarative semantics, and for completeness of the operational semantics are given

Higher-order Linear Logic Programming of Categorial Deduction

We show how categorial deduction can be implemented in higher-order (linear) logic programming, thereby realising parsing as deduction for the associative and non-associative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variety of its extensions.Comment: 8 pages LaTeX, uses eaclap.sty, to appear EACL9