Gernot Salzer

The Unification of Infinite Sets of Terms and Its Applications

In various fields using first order terms — like automated reasoning, logic programming or term rewriting — we encounter infinite sequences of structurally similar terms, leading to non-terminating or at least time and space consuming computations. As a remedy we introduce a meta-language consisting of recursive terms (R-terms), which are a finite representation of infinite sets of terms.

A non-ground realization of the stable and well-founded semantics

Abstract The declarative semantics of nonmonotonic logic programming has largely been based on propositional programs. However, the ground instantiation of a logic program may be very large, and likewise, a ground stable model may also be very large. We develop a non-ground semantic theory for non-monotonic logic programming. Its principal advantage is that stable models and well-founded models can be represented as sets of atoms, rather than as sets of ground atoms. A set SI of atoms may be viewed as a compact representation of the Herbrand interpretation consisting of all ground instances of atoms in SI . We develop generalizations of the stable and well-founded semantics based on such non-ground interpretations SI . The key notions for our theory are those of covers and anticovers . A cover as well as its anticover are sets of substitutions — non-ground in general — representing all substitutions obtained by ground instantiating some substitution in the (anti)cover, with the additional requirement that each ground substitution is represented either by the cover or by the anticover, but not by both. We develop methods for computing anticovers for a given cover, show that membership in so-called optimal covers is decidable, and investigate the complexity in the Datalog case.

Optimal Axiomatizations for Multiple-Valued Operators and Quantifiers Based on Semi-lattices

We investigate the problem of finding optimal axiomatizations for operators and distribution quantifiers in finitely-valued first-order logics. We show that the problem can be viewed as the minimization of certain two-valued prepositional formulas. We outline a general procedure leading to optimized quantifier rules for the sequent calculus, for natural deduction and for clause formation. In the case of operators and quantifiers based on semi-lattices, rules with a minimal branching degree can be obtained by instantiating a schema, which can also be used for optimal tableaux with sets-as-signs.

MUltlog 1.0: Towards an Expert System for Many-Valued Logics

MUltlog is a system which takes as input the specification of a finitely-valued first-order logic and produces a sequent calculus, a natural deduction system, and a calculus for transforming a many-valued formula to clauses suitable for many-valued resolution. All generated rules are optimized regarding their branching degree. The output is in the form of a scientific paper, written in LATEX.

Algebraic foundation of a data model for an extensible space-based collaboration protocol

Space-based computing middleware offers a data driven style for the coordination of processes. The interaction requirements between these processes can be complex, and the template matching coordination law of the Linda and JavaSpaces model is not sufficient. Moreover, the usage should not be limited to a single platform. Several authors have proposed coordination extensions, but besides the suggestion to use XML or RDF based query facilities, a formalization of a general and extensible space-based coordination model has not yet been realized. In this paper we present the algebraic data structures and the coordination model based on a navigational query language for the extensible virtual shared memory architecture, and show how they can be adapted to support arbitrary coordination laws by the introduction of user-definable matchmaker and selector functions. The platform independence is achieved through a language independent communication protocol. The formal specification of the data model is the necessary basis for this protocol.

Ordered Paramodulation and Resolution as Decision Procedure

In recent years interesting decidability results for syntactically specified classes of clause sets have been achieved by employing resolution as a decision procedure. We extend this line of research by considering also clauses with equality literals. We use a special version of ordered paramodulation and resolution to decide a class of clause sets that corresponds to an extension of the Ackermann class with equality (i.e., prenex formulas with prefixes of type ∃*∀∃*). By encoding Turing machines we also show that slight modifications of the defining conditions for this class lead to undecidability.

Computing product configurations via UML and integer linear programming

The Unified Modelling Language (UML) can be used to specify complex systems: component types are modelled as classes, interdependencies as associations with multiplicities and labels. This paper describes how to handle constraints on associations and multiplicities declaratively by translating them to inequalities over integers without adding complexity. This method provides well-defined semantics and allows for efficient algorithms for reasoning tasks like detecting inconsistencies. We identify some challenges arising from the use of class diagrams for product configuration, and propose solutions for some of them. The paper concludes with the discussion of an example derived from a real-world configuration problem in the railway domain.

Consistency and Minimality of UML Class Specifications with Multiplicities and Uniqueness Constraints

The Unified Modeling Language (UML) has become a universal tool for the formal object-oriented specification of hard- and software. In particular, UML class diagrams and so-called multiplicities, which restrict the number of links between objects, are essential when using UML for applications like the specification of admissible configurations of components. In this paper we give a formal definition of the semantics of UML class diagrams and multiplicities. We extend results obtained in the context of entity relationship diagrams to cover UML specific extensions like the (non-)uniqueness attribute of binary associations. We show that the consistency of such specifications can be checked in polynomial time, and give an algorithm for computing minimal configurations (models). The core of our approach is a translation of UML class diagrams to Diophantine inequations.

Proving Properties of Term Rewrite Systems via Logic Programs

We present a general translation of term rewrite systems (TRS) to logic programs such that basic rewriting derivations become logic deductions. Certain TRS result in so-called cs-programs, which were originally studied in the context of constraint systems and tree tuple languages. By applying decidability and computability results of cs-programs we obtain new classes of TRS that have nice properties like decidability of unification, regular sets of descendants or finite representations of R-unifiers. Our findings generalize former results in the field of term rewriting.

Reducing multiplicities in class diagrams

In class diagrams, so-called multiplicities are integer ranges attached to association ends. They constrain the number of instances of the associated class that an instance may be linked to, or in an alternative reading, the number of links to instances of the associated class. In complex diagrams with several chains of associations between two classes (arising e.g. in configuration management) it may happen that the lower or upper bound of a range can never be attained because of restrictions imposed by a parallel chain. In this paper we investigate how multiplicities behave when chaining associations together, and we characterise situations where intervals can be tightened due to information from other chains. Detecting and eliminating such redundancies provides valuable feedback to the user, as redundancies may hint at some underlying misconception.