Uwe Waldmann

Refutational Theorem Proving for Hierarchic First-Order Theories

We extend previous results on theorem proving for first-order clauses with equality to hierarchic first-order theories. Semantically such theories are confined to conservative extensions of the base models. It is shown that superposition together with variable abstraction and constraint refutation is refutationally complete for theories that are sufficiently complete with respect to simple instances. For the proof we introduce a concept of approximation between theorem proving systems, which makes it possible to reduce the problem to the known case of (flat) first-order theories. These results allow the modular combination of a superposition-based theorem prover with an arbitrary refutational prover for the primitive base theory, whose axiomatic representation in some logic may remain hidden. Furthermore they can be used to eliminate existentially quantified predicate symbols from certain second-order formulae.

Termination Proofs of Well-Moded Logic Programs via Conditional Rewrite Systems

In this paper, it is shown that a translation from logic programs to conditional rewrite rules can be used in a straightforward way to check (semi-automatically) whether a program is terminating under the prolog selection rule.

Exact state set representations in the verification of linear hybrid systems with large discrete state space

We propose algorithms significantly extending the limits for maintaining exact representations in the verification of linear hybrid systems with large discrete state spaces. We use AND-Inverter Graphs (AIGs) extended with linear constraints (LinAIGs) as symbolic representation of the hybrid state space, and show how methods for maintaining compactness of AIGs can be lifted to support model-checking of linear hybrid systems with large discrete state spaces. This builds on a novel approach for eliminating sets of redundant constraints in such rich hybrid state representations by a suitable exploitation of the capabilities of SMT solvers, which is of independent value beyond the application context studied in this paper. We used a benchmark derived from an Airbus flap control system (containing 220 discrete states) to demonstrate the relevance of the approach.

Modular proof systems for partial functions with Evans equality

The paper presents a modular superposition calculus for the combination of first-order theories involving both total and partial functions. The modularity of the calculus is a consequence of the fact that all the inferences are pure-only involving clauses over the alphabet of either one, but not both, of the theories-when refuting goals represented by sets of pure formulae. The calculus is shown to be complete provided that functions that are not in the intersection of the component signatures are declared as partial. This result also means that if the unsatisfiability of a goal modulo the combined theory does not depend on the totality of the functions in the extensions, the inconsistency will be effectively found. Moreover, we consider a constraint superposition calculus for the case of hierarchical theories and show that it has a related modularity property. Finally, we identify cases where the partial models can always be made total so that modular superposition is also complete with respect to the standard (total function) semantics of the theories.

Semantics of order-sorted specifications

Abstract Order-sorted specifications (i.e. many-sorted specifications with subsort relations) have been proved to be a useful tool for the description of partially defined functions and error handling in abstract data types. Several definitions for order-sorted algebras have been proposed. In some papers an operator symbol, which may be multiply declared, is interpreted by a family of functions (“overloaded” algebras). In other papers it is always interpreted by a single function (“nonoverloaded” algebras). On the one hand, we try to demonstrate the differences between these two approaches with respect to equality, rewriting and completion; on the other hand, we prove that in fact both theories can be studied in parallel provided that certain notions are suitably defined. The overloaded approach differs from the many-sorted and the nonoverloaded one in that the overloaded term algebra is not necessarily initial. We give a decidable sufficient criterion for the initiality of the term algebra, which is less restrictive than GJM-regularity as proposed by Goguen, Jouannaud and Meseguer. Sort-decreasingness is an important property of rewrite systems since it ensures that confluence and Church-Rosser property are equivalent, that the overloaded and nonoverloaded rewrite relations agree, and that variable overlaps do not yield critical pairs. We prove that it is decidable whether or not a rewrite rule is sort-decreasing, even if the signature is not regular. Finally, we demonstrate that every overloaded completion procedure may also be used in the nonoverloaded world, but not conversely, and that specifications exist that can only be completed using the nonoverloaded semantics.

Superposition with Simplification as a Desision Procedure for the Monadic Class with Equality

We show that superposition, a restricted form of paramodulation, can be combined with specifically designed simplification rules such that it becomes a decision procedure for the monadic class with equality. The completeness of the method follows from a general notion of redundancy for clauses and superposition inferences.

An extension of the Knuth-Bendix ordering with LPO-like properties

The Knuth-Bendix ordering is usually preferred over the lexicographic path ordering in successful implementations of resolution and superposition, but it is incompatible with certain requirements of hierarchic superposition calculi. Moreover, it does not allow non-linear definition equations to be oriented in a natural way. We present an extension of the Knuth-Bendix ordering that makes it possible to overcome these restrictions.

Theorem Proving for Hierarchic First-Order Theories

We extend previous results on theorem proving for first-order clauses with equality to hierarchic first-order theories. Semantically such theories are confined to conservative extensions of the base models. It is shown that superposition together with variable abstraction and constraint refutation is refutationally complete for sufficiently complete theories. For the proof we introduce a concept of approximation between theorem proving systems, which makes it possible to reduce the problem to the known case of (flat) first-order theories. These results allow the modular combination of a superposition-based theorem prover with an arbitrary refutational prover for the primitive base theory, whose axiomatic repesentation in some logic may remain hidden. Furthermore they can be used to eliminate existentially quantified predicate symbols from certain second-order formulae.

Modular Proof Systems for Partial Functions with Weak Equality

The paper presents a modular superposition calculus for the combination of first-order theories involving both total and partial functions. Modularity means that inferences are pure, only involving clauses over the alphabet of either one, but not both, of the theories. The calculus is shown to be complete provided that functions that are not in the intersection of the component signatures are declared as partial. This result also means that if the unsatisfiability of a goal modulo the combined theory does not depend on the totality of the functions in the extensions, the inconsistency will be effectively found. Moreover, we consider a constraint superposition calculus for the case of hierarchical theories and show that it has a related modularity property. Finally we identify cases where the partial models can always be made total so that modular superposition is also complete with respect to the standard (total function) semantics of the theories.

Superposition and Model Evolution Combined

We present a new calculus for first-order theorem proving with equality,