Harald Ganzinger

Rewrite-based equational theorem proving with selection and simplification

We present various refutationally complete calculi for first-order clauses with equality that allow for arbitrary selection of negative atoms in clauses. Refutation completeness is established via the use of well-founded orderings on clauses for defining a Herbrand model for a consistent set of clauses. We also formulate an abstract notion of redundancy and show that the deletion of redundant clauses during the theorem proving process preserves refutation completeness. It is often possible to compute the closure of nontrivial sets of clauses under application of non-redundant inferences. The refutation of goals for such complete sets of clauses is simpler than for arbitrary sets of clauses, in particular one can restrict attention to proofs that have support from the goals without compromising refutation completeness. Additional syntactic properties allow to restrict the search space even further, as we demonstrate for so-called quasiHorn clauses. The results in this paper contain as special cases or generalize many known results about Knuth-Bendix-like completion procedures (for equations, Horn clauses, and Horn clauses over built-in Booleans), completion of first-order clauses by clausal rewriting, and inductive theorem proving for Horn clauses.

DPLL(T): Fast Decision Procedures

The logic of equality with uninterpreted functions (EUF) and its extensions have been widely applied to processor verification, by means of a large variety of progressively more sophisticated (lazy or eager) translations into propositional SAT. Here we propose a new approach, namely a general DPLL(X) engine, whose parameter X can be instantiated with a specialized solver Solver T for a given theory T, thus producing a system DPLL(T). We describe this DPLL(T) scheme, the interface between DPLL(X) and Solver T , the architecture of DPLL(X), and our solver for EUF, which includes incremental and backtrackable congruence closure algorithms for dealing with the built-in equality and the integer successor and predecessor symbols. Experiments with a first implementation indicate that our technique already outperforms the previous methods on most benchmarks, and scales up very well.

Basic paramodulation

We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the basic strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. In addition we introduce restrictions based on term selection rules and redex orderings, which are general criteria for delimiting the terms which are available for inferences. These refinements are compatible with standard ordering restrictions and are complete without paramodulation into variables or using functional reflexivity axioms. We prove refutational completeness in the context of deletion rules, such as simplification by rewriting (demodulation) and subsumption, and of techniques for eliminating redundant inferences.

A superposition decision procedure for the guarded fragment with equality

We give a new decision procedure for the guarded fragment with equality. The procedure is based on resolution with superposition. We argue that this method will be more useful in practice than methods based on the enumeration of certain finite structures. It is surprising to see that one does not need any sophisticated simplification and redundancy elimination method to make superposition terminate on the class of clauses that is obtained from the clausification of guarded formulas. Yet the decision procedure obtained is optimal with regard to time complexity. We also show that the method can be extended to the loosely guarded fragment with equality.

Attribute coupled grammars

In this paper, attribute grammars are viewed as specifying translations from source language terms into target language terms. The terms are constructed over a hierarchical signature consisting of a semantic and a syntactic part. Attribute grammars are redefined to become morphisms in the category of such signatures, called attribute coupled grammars, such that they come with an associative composition operation. The composition allows for a new kind of modularity in compiler specifications. The paper also discusses properties of the concept with respect to attribute evaluation and application as a tree transformation device.

On restrictions of ordered paramodulation with simplification

We consider a restricted version of ordered paramodulation, called strict superposition. We show that strict superposition (together with equality resolution) is refutationally complete for Horn clauses, but not for general first-order clauses. Two moderate enrichments of the strict superposition calculus are, however, sufficient to establish refutation completeness. This strictly improves previous results. We also propose a simple semantic notion of redundancy for clauses which covers most simplification and elimination techniques used in practice yet preserves completeness of the proposed calculi. The paper introduces a new and comparatively simple technique for completeness proofs based on the use of canonical rewrite systems to represent equality interpretations.

Basic Paramodulation and Superposition

We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the basic strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. These refinements are compatible with standard ordering restrictions and are complete without paramodulation into variables or using functional reflexivity axioms. We prove refutational completeness in the context of deletion rules, such as simplification by rewriting (demodulation) and subsumption, and of techniques for eliminating redundant inferences. Finally, we discuss experimental data obtained from a modification of Otter.

New directions in instantiation-based theorem proving

We consider instantiation-based theorem proving whereby instances of clauses are generated by certain inferences, and where inconsistency is detected by proposition tests. We give a model construction proof of completeness by which restrictive inference systems as well as admissible simplification techniques can be justified. Another contribution of the paper are inference systems that allow one to also employ decision procedures for first-order fragments more complex than propositional logic. The decision provides for an approximate consistency test, and the instance generation inference system is a means of successively refining the approximation.

A completion procedure for conditional equationst

The paper presents a new completion procedure for conditional equations. The work is based on the notion of reductive conditional rewriting and the procedure has been designed to handle in particular non-reductive equations that are generated during completion. The paper also describes techniques for simplification of conditional equations and rules, so that the procedure terminates on more specifications. The correctness proofs which form a substantial part of this paper employ recursive path orderings on the proof trees of conditional equational logic, an extension of the ideas of Bachmair, Dershowitz & Hsiang to the conditional case.

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.