Extensional Higher-Order Paramodulation in Leo-III

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is compatible with the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized communication between reasoning systems, enabling e.g. proof reconstruction from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic first-order and higher-order logic, in all normal quantified modal logics, as well as in different deontic logics. Its development had initiated the ongoing extension of the TPTP infrastructure to reasoning within non-classical logics.Comment: 34 pages, 7 Figures, 1 Table; submitted articl

Universal (Meta-)Logical Reasoning: Recent Successes

Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal logical reasoning engine. Such an engine may be employed, as already envisioned by Leibniz, to support the rigorous formalisation and deep logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed

GRUNGE: A Grand Unified ATP Challenge

This paper describes a large set of related theorem proving problems obtained by translating theorems from the HOL4 standard library into multiple logical formalisms. The formalisms are in higher-order logic (with and without type variables) and first-order logic (possibly with multiple types, and possibly with type variables). The resultant problem sets allow us to run automated theorem provers that support different logical formats on corresponding problems, and compare their performances. This also results in a new "grand unified" large theory benchmark that emulates the ITP/ATP hammer setting, where systems and metasystems can use multiple ATP formalisms in complementary ways, and jointly learn from the accumulated knowledge.Comment: CADE 27 -- 27th International Conference on Automated Deductio

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

An approximation and refinement approach to first-order automated reasoning

With the goal of lifting model-based guidance from the propositional setting to first order logic, I have developed an approximation theorem proving approach based on counterexample-guided abstraction refinement. A given clause set is transformed into a simplified form where satisfiability is decidable. This approximation extends the signature and preserves unsatisfiability: if the simplified clause set is satisfiable, so is the original clause set. A resolution refutation generated by a decision procedure on the simplified clause set can then either be lifted to a refutation in the original clause set, or it guides a refinement excluding the previously found unliftable refutation. This way the approach is refutationally complete. The monadic shallow linear Horn fragment, which is the initial target of the approximation, is well-known to be decidable. It was a long standing open problem how to extend the fragment to the non-Horn case, preserving decidability, that would, e.g., enable to express non-determinism in protocols. I have now proven decidability of the non-Horn monadic shallow linear fragment via ordered resolution. I further extend the clause language with a new type of constraints, called straight dismatching constraints. The extended clause language is motivated by an improved refinement of the approximation-refinement framework. All needed operations on straight dismatching constraints take linear or linear logarithmic time in the size of the constraint. Ordered resolution with straight dismatching constraints is sound and complete and the monadic shallow linear fragment with straight dismatching constraints is decidable. I have implemented my approach based on the SPASS theorem prover. On certain satisfiable problems, the implementation shows the ability to beat established provers such as SPASS, iProver, and Vampire.Mit dem Ziel die Modell-basierten Methoden der Aussagenlogik auf die Logik erster Stufe anzuwenden habe ich ein approximations Beweis-System entwickelt, das auf der Idee der ’Gegenbeispiel-gelenkten Abstraktions-Verfeinerung’ beruht. Eine gegebene Klausel-Menge wird zunächst in eine vereinfachte Form übersetzt, in der die Erfüllbarkeit entscheidbar ist. Diese sogenannte Approximation erweitert die Signatur, aber erhält Unerfüllbarkeit: Falls die approximierte Klauseln erfüllbar sind, so ist es auch die ursprüngliche Menge. Ein Resolutions-Beweis, der von einer Entscheidungs-Prozedur auf der Approximation erzeugt wurde, kann dann entweder als Basis eines Unerfüllbarkeits Beweises der ursprünglichen Menge dienen oder aber eine Verfeinerung der Approximation aufzeigen, welche den gefundenen Beweis davon ausschließt noch einmal gefunden zu werden. Damit ist der Ansatz widerlegungs vollständig. Das monadisch flache lineare Horn Fragment, das als anfängliches Ziel der Approximation dient, ist bereits seit längerem als entscheidbar bekannt. Es war ein lange offenes Problem, wie man das Fragment auf den nicht-Horn Fall erweitern kann ohne Entscheidbarkeit zu verlieren. Damit lassen sich unter anderem nicht-deterministische Protokolle ausdrücken. Ich habe nun die Entscheidbarkeit des nicht-Horn monadisch flachen linearen Fragments mittels geordneter Resolution bewiesen. Zusätzlich habe ich die Klausel-Sprache durch eine neue Art von Constraints erweitert, die ich als ’straight dismatching constraints’ bezeichne. Diese Erweiterung ist dadurch motiviert dass sie eine Verbesserung der Approximations-Verfeinerung des vorgestellten Systems erlaubt. Alle benötigten Operationen auf diesen Constraints nehmen lediglich lineare oder linear-logarithmische Zeit und Platz in Anspruch. Ich zeige, dass geordnete Resolution mit Constraints korrekt und vollständig ist und dass das monadische flache lineare Fragment mit Constraints entscheidbar ist. Ich habe meinen Ansatz auf dem Theorem-Beweiser SPASS basierend implementiert. Auf bestimmten erfüllbaren Problem schlägt meine Implementierung sogar etablierte Beweiser wie SPASS, iProver und Vampire