27 research outputs found
More SPASS with Isabelle: superposition with hard sorts and configurable simplification
Sledgehammer for Isabelle/HOL integrates automatic theorem provers to discharge interactive proof obligations. This paper considers a tighter integration of the superposition prover SPASS to increase Sledgehammer’s success rate. The main enhancements are native support for hard sorts (simple types) in SPASS, simplification that honors the orientation of Isabelle simp rules, and a pair of clause-selection strategies targeted at large lemma libraries. The usefulness of this integration is confirmed by an evaluation on a vast benchmark suite and by a
case study featuring a formalization of language-based security
First Experiments with a Flexible Infrastructure for Normative Reasoning
A flexible infrastructure for normative reasoning is outlined. A small-scale
demonstrator version of the envisioned system has been implemented in the proof
assistant Isabelle/HOL by utilising the first authors universal logical
reasoning approach based on shallow semantical embeddings in meta-logic HOL.
The need for such a flexible reasoning infrastructure is motivated and
illustrated with a contrary-to-duty example scenario selected from the General
Data Protection Regulation.Comment: 9 pages, 4 figure
Beagle as a HOL4 external ATP method
International audienceThis paper presents BEAGLE TAC, a HOL4 tactic for using Beagle as an external ATP for discharging HOL4 goals. We implement a translation of the higher-order goals to the TFA format of TPTP and add trace output to Beagle to reconstruct the intermediate steps derived by the ATP in HOL4. Our translation combines the characteristics of existing successful translations from HOL to FOL and SMT-LIB; however, we needed to adapt certain stages of the translation in order to benefit form the expressiveness of the TFA format and the power of Beagle. In our initial experiments, we demonstrate that our system can prove, without any arithmetic lemmas, 81% of the goals solved by Metis
Learning-Assisted Automated Reasoning with Flyspeck
The considerable mathematical knowledge encoded by the Flyspeck project is
combined with external automated theorem provers (ATPs) and machine-learning
premise selection methods trained on the proofs, producing an AI system capable
of answering a wide range of mathematical queries automatically. The
performance of this architecture is evaluated in a bootstrapping scenario
emulating the development of Flyspeck from axioms to the last theorem, each
time using only the previous theorems and proofs. It is shown that 39% of the
14185 theorems could be proved in a push-button mode (without any high-level
advice and user interaction) in 30 seconds of real time on a fourteen-CPU
workstation. The necessary work involves: (i) an implementation of sound
translations of the HOL Light logic to ATP formalisms: untyped first-order,
polymorphic typed first-order, and typed higher-order, (ii) export of the
dependency information from HOL Light and ATP proofs for the machine learners,
and (iii) choice of suitable representations and methods for learning from
previous proofs, and their integration as advisors with HOL Light. This work is
described and discussed here, and an initial analysis of the body of proofs
that were found fully automatically is provided
Seventeen Provers under the Hammer
International audienceOne of the main success stories of automatic theorem provers has been their integration into proof assistants. Such integrations, or "hammers," increase proof automation and hence user productivity. In this paper, we use Isabelle/HOL's Sledgehammer tool to find out how useful modern provers are at proving formulas in higher-order logic. Our evaluation follows in the steps of Böhme and Nipkow's Judgment Day study from 2010, but instead of three provers we use 17, including SMT solvers and higher-order provers. Our work offers an alternative yardstick for comparing modern provers, next to the benchmarks and competitions emerging from the TPTP World and SMT-LIB
A Deontic Logic Reasoning Infrastructure
A flexible infrastructure for the automation of deontic and normative reasoning is presented. Our motivation is the development, study and provision of legal and moral reasoning competencies in future intelligent machines. Since there is no consensus on the “best” deontic logic formalisms and since the answer may be application specific, a flexible infrastructure is proposed in which candidate logic formalisms can be varied, assessed and compared in experimental ethics application studies. Our work thus links the historically rich research areas of classical higher-order logic, deontic logics, normative reasoning and formal ethics
Superposition: Types and Induction
Proof assistants are becoming widespread for formalization of theories both in computer science and mathematics. They provide rich logics with powerful type systems and machine-checked proofs which increase the confidence in the correctness in complicated and detailed proofs.
However, they incur a significant overhead compared to pen-and-paper proofs.
This thesis describes work on bridging the gap between high-order proof assistants and first-order automated theorem provers by extending the capabilities of the automated theorem provers to provide features usually found in proof assistants.
My first contribution is the development and implementation of a first-order superposition calculus with a polymorphic type system that supports type classes and the accompanying refutational completeness proof for that calculus. The inclusion of the type system into the superposition calculus and solvers completely removes the type encoding overhead when encoding problems from many proof assistants.
My second contribution is the development of SupInd, an extension of the typed superposition calculus that supports data types and structural induction over those data types. It includes heuristics that guide the induction and conjecture strengthening techniques, which can be applied independently of the underlying calculus.
I have implemented the contributions in a tool called Pirate. The evaluations of both contributions show promising results.Beweisassistenten werden zunehmend in der Formalisierung von Theorien, sowohl in der Informatik als auch in der Mathematik, genutzt. Ihre vielseitigen Logiken mit ausdrucksstarken Typsystemen ermöglichen Maschinenkontrollierte Beweise. Dies erhöht die Vertrauenswürdigkeit von komplizierten und detaillierten Beweisen. Im Gegensatz zu Papierbeweisen ist ihr Gebrauch jedoch aufwendiger.
Diese Dissertation beschreibt Fortschritte darin, den Abstand zwischen Beweisassistenten höherer Stufe und automatischen Beweissystemen erster Stufe zu schließen, indem automatische Beweissysteme so erweitert werden, dass sie die Möglichkeiten die Beweisassistenten bieten auch bereit stellen.
Der erste Beitrag ist die Erweiterung des Superpositionskalküls erster Stufe um ein polymorphes Typsystem, das Typklassen unterstützt. Die Erweiterung beinhaltet einen Beweis der Widerlegungsvollständigkeit. Das Typsystem als Teil des Superpositionskalkül ermöglicht die Übertragung von Problemen aus Beweisassistenten ohne den sonst üblichen Mehraufwand durch
Typenenkodierung.
Der zweite Beitrag ist die Entwicklung von SupInd, einer Erweiterung von Superposition, die Datentypen und strukturelle Induktion über diese Datentypen ermöglicht. SupInd beinhaltet Heuristiken, die die Induktion lenken und Annahmenverstärkungstechniken, die auch unabhängig des Kalküls benutzt werden können.
Die Beiträge wurden im Tool Pirate umgesetzt, die Evaluationen zeigen vielversprechende Ergebnisse
Automating Free Logic in HOL, with an Experimental Application in Category Theory
A shallow semantical embedding of free logic in classical higher-order logic is presented, which enables the off-the-shelf application of higher-order interactive and automated theorem provers for the formalisation andverification of free logic theories. Subsequently, this approach is applied to aselected domain of mathematics: starting from a generalization of the standardaxioms for a monoid we present a stepwise development of various, mutuallyequivalent foundational axiom systems for category theory. As a side-effect ofthis work some (minor) issues in a prominent category theory textbook havebeen revealed.The purpose of this article is not to claim any novel results in category the-ory, but to demonstrate an elegant way to “implement” and utilize interactiveand automated reasoning in free logic, and to present illustrative experiments