12 research outputs found
Smart matching
One of the most annoying aspects in the formalization of mathematics is the
need of transforming notions to match a given, existing result. This kind of
transformations, often based on a conspicuous background knowledge in the given
scientific domain (mostly expressed in the form of equalities or isomorphisms),
are usually implicit in the mathematical discourse, and it would be highly
desirable to obtain a similar behavior in interactive provers. The paper
describes the superposition-based implementation of this feature inside the
Matita interactive theorem prover, focusing in particular on the so called
smart application tactic, supporting smart matching between a goal and a given
result.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
Superposition as a logical glue
The typical mathematical language systematically exploits notational and
logical abuses whose resolution requires not just the knowledge of domain
specific notation and conventions, but not trivial skills in the given
mathematical discipline. A large part of this background knowledge is expressed
in form of equalities and isomorphisms, allowing mathematicians to freely move
between different incarnations of the same entity without even mentioning the
transformation. Providing ITP-systems with similar capabilities seems to be a
major way to improve their intelligence, and to ease the communication between
the user and the machine. The present paper discusses our experience of
integration of a superposition calculus within the Matita interactive prover,
providing in particular a very flexible, "smart" application tactic, and a
simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311
Proceedings of the Deduktionstreffen 2019
The annual meeting Deduktionstreffen is the prime activity of the Special Interest Group on Deduction Systems (FG DedSys) of the AI Section of the German Society for Informatics (GI-FBKI). It is a meeting with a familiar, friendly atmosphere, where everyone interested in deduction can report on their work in an informal setting
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
Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL
The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic
Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL
The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic
How To Efficiently Implement An OSHL-Based Automatic Theorem Prover
Ordered Semantic Hyper-linking (OSHL) is a general-purpose instance-based first-order automated theorem proving algorithm. Although OSHL has many useful properties, previous implementations of OSHL were not very efficient. The implementation of such a theorem prover differs from other more traditional programs in that a lot of its subroutines are more mathematical than procedural. The low performance of previous implementations prevents us from evaluating how the proof strategy used in OSHL matches up against other theorem proving strategies. This dissertation addresses this problem on three levels. First, an abstract, generalized version genOSHL is defined which captures the essential features of OSHL and for which the soundness and completeness are proved. This gives genOSHL the flexibility to be tweaked while still preserving soundness and completeness. A type inference algorithm is introduced which allows genOSHL to possibly reduce its search space while still preserving the soundness and completeness. Second, incOSHL, a specialized version of genOSHL, which differs from the original OSHL algorithm, is defined by specializing genOSHL. Its soundness of completeness follows from that of genOSHL. Third, an embedded programming language called STACK EL, which allows managing program states and their dependencies on global mutable data, is designed and implemented. STACK EL allows our prover to generate instances incrementally. We also study the performance of our incremental theorem prover that implements incOSHL.Doctor of Philosoph