95 research outputs found
Sharing HOL4 and HOL Light proof knowledge
New proof assistant developments often involve concepts similar to already
formalized ones. When proving their properties, a human can often take
inspiration from the existing formalized proofs available in other provers or
libraries. In this paper we propose and evaluate a number of methods, which
strengthen proof automation by learning from proof libraries of different
provers. Certain conjectures can be proved directly from the dependencies
induced by similar proofs in the other library. Even if exact correspondences
are not found, learning-reasoning systems can make use of the association
between proved theorems and their characteristics to predict the relevant
premises. Such external help can be further combined with internal advice. We
evaluate the proposed knowledge-sharing methods by reproving the HOL Light and
HOL4 standard libraries. The learning-reasoning system HOL(y)Hammer, whose
single best strategy could automatically find proofs for 30% of the HOL Light
problems, can prove 40% with the knowledge from HOL4
ENIGMA: Efficient Learning-based Inference Guiding Machine
ENIGMA is a learning-based method for guiding given clause selection in
saturation-based theorem provers. Clauses from many proof searches are
classified as positive and negative based on their participation in the proofs.
An efficient classification model is trained on this data, using fast
feature-based characterization of the clauses . The learned model is then
tightly linked with the core prover and used as a basis of a new parameterized
evaluation heuristic that provides fast ranking of all generated clauses. The
approach is evaluated on the E prover and the CASC 2016 AIM benchmark, showing
a large increase of E's performance.Comment: Submitted to LPAR 201
Matching concepts across HOL libraries
Many proof assistant libraries contain formalizations of the same
mathematical concepts. The concepts are often introduced (defined) in different
ways, but the properties that they have, and are in turn formalized, are the
same. For the basic concepts, like natural numbers, matching them between
libraries is often straightforward, because of mathematical naming conventions.
However, for more advanced concepts, finding similar formalizations in
different libraries is a non-trivial task even for an expert.
In this paper we investigate automatic discovery of similar concepts across
libraries of proof assistants. We propose an approach for normalizing
properties of concepts in formal libraries and a number of similarity measures.
We evaluate the approach on HOL based proof assistants HOL4, HOL Light and
Isabelle/HOL, discovering 398 pairs of isomorphic constants and types
Conceptual modelling: Towards detecting modelling errors in engineering applications
Rapid advancements of modern technologies put high demands on mathematical modelling of engineering systems. Typically, systems are no longer âsimpleâ objects, but rather coupled systems involving multiphysics phenomena, the modelling of which involves coupling of models that describe different phenomena. After constructing a mathematical model, it is essential to analyse the correctness of the coupled models and to detect modelling errors compromising the final modelling result. Broadly, there are two classes of modelling errors: (a) errors related to abstract modelling, eg, conceptual errors concerning the coherence of a model as a whole and (b) errors related to concrete modelling or instance modelling, eg, questions of approximation quality and implementation. Instance modelling errors, on the one hand, are relatively well understood. Abstract modelling errors, on the other, are not appropriately addressed by modern modelling methodologies. The aim of this paper is to initiate a discussion on abstract approaches and their usability for mathematical modelling of engineering systems with the goal of making it possible to catch conceptual modelling errors early and automatically by computer assistant tools. To that end, we argue that it is necessary to identify and employ suitable mathematical abstractions to capture an accurate conceptual description of the process of modelling engineering systems
A Vernacular for Coherent Logic
We propose a simple, yet expressive proof representation from which proofs
for different proof assistants can easily be generated. The representation uses
only a few inference rules and is based on a frag- ment of first-order logic
called coherent logic. Coherent logic has been recognized by a number of
researchers as a suitable logic for many ev- eryday mathematical developments.
The proposed proof representation is accompanied by a corresponding XML format
and by a suite of XSL transformations for generating formal proofs for
Isabelle/Isar and Coq, as well as proofs expressed in a natural language form
(formatted in LATEX or in HTML). Also, our automated theorem prover for
coherent logic exports proofs in the proposed XML format. All tools are
publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014
Concrete Semantics with Coq and CoqHammer
The "Concrete Semantics" book gives an introduction to imperative programming
languages accompanied by an Isabelle/HOL formalization. In this paper we
discuss a re-formalization of the book using the Coq proof assistant. In order
to achieve a similar brevity of the formal text we extensively use CoqHammer,
as well as Coq Ltac-level automation. We compare the formalization efficiency,
compactness, and the readability of the proof scripts originating from a Coq
re-formalization of two chapters from the book
Premise Selection and External Provers for HOL4
Learning-assisted automated reasoning has recently gained popularity among
the users of Isabelle/HOL, HOL Light, and Mizar. In this paper, we present an
add-on to the HOL4 proof assistant and an adaptation of the HOLyHammer system
that provides machine learning-based premise selection and automated reasoning
also for HOL4. We efficiently record the HOL4 dependencies and extract features
from the theorem statements, which form a basis for premise selection.
HOLyHammer transforms the HOL4 statements in the various TPTP-ATP proof
formats, which are then processed by the ATPs. We discuss the different
evaluation settings: ATPs, accessible lemmas, and premise numbers. We measure
the performance of HOLyHammer on the HOL4 standard library. The results are
combined accordingly and compared with the HOL Light experiments, showing a
comparably high quality of predictions. The system directly benefits HOL4 users
by automatically finding proofs dependencies that can be reconstructed by
Metis
Hammering towards QED
This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong âone-strokeâ tool for discharging difficult lemmas without the need for careful and detailed manual programming of proof search. The main ingredients underlying this approach are efficient automatic theorem provers that can cope with hundreds of axioms, suitable translations of the proof assistantâs logic to the logic of the automatic provers, heuristic and learning methods that select relevant facts from large libraries, and methods that reconstruct the automatically found proofs inside the proof assistants. We outline the history of these methods, explain the main issues and techniques, and show their strength on several large benchmarks. We also discuss the relation of this technology to the QED Manifesto and consider its implications for QED-like efforts.Blanchetteâs Sledgehammer research was supported by the Deutsche Forschungs-
gemeinschaft projects Quis Custodiet (grants NI 491/11-1 and NI 491/11-2) and
Hardening the Hammer (grant NI 491/14-1). Kaliszyk is supported by the Austrian
Science Fund (FWF) grant P26201. Sledgehammer was originally supported by the
UKâs Engineering and Physical Sciences Research Council (grant GR/S57198/01).
Urbanâs work was supported by the Marie-Curie Outgoing International Fellowship
project AUTOKNOMATH (grant MOIF-CT-2005-21875) and by the Netherlands
Organisation for Scientific Research (NWO) project Knowledge-based Automated
Reasoning (grant 612.001.208).This is the final published version. It first appeared at http://jfr.unibo.it/article/view/4593/5730?acceptCookies=1
ProofWatch: Watchlist Guidance for Large Theories in E
Watchlist (also hint list) is a mechanism that allows related proofs to guide
a proof search for a new conjecture. This mechanism has been used with the
Otter and Prover9 theorem provers, both for interactive formalizations and for
human-assisted proving of open conjectures in small theories. In this work we
explore the use of watchlists in large theories coming from first-order
translations of large ITP libraries, aiming at improving hammer-style
automation by smarter internal guidance of the ATP systems. In particular, we
(i) design watchlist-based clause evaluation heuristics inside the E ATP
system, and (ii) develop new proof guiding algorithms that load many previous
proofs inside the ATP and focus the proof search using a dynamically updated
notion of proof matching. The methods are evaluated on a large set of problems
coming from the Mizar library, showing significant improvement of E's standard
portfolio of strategies, and also of the previous best set of strategies
invented for Mizar by evolutionary methods.Comment: 19 pages, 10 tables, submitted to ITP 2018 at FLO
Capturing Hiproofs in HOL Light
Hierarchical proof trees (hiproofs for short) add structure to ordinary proof
trees, by allowing portions of trees to be hierarchically nested. The
additional structure can be used to abstract away from details, or to label
particular portions to explain their purpose. In this paper we present two
complementary methods for capturing hiproofs in HOL Light, along with a tool to
produce web-based visualisations. The first method uses tactic recording, by
modifying tactics to record their arguments and construct a hierarchical tree;
this allows a tactic proof script to be modified. The second method uses proof
recording, which extends the HOL Light kernel to record hierachical proof trees
alongside theorems. This method is less invasive, but requires care to manage
the size of the recorded objects. We have implemented both methods, resulting
in two systems: Tactician and HipCam
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