3,407 research outputs found
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
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
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
A Survey on Retrieval of Mathematical Knowledge
We present a short survey of the literature on indexing and retrieval of
mathematical knowledge, with pointers to 72 papers and tentative taxonomies of
both retrieval problems and recurring techniques.Comment: CICM 2015, 20 page
Mathematical practice, crowdsourcing, and social machines
The highest level of mathematics has traditionally been seen as a solitary
endeavour, to produce a proof for review and acceptance by research peers.
Mathematics is now at a remarkable inflexion point, with new technology
radically extending the power and limits of individuals. Crowdsourcing pulls
together diverse experts to solve problems; symbolic computation tackles huge
routine calculations; and computers check proofs too long and complicated for
humans to comprehend.
Mathematical practice is an emerging interdisciplinary field which draws on
philosophy and social science to understand how mathematics is produced. Online
mathematical activity provides a novel and rich source of data for empirical
investigation of mathematical practice - for example the community question
answering system {\it mathoverflow} contains around 40,000 mathematical
conversations, and {\it polymath} collaborations provide transcripts of the
process of discovering proofs. Our preliminary investigations have demonstrated
the importance of "soft" aspects such as analogy and creativity, alongside
deduction and proof, in the production of mathematics, and have given us new
ways to think about the roles of people and machines in creating new
mathematical knowledge. We discuss further investigation of these resources and
what it might reveal.
Crowdsourced mathematical activity is an example of a "social machine", a new
paradigm, identified by Berners-Lee, for viewing a combination of people and
computers as a single problem-solving entity, and the subject of major
international research endeavours. We outline a future research agenda for
mathematics social machines, a combination of people, computers, and
mathematical archives to create and apply mathematics, with the potential to
change the way people do mathematics, and to transform the reach, pace, and
impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent
Computer Mathematics, CICM 2013, July 2013 Bath, U
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
A Graphical Language for Proof Strategies
Complex automated proof strategies are often difficult to extract, visualise,
modify, and debug. Traditional tactic languages, often based on stack-based
goal propagation, make it easy to write proofs that obscure the flow of goals
between tactics and are fragile to minor changes in input, proof structure or
changes to tactics themselves. Here, we address this by introducing a graphical
language called PSGraph for writing proof strategies. Strategies are
constructed visually by "wiring together" collections of tactics and evaluated
by propagating goal nodes through the diagram via graph rewriting. Tactic nodes
can have many output wires, and use a filtering procedure based on goal-types
(predicates describing the features of a goal) to decide where best to send
newly-generated sub-goals.
In addition to making the flow of goal information explicit, the graphical
language can fulfil the role of many tacticals using visual idioms like
branching, merging, and feedback loops. We argue that this language enables
development of more robust proof strategies and provide several examples, along
with a prototype implementation in Isabelle
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