41,480 research outputs found
Dependencies in Formal Mathematics: Applications and Extraction for Coq and Mizar
Two methods for extracting detailed formal dependencies from the Coq and
Mizar system are presented and compared. The methods are used for dependency
extraction from two large mathematical repositories: the Coq Repository at
Nijmegen and the Mizar Mathematical Library. Several applications of the
detailed dependency analysis are described and proposed. Motivated by the
different applications, we discuss the various kinds of dependencies that we
are interested in,and the suitability of various dependency extraction methods
Large Formal Wikis: Issues and Solutions
We present several steps towards large formal mathematical wikis. The Coq
proof assistant together with the CoRN repository are added to the pool of
systems handled by the general wiki system described in
\cite{DBLP:conf/aisc/UrbanARG10}. A smart re-verification scheme for the large
formal libraries in the wiki is suggested for Mizar/MML and Coq/CoRN, based on
recently developed precise tracking of mathematical dependencies. We propose to
use features of state-of-the-art filesystems to allow real-time cloning and
sandboxing of the entire libraries, allowing also to extend the wiki to a true
multi-user collaborative area. A number of related issues are discussed.Comment: To appear in The Conference of Intelligent Computer Mathematics: CICM
201
Premise Selection for Mathematics by Corpus Analysis and Kernel Methods
Smart premise selection is essential when using automated reasoning as a tool
for large-theory formal proof development. A good method for premise selection
in complex mathematical libraries is the application of machine learning to
large corpora of proofs. This work develops learning-based premise selection in
two ways. First, a newly available minimal dependency analysis of existing
high-level formal mathematical proofs is used to build a large knowledge base
of proof dependencies, providing precise data for ATP-based re-verification and
for training premise selection algorithms. Second, a new machine learning
algorithm for premise selection based on kernel methods is proposed and
implemented. To evaluate the impact of both techniques, a benchmark consisting
of 2078 large-theory mathematical problems is constructed,extending the older
MPTP Challenge benchmark. The combined effect of the techniques results in a
50% improvement on the benchmark over the Vampire/SInE state-of-the-art system
for automated reasoning in large theories.Comment: 26 page
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
Proof in Context -- Web Editing with Rich, Modeless Contextual Feedback
The Agora system is a prototypical Wiki for formal mathematics: a web-based
system for collaborating on formal mathematics, intended to support informal
documentation of formal developments. This system requires a reusable proof
editor component, both for collaborative editing of documents, and for
embedding in the resulting documents. This paper describes the design of
Agora's asynchronous editor, that is generic enough to support different tools
working on editor content and providing contextual information, with
interactive theorem proverss being a special, but important, case described in
detail for the Coq theorem prover.Comment: In Proceedings UITP 2012, arXiv:1307.152
HOL(y)Hammer: Online ATP Service for HOL Light
HOL(y)Hammer is an online AI/ATP service for formal (computer-understandable)
mathematics encoded in the HOL Light system. The service allows its users to
upload and automatically process an arbitrary formal development (project)
based on HOL Light, and to attack arbitrary conjectures that use the concepts
defined in some of the uploaded projects. For that, the service uses several
automated reasoning systems combined with several premise selection methods
trained on all the project proofs. The projects that are readily available on
the server for such query answering include the recent versions of the
Flyspeck, Multivariate Analysis and Complex Analysis libraries. The service
runs on a 48-CPU server, currently employing in parallel for each task 7 AI/ATP
combinations and 4 decision procedures that contribute to its overall
performance. The system is also available for local installation by interested
users, who can customize it for their own proof development. An Emacs interface
allowing parallel asynchronous queries to the service is also provided. The
overall structure of the service is outlined, problems that arise and their
solutions are discussed, and an initial account of using the system is given
Learning-assisted Theorem Proving with Millions of Lemmas
Large formal mathematical libraries consist of millions of atomic inference
steps that give rise to a corresponding number of proved statements (lemmas).
Analogously to the informal mathematical practice, only a tiny fraction of such
statements is named and re-used in later proofs by formal mathematicians. In
this work, we suggest and implement criteria defining the estimated usefulness
of the HOL Light lemmas for proving further theorems. We use these criteria to
mine the large inference graph of the lemmas in the HOL Light and Flyspeck
libraries, adding up to millions of the best lemmas to the pool of statements
that can be re-used in later proofs. We show that in combination with
learning-based relevance filtering, such methods significantly strengthen
automated theorem proving of new conjectures over large formal mathematical
libraries such as Flyspeck.Comment: journal version of arXiv:1310.2797 (which was submitted to LPAR
conference
Prospects and Challenges in R Package Development
R, a software package for statistical computing and graphics, has evolved into the lingua franca of (computational) statistics. One of the cornerstones of R's success is the decentralized and modularized way of creating software using a multi-tiered development model: The R Development Core Team provides the "base system", which delivers basic statistical functionality, and many other developers contribute code in the form of extensions in a standardized format via so-called packages. In order to be accessible by a broader audience, packages are made available via standardized source code repositories. To support such a loosely coupled development model, repositories should be able to verify that the provided packages meet certain formal quality criteria and "work": both relative to the development of the base R system as well as with other packages (interoperability). However, established quality assurance systems and collaborative infrastructures typically face several challenges, some of which we will discuss in this paper.Series: Research Report Series / Department of Statistics and Mathematic
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