28 research outputs found
ATP and Presentation Service for Mizar Formalizations
This paper describes the Automated Reasoning for Mizar (MizAR) service, which
integrates several automated reasoning, artificial intelligence, and
presentation tools with Mizar and its authoring environment. The service
provides ATP assistance to Mizar authors in finding and explaining proofs, and
offers generation of Mizar problems as challenges to ATP systems. The service
is based on a sound translation from the Mizar language to that of first-order
ATP systems, and relies on the recent progress in application of ATP systems in
large theories containing tens of thousands of available facts. We present the
main features of MizAR services, followed by an account of initial experiments
in finding proofs with the ATP assistance. Our initial experience indicates
that the tool offers substantial help in exploring the Mizar library and in
preparing new Mizar articles
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
Developing Corpus-based Translation Methods between Informal and Formal Mathematics: Project Description
The goal of this project is to (i) accumulate annotated informal/formal
mathematical corpora suitable for training semi-automated translation between
informal and formal mathematics by statistical machine-translation methods,
(ii) to develop such methods oriented at the formalization task, and in
particular (iii) to combine such methods with learning-assisted automated
reasoning that will serve as a strong semantic component. We describe these
ideas, the initial set of corpora, and some initial experiments done over them
Machine Learning of Coq Proof Guidance: First Experiments
We report the results of the first experiments with learning proof
dependencies from the formalizations done with the Coq system. We explain the
process of obtaining the dependencies from the Coq proofs, the characterization
of formulas that is used for the learning, and the evaluation method. Various
machine learning methods are compared on a dataset of 5021 toplevel Coq proofs
coming from the CoRN repository. The best resulting method covers on average
75% of the needed proof dependencies among the first 100 predictions, which is
a comparable performance of such initial experiments on other large-theory
corpora
Improving legibility of natural deduction proofs is not trivial
In formal proof checking environments such as Mizar it is not merely the
validity of mathematical formulas that is evaluated in the process of adoption
to the body of accepted formalizations, but also the readability of the proofs
that witness validity. As in case of computer programs, such proof scripts may
sometimes be more and sometimes be less readable. To better understand the
notion of readability of formal proofs, and to assess and improve their
readability, we propose in this paper a method of improving proof readability
based on Behaghel's First Law of sentence structure. Our method maximizes the
number of local references to the directly preceding statement in a proof
linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page
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