Jiří Vyskočil

MaLARea SG1 - Machine Learner for Automated Reasoning with Semantic Guidance

This paper describes a system combining model-based and learning-based methods for automated reasoning in large theories, i.e. on a large number of problems that use many axioms, lemmas, theorems, definitions, and symbols, in a consistent fashion. The implementation is based on the existing MaLARea system, which cycles between theorem proving attempts and learning axiom relevance from successes. This system is extended by taking into account semantic relevance of axioms, in a way similar to that of the SRASS system. The resulting combined system significantly outperforms both MaLARea and SRASS on the MPTP Challenge large theory benchmark, in terms of both the number of problems solved and the time taken to find solutions. The design, implementation, and experimental testing of the system are described here.

MaLeCoP: machine learning connection prover

Probabilistic guidance based on learned knowledge is added to the connection tableau calculus and implemented on top of the lean-CoP theorem prover, linking it to an external advisor system. In the typical mathematical setting of solving many problems in a large complex theory, learning from successful solutions is then used for guiding theorem proving attempts in the spirit of the MaLARea system. While in MaLARea learning-based axiom selection is done outside unmodified theorem provers, in MaLeCoP the learning-based selection is done inside the prover, and the interaction between learning of knowledge and its application can be much finer. This brings interesting possibilities for further construction and training of self-learning AI mathematical experts on large mathematical libraries, some of which are discussed. The initial implementation is evaluated on the MPTP Challenge large theory benchmark.

Automated proof compression by invention of new definitions

State-of-the-art automated theorem provers (ATPs) are today able to solve relatively complicated mathematical problems. But as ATPs become stronger and more used by mathematicians, the length and human unreadability of the automatically found proofs become a serious problem for the ATP users. One remedy is automated proof compression by invention of new definitions. We propose a new algorithm for automated compression of arbitrary sets of terms (like mathematical proofs) by invention of new definitions, using a heuristics based on substitution trees. The algorithm has been implemented and tested on a number of automatically found proofs. The results of the tests are included.

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.

System Description: E.T. 0.1

E.T. 0.1 is a meta-system specialized for theorem proving over large first-order theories containing thousands of axioms. Its design is motivated by the recent theorem proving experiments over the Mizar, Flyspeck and Isabelle data-sets. Unlike other approaches, E.T. does not learn from related proofs, but assumes a situation where previous proofs are not available or hard to get. Instead, E.T. uses several layers of complementary methods and tools with different speed and precision that ultimately select small sets of the most promising axioms for a given conjecture. Such filtered problems are then passed to E, running a large number of suitable automatically invented theorem-proving strategies. On the large-theory Mizar problems, E.T. considerably outperforms E, Vampire, and any other prover that does not learn from related proofs. As a general ATP, E.T. improved over the performance of unmodified E in the combined FOF division of CASC 2014 by 6 %.

Learning to Parse on Aligned Corpora (Rough Diamond)

One of the first big hurdles that mathematicians encounter when considering writing formal proofs is the necessity to get acquainted with the formal terminology and the parsing mechanisms used in the large ITP libraries. This includes the large number of formal symbols, the grammar of the formal languages and the advanced mechanisms instrumenting the proof assistants to correctly understand the formal expressions in the presence of ubiquitous overloading.

Automating Formalization by Statistical and Semantic Parsing of Mathematics.

We discuss the progress in our project which aims to automate formalization by combining natural language processing with deep semantic understanding of mathematical expressions. We introduce the overall motivation and ideas behind this project, and then propose a context-based parsing approach that combines efficient statistical learning of deep parse trees with their semantic pruning by type checking and large-theory automated theorem proving. We show that our learning method allows efficient use of large amount of contextual information, which in turn significantly boosts the precision of the statistical parsing and also makes it more efficient. This leads to a large improvement of our first results in parsing theorems from the Flyspeck corpus.

Encoding of Planning Problems and Their Optimizations in Linear Logic

Girards Linear Logic is a formalism which can be used to manage a lot of problems with consumable resources. Its expressiveness is quite good for an easily understandable encoding of many problems. We concentrated on expressing planning problems by linear logic in this paper. We observed a rich usage of a construct of consumable resources in planning problem formulations. This fact motivates us to provide a possible encoding of planning problems in linear logic. This paper shows how planning problems can be encoded in Linear Logic and how some optimizations of planning problems can be encoded. These optimizations can help planners to improve the efficiency of finding solutions (plans).

Bunch modulation in LWFA blowout regime

Laser wakefield acceleration (LWFA) is able to produce high quality electron bunches interesting for many applications ranging from coherent light sources to high energy physics. The blow-out regime of LWFA provides excellent accelerating structure able to maintain small transverse emittance and energy spread of the accelerating electron beam if combined with localised injection. A modulation of the back of a self-injected electron bunch in the blowout regime of Laser Wakefield Acceleration appears 3D Particle-in-Cell simulations with the code OSIRIS. The shape of the modulation is connected to the polarization of the driving laser pulse, although the wavelength of the modulation is longer than that of the pulse. Nevertheless a circularly polarized laser pulse leads to a corkscrew-like modulation, while in the case of linear polarization, the modulation lies in the polarization plane.