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

Automated Reasoning and Presentation Support for Formalizing Mathematics in Mizar

This paper presents a combination of several automated reasoning and proof presentation tools with the Mizar system for formalization of mathematics. The combination forms an online service called MizAR, similar to the SystemOnTPTP service for first-order automated reasoning. The main differences to SystemOnTPTP are the use of the Mizar language that is oriented towards human mathematicians (rather than the pure first-order logic used in SystemOnTPTP), and setting the service in the context of the large Mizar Mathematical Library of previous theorems,definitions, and proofs (rather than the isolated problems that are solved in SystemOnTPTP). These differences poses new challenges and new opportunities for automated reasoning and for proof presentation tools. This paper describes the overall structure of MizAR, and presents the automated reasoning systems and proof presentation tools that are combined to make MizAR a useful mathematical service.Comment: To appear in 10th International Conference on. Artificial Intelligence and Symbolic Computation AISC 201

mizar-items: Exploring fine-grained dependencies in the Mizar Mathematical Library

The Mizar Mathematical Library (MML) is a rich database of formalized mathematical proofs (see http://mizar.org). Owing to its large size (it contains more than 1100 "articles" summing to nearly 2.5 million lines of text, expressing more than 50000 theorems and 10000 definitions using more than 7000 symbols), the nature of its contents (the MML is slanted toward pure mathematics), and its classical foundations (first-order logic, set theory, natural deduction), the MML is an especially attractive target for research on foundations of mathematics. We have implemented a system, mizar-items, on which a variety of such foundational experiements can be based. The heart of mizar-items is a method for decomposing the contents of the MML into fine-grained "items" (e.g., theorem, definition, notation, etc.) and computing dependency relations among these items. mizar-items also comes equipped with a website for exploring these dependencies and interacting with them.Comment: Accepted at CICM 2011: Conferences in Intelligent Computer Mathematics, Track C: Systems and Project

Discovery of a Faint Companion to Alcor Using MMT/AO 5 μ\mum Imaging

We report the detection of a faint stellar companion to the famous nearby A5V star Alcor (80 UMa). The companion has M-band ($\lambda$ = 4.8 $\mu$m) magnitude 8.8 and projected separation 1".11 (28 AU) from Alcor. The companion is most likely a low-mass ($\sim$0.3 \msun) active star which is responsible for Alcor's X-ray emission detected by ROSAT (L$_{\rm X}$ $\simeq$ 10$^{28.3}$ erg/s). Alcor is a nuclear member of the Ursa Major star cluster (UMa; d $\simeq$ 25 pc, age $\simeq$ 0.5 Gyr), and has been occasionally mentioned as a possible distant (709") companion of the stellar quadruple Mizar ($\zeta$ UMa). Comparing the revised Hipparcos proper motion for Alcor with the mean motion for other UMa nuclear members shows that Alcor has a peculiar velocity of 1.1 km/s, which is comparable to the predicted velocity amplitude induced by the newly-discovered companion ($\sim$1 km/s). Using a precise dynamical parallax for Mizar and the revised Hipparcos parallax for Alcor, we find that Mizar and Alcor are physically separated by 0.36 $\pm$ 0.19 pc (74 $\pm$ 39 kAU; minimum 18 kAU), and their velocity vectors are marginally consistent ($\chi^2$ probability 6%). Given their close proximity and concordant motions we suggest that the Mizar quadruple and the Alcor binary be together considered the 2nd closest stellar sextuplet. The addition of Mizar-Alcor to the census of stellar multiples with six or more components effectively doubles the local density of such systems within the local volume (d $<$ 40 pc).Comment: 28 pages, 2 figures, AJ, in press; emulateapj short version at http://www.pas.rochester.edu/~emamajek/alcor.pd

MizAR 60 for Mizar 50

As a present to Mizar on its 50th anniversary, we develop an AI/TP system that automatically proves about 60% of the Mizar theorems in the hammer setting. We also automatically prove 75% of the Mizar theorems when the automated provers are helped by using only the premises used in the human-written Mizar proofs. We describe the methods and large-scale experiments leading to these results. This includes in particular the E and Vampire provers, their ENIGMA and Deepire learning modifications, a number of learning-based premise selection methods, and the incremental loop that interleaves growing a corpus of millions of ATP proofs with training increasingly strong AI/TP systems on them. We also present a selection of Mizar problems that were proved automatically