Piotr Rudnicki

A fast algorithm for constructing trees from distance matrices

Abstract We present an algorithm which, given a tree-realizable distance matrix, constructs the tree in optimal O ( n 2 ) time. For trees of bounded degree k , the algorithm runs in O ( kn log k n ) time, and for random trees it apparently runs in O ( n ) average time. We show how the algorithm can be used to test tree-realizability of a distance matrix.

On Equivalents of Well-Foundedness

Four statements equivalent to well-foundedness (well-founded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending ω-chains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies well-foundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project. 12pt ENOD – Experience, Not Only DoctrineG. Kreisel

ATP and Presentation Service for Mizar Formalizations

This paper describes the Automated Reasoning for Mizar (

Information Retrieval in MML

\textsf{Miz}\mathbb{AR}

Commutative Algebra in the Mizar System

) 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

A Compendium of Continuous Lattices in MIZAR

\textsf{Miz}\mathbb{AR}

A wiki for Mizar: motivation, considerations, and initial prototype

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.

Licensing the Mizar mathematical library

Mizar, a proof-checking system, is used to build the Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. We describe issues concerning information retrieval, i.e., searching, browsing and presentation of MML contents. A web-based tool providing such functionalities is being implemented by G. Bancerek. We hope that our observations are helpful when solving similar problems for other repositories of formalized mathematics.

On the Integrity of a Repository of Formalized Mathematics

We report on the development of algebra in the Mizar system. This includes the construction of formal multivariate power series and polynomials as well as the definition of ideals up to a proof of the Hilbert basis theorem. We present how the algebraic structures are handled and how we inherited the past developments from the Mizar Mathematical Library (MML). The MML evolves and past contributions are revised and generalized. Our work on formal power series caused a number of such revisions. It seems that revising past developments with an intent to generalize them is a necessity when building a database of formalized mathematics. This poses a question: how much generalization is best?

LANSF: a protocol modelling environment and its implementation

This paper reports on the MIZAR formalization of the theory of continuous lattices as presented in Gierz et al.: A Compendium of Continuous Lattices, 1980. By a MIZAR formalization we mean a formulation of theorems, definitions, and proofs written in the MIZAR language whose correctness is verified by the MIZAR processor. This effort was originally motivated by the question of whether or not the MIZAR system was sufficiently developed for the task of expressing advanced mathematics. The current state of the formalization – 57 MIZAR articles written by 16 authors – indicates that in principle the MIZAR system has successfully met the challenge. To our knowledge it is the most sizable effort aimed at mechanically checking some substantial and relatively recent field of advanced mathematics. However, it does not mean that doing mathematics in MIZAR is as simple as doing mathematics traditionally (if doing mathematics is simple at all). The work of formalizing the material of the Gierz et al. compendium has (i) prompted many improvements of the MIZAR proof checking system, (ii) caused numerous revisions of the the MIZAR data base, and (iii) contributed to the “to do” list of further changes to the MIZAR system.