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NDIGOZ016756.Pdf Trends in Contemporary Computer Science Podlasie 2014 Edited by Anna Gomolińska, Adam Grabowski, Małgorzata Hryniewicka, Magdalena Kacprzak, and Ewa Schmeidel Białystok 2014 Editors of the Volume: Anna Gomolińska Adam Grabowski Małgorzata Hryniewicka Magdalena Kacprzak Ewa Schmeidel Editorial Advisory Board: Agnieszka Dardzińska-Głębocka Pauline Kawamoto Norman Megill Hiroyuki Okazaki Christoph Schwarzweller Yasunari Shidama Andrzej Szałas Josef Urban Hiroshi Yamazaki Bożena Woźna-Szcześniak Typesetting: Adam Grabowski Cover design: Anna Poskrobko Supported by the Polish Ministry of Science and Higher Education Distributed under the terms of Creative Commons CC-BY License Bialystok University of Technology Publishing Office Białystok 2014 ISBN 978-83-62582-58-7 Table of Contents Preface .......................................................... 5 I Computer-Assisted Formalization of Mathematics Formal Characterization of Almost Distributive Lattices ............... 11 Adam Grabowski Formalization of Fundamental Theorem of Finite Abelian Groups in Mizar 23 Kazuhisa Nakasho, Hiroyuki Okazaki, Hiroshi Yamazaki, and Yasunari Shidama Budget Imbalance Criteria for Auctions: A Formalized Theorem ........ 35 Marco B. Caminati, Manfred Kerber, and Colin Rowat Feasible Analysis of Algorithms with Mizar .......................... 45 Grzegorz Bancerek II Interactive Theorem Proving Equalities in Mizar ................................................ 59 Artur Korniłowicz Improving Legibility of Proof Scripts Based on Quantity of Introduced Labels ........................................................... 71 Karol Pąk On Logical and Semantic Investigations in Kyiv School of Automated Reasoning ........................................................ 83 Alexander Lyaletski and Mykola Nikitchenko Towards Standard Environments for Formalizing Mathematics .......... 99 Adam Grabowski and Christoph Schwarzweller Parallelizing Mizar ................................................ 109 Josef Urban 4 III Optimization Techniques and Decision-making Processes Optimization Problems and Methods Applicable in Intelligent Tourist Travel Planners ................................................... 127 Jolanta Koszelew Combining Genetic Algorithm and Path Relinking for Solving Orienteering Problem with Time Windows ........................... 135 Joanna Karbowska-Chilińska and Paweł Zabielski Comparison of Different Graph Weights Representations Used to Solve the Time-Dependent Orienteering Problem ........................... 147 Krzysztof Ostrowski An Approach to Making Decisions with Metasets ..................... 159 Magdalena Kacprzak and Bartłomiej Starosta IV Formal Methods and Data Mining On the SAT-based Verification of Communicative Commitments ........ 175 Bożena Woźna-Szcześniak Action Rules Mining in Laryngological Disorders ...................... 187 Agnieszka Dardzińska-Głębocka, Bożena Kosztyła-Hojna, and Anna Łobaczuk-Sitnik Similarity-based Searching in Structured Spaces of Music Information .. 195 Mariusz Rybnik and Władysław Homenda Author Index ................................................ 206 Preface This monograph presents the recent results obtained by the Podlasie com- puter scientists and their colleagues working among the wide world. The book consists of sixteen chapters and is divided into four parts. In Part I, comprising four chapters, some aspects of formalization of math- ematics are presented. In Chapter 1, almost distributive lattices (ADL) being structures defined by Swamy and Rao in 1981 as the generalization of ordinary distributive lattices by removing certain conditions from the well-known axioma- tization of these are considered. As distributive lattices are pretty well present in the Mizar Mathematical Library (MML), also formal characterization of ADL in terms of Mizar attributes and clusters are given. Many of the lemmas and coun- terexamples can be obtained semiautomatically with the help of external provers (e.g. Prover9 and MACE), and also internal Mizar utilities can improve human work in this area. A new formalization of crucial parts of the chosen results obtained by Rao is presented. The characterization of the class of generalized almost distributive lattices is obtained as well. The fundamental theorem of finite abelian groups, which describes the struc- ture of these groups, is formalized in Chapter 2. Since the theorem underlies the study of finite abelian groups, it is expected to become widely applied in formalizing fields such as number theory and cryptology as future Mizar results. In Chapter 3, an original theorem in auction theory is provided. It specifies general conditions under which the sum of the payments of all bidders is granted not to be identical to zero, and more generally – not to be constant. Moreover, it explicitly supplies a construction for a finite minimal set of possible bids on which such sum is not constant. In particular, the theorem presented applies to the important case of the second-price Vickrey auction, where it reduces itself to a basic result of which a novel, alternative proof is given. In order to enhance the confidence in this new theorem, it has been formalized in Isabelle/HOL: the main results and definitions of the formal proof are reproduced here in the common mathematical language, and accompanied with an informal discussion about the underlying ideas. Chapter 4 is devoted to the analysis of algorithms with Mizar. The approach proposed consists in the Mizar formalization of abstract concepts like algebra of instructions, execution function, termination, and their substantiation in a model with integers as the only data type and in models with abstract data types. The proof of correctness of the algorithm Exponentiation by Squaring is described. Part II is devoted to the memory of Andrzej Trybulec, a pioneer of computer– assisted formalization of mathematical problems. Over the last decades we wit- nessed a number of successful instances of such formalization. Research in this field has been boosted by the development of systems for practical formalization of mathematics (proof assistants), creation of large repositories of computer– verified formal mathematics, and integration of interactive and automated meth- 6 ods of theorem proving. Proof assistants provide a very useful teaching tool suit- able for undergraduate instruction, in particular for training beginning students in writing rigorous proofs. Thus, interactive theorem proving is the main subject of this part of the monograph. The part comprises five chapters. The usage of extensionality of sets, possibly satisfying some additional prop- erties, by proof checkers is presented in Chapter 5. In particular, it is shown how extensionality influences proof tactics and equational calculus. Short descrip- tions of Mizar constructions, having an impact on two basic Mizar modules: Reasoner for supporting definitional expansions and Equalizer for computing of the congruence closure of a given set of equalities, are collected. Chapter 6 is devoted to the formal proof checking systems such as Mizar and Isabelle/Isar. These systems can verify the correctness of proof scripts, both easily readable and obscure. However, for humans like those analysing of the main idea of a formal proof or redeveloping of fragments of reasoning to make them stronger, the legibility has a substantial significance. Furthermore, proof writers create still more and more complex deductions that cannot be shortened to several steps by any tools currently available. Therefore, it is important to understand better how we can facilitate the work of script readers modifying the order of independent deduction steps or how we can reorganize the proof structure by extracting lemmas that are obtained automatically. Experimental results are presented, obtained with a method that improves proof legibility and is based on human short-term memory. Chapter 7 is devoted to a brief description of the logical and semantic ap- proaches being developed in the Kyiv school of automated reasoning. This ap- proach can be traced back to 1970, when academician V. Glushkov initiated research on automated theorem proving in mathematics, which is known as the Evidence Algorithm programme (EA) having many common theses with the Polish Mizar project. A significant attention has been paid to the study and the development of logic and semantics as well as a technique for logical inference search. Carried out first at the Institute of Cybernetics of NASU, these investiga- tions moved in 1987 to the Faculty of Cybernetics of the National University of Kyiv, where now they are developed in two directions. The first direction is the traditional one, centered mainly on the construction of proof methods in various first-order logics. The second direction aims at developing of logics oriented on semantic models of programs. It is expected that the results obtained will give us a possibility to extend SAD with more expressive logical languages and more powerful reasoning tools. Though more and more advanced theorems have been formalized in proof systems, their presentation still lacks the elegance of the mathematical writing. The reason is that proof systems have to state much more details – a large number of which is usually omitted by mathematicians. In Chapter 8, the au- thors argue that proof languages should be improved into this direction to make proof systems more attractive and usable – the ultimate goal of course being a like-on-paper presentation. It has been shown that using advanced
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