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

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

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

A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving

We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the advantages of the declarative style - the possibility to write formal proofs like normal mathematical text - and the procedural style - strong automation and help with shaping the proofs, including determining the statements of intermediate steps. Our approach is new, and differs significantly from the ways in which the procedural and declarative proof styles have been combined before in the Isabelle, Ssreflect and Matita systems. Our approach is generic and can be implemented on top of any procedural interactive theorem prover, regardless of its architecture and logical foundations. To show the viability of our proposed approach, we fully implemented it as a proof interface called miz3, on top of the HOL Light interactive theorem prover. The declarative language that this interface uses is a slight variant of the language of the Mizar system, and can be used for any interactive theorem prover regardless of its logical foundations. The miz3 interface allows easy access to the full set of tactics and formal libraries of HOL Light, and as such has "industrial strength". Our approach gives a way to automatically convert any procedural proof to a declarative counterpart, where the converted proof is similar in size to the original. As all declarative systems have essentially the same proof language, this gives a straightforward way to port proofs between interactive theorem provers

Querying Proofs (Work in Progress)

We motivate and introduce the basis for a query language designed for inspecting electronic representations of proofs. We argue that there is much to learn from large proofs beyond their validity, and that a dedicated query language can provide a principled way of implementing a family of useful operations

Algorytmy poprawy czytelności formalnych rozumowań zapisanych w systemie naturalnej dedukcji

Przedmiotem badań opisanych w rozprawie doktorskiej są metody poprawy czytelności formalnych rozumowań zapisanych w systemie naturalnej dedukcji. Wykorzystanie komputerowej weryfikacji jest znanym narzędziem ułatwiającym sprawdzanie poprawności formułowanych rozumowań, aczkolwiek jakiekolwiek próby analizowania tak uszczegółowionych rozumowań są wyjątkowo trudne, a zdaniem niektórych niemożliwe. Czytelność takich wywodów jest pojęciem subiektywnym, różnie rozumianym przez poszczególnych autorów rozumowań. Analiza ich potrzeb przyczyniła się jednak do wyodrębnienia grupy kryteriów umożliwiających uczytelnienie formalnych rozumowań, poprzez upodabnienie ich postaci do takiej, która występuje w nieformalnych dowodach matematycznych. W pierwszej części rozprawy został przedstawiony model abstrakcyjnego dowodu matematycznego odzwierciedlający rzeczywistą strukturę dowodów zapisanych w języku Mizar. Model ten umożliwia interpretowanie przepływu informacji w rozumowaniu jako szczególnego rodzaju skierowanych grafów acyklicznych. W oparciu o ten model w drugiej części rozprawy zostały formalnie opracowane pojęcia oraz wyznaczniki poprawy czytelności. Uczytelnianie formalnych rozumowań zostało zbadane pod kątem zastosowania dwóch rodzajów środków stosowanych w praktyce matematycznej. Jako pierwszy z nich, zostały zbadane metody odnajdywania lokalnych podrozumowań, a następnie ich wyizolowywania (wyodrębniania) w postaci lematów lub kapsułkowaniu na głębszych poziomach zagnieżdżenia. Drugim zaś analizowanym środkiem była reorganizacja niezależnych od siebie kroków rozumowań w sposobie ich uporządkowania w dowodzie, mająca na celu poprawę wybranych własności linearyzacji dowodu. W wyniku przeprowadzonych badań w zakresie pierwszego środka została skonstruowana metoda wyizolowywania i kapsułkowania fragmentów rozumowania przy zachowaniu poprawności modyfikowanego skryptu dowodowego oraz zostały zbadane własności fragmentów dowodu, które determinują budowę stwierdzenia opisującego rozumowanie zawarte w tych fragmentach. W zakresie zaś drugiego środka zostało opracowane pięć, najczęściej wskazywanych przez użytkowników bazy Mizar Mathematical Library wskaźników czytelności. Przeprowadzone badania nad złożonością problemu optymalizacji wartości przyjętych wskaźników wykazały, że optymalizacja czterech z nich wiąże się z rozwiązywaniem problemów NP--trudnych. Dodatkowo, zostały stworzone programy umożliwiające automatyczną poprawę czytelności skryptów dowodowych zapisanych w języku Mizar, których działanie opiera się na optymalizacji wartości opracowanych wskaźników przy zadanej przez użytkownika hierarchii ich ważności.In this dissertation the methods to improve legibility of existing formal reasonings written in natural deduction are presented. Computer assisted proof development frameworks can check the correctness of such reasonings, but any attempt to analyze details of the proofs scripts created in this way, according to opinion of some proof writers, is extremely difficult or even impossible. The readability of such arguments is a~subjective quality which is understood by different proof writers in different ways. Still the analysis of their needs led to a~distinguished set of criteria that facilitate making the formal deductions closer to the informal mathematical proofs. First part of the dissertation describes an abstract model of mathematical proofs written in the Mizar language. This model expresses the intuitions connected with the reasonings, where the information flow in proof is regarded as a special kind of digraph. Based on this model notion and parameters associated with legibility criteria are formally defined in the second part of the dissertation. Improvement of readability has been realised by two separate approaches that are used in informal mathematical practice. The first approach is based on the finding fragments of reasoning and consists in isolation (extraction) of these fragments in the form of\break lemmas or encapsulation at the deeper levels of nested proof. The second approach to improvement of the readability consists in the modification of the order of independent steps written in the proof script. The methods that reorganize the order of steps focus mainly on the location of information used to justify a step. As a result of research based on the first approach, methods to extract or encapsulate reasoning fragments from existing deductions were elaborated. Also properties of reasoning fragments that determine the structure of statements which describing the information about reasoning contained in these fragments were described. In the second approach five parameters of legibility that are indicated as most important by the users users of Mizar database has been formally defined. Analysis of the proposed parameters related to improvement of proof readability revealed that four of the considered problems are NP-complete. Additionally, an auxiliary application to improve the readability of articles distributed in MML based on the most popular hierarchy of the considered parameters were created

Hammering towards QED

This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong “one-stroke” tool for discharging difficult lemmas without the need for careful and detailed manual programming of proof search. The main ingredients underlying this approach are efficient automatic theorem provers that can cope with hundreds of axioms, suitable translations of the proof assistant’s logic to the logic of the automatic provers, heuristic and learning methods that select relevant facts from large libraries, and methods that reconstruct the automatically found proofs inside the proof assistants. We outline the history of these methods, explain the main issues and techniques, and show their strength on several large benchmarks. We also discuss the relation of this technology to the QED Manifesto and consider its implications for QED-like efforts.Blanchette’s Sledgehammer research was supported by the Deutsche Forschungs- gemeinschaft projects Quis Custodiet (grants NI 491/11-1 and NI 491/11-2) and Hardening the Hammer (grant NI 491/14-1). Kaliszyk is supported by the Austrian Science Fund (FWF) grant P26201. Sledgehammer was originally supported by the UK’s Engineering and Physical Sciences Research Council (grant GR/S57198/01). Urban’s work was supported by the Marie-Curie Outgoing International Fellowship project AUTOKNOMATH (grant MOIF-CT-2005-21875) and by the Netherlands Organisation for Scientific Research (NWO) project Knowledge-based Automated Reasoning (grant 612.001.208).This is the final published version. It first appeared at http://jfr.unibo.it/article/view/4593/5730?acceptCookies=1