14 research outputs found
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