Mizar system
Encyclopedia
The Mizar system consists of a language for writing strictly formalized mathematical definitions and proofs
, a computer program
which is able to check proofs written in this language, and a library of definitions and proved theorems which can be referenced and used in new articles. Mizar has goals similar to those of the QED project
proposed by Bob Boyer
around 1993. Mizar is proprietary.
and is maintained at Białystok University, Poland
, the University of Alberta
, Canada
, and Shinshu University
, Japan
.
Mizar articles are written in ordinary ASCII
. The Mizar language is close enough to the mathematical vernacular that mathematicians can read and understand Mizar articles almost immediately; it is formal enough that proofs can be checked automatically. All steps in a proof have to be justified, and it has been estimated that a Mizar article is about four times as long as an equivalent mathematical paper written in ordinary style.
The proof checker uses classical logic, is written in Pascal, and can be downloaded and freely used for non-commercial purposes. It runs on PC platform
s, Windows
, Solaris, FreeBSD
and Linux
, and Mac OS X
/Darwin
. The source code is available only to members of the Association of Mizar Users.
The Mizar distribution includes the Mizar Mathematical Library (MML) consisting of many definitions and theorems which can be referred to in newly written articles. These new articles, after having been reviewed and checked automatically, can be published in the associated Journal of Formalized Mathematics and then become part of the MML.
The MML is built on the axioms of the Tarski-Grothendieck set theory
. As of February 2010, it contained about 9500 definitions and 49,500 theorems. Examples are the Hahn–Banach theorem
, König's lemma
, Brouwer fixed point theorem
, Gödel's completeness theorem
, and facts about the Cantor set
.
Even though semantically all objects MML talks about are sets, the language nevertheless allows one to define and use syntactical types: a variable may for example be declared of type Nat if it stands for a natural number
, or of type Group if it denotes a group
. This makes the notation more convenient and closer to the way mathematicians think of symbols.
Browsable abstracts of MML articles are available as the Journal of Formalized Mathematics and MML Query implements a search engine for MML.
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
, a computer program
Computer program
A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...
which is able to check proofs written in this language, and a library of definitions and proved theorems which can be referenced and used in new articles. Mizar has goals similar to those of the QED project
QED project
The QED manifesto was a proposal for a computer-based database of all mathematical knowledge, strictly formalized and with all proofs having been checked automatically. The idea for the project arose in 1993, mainly under the impetus of Robert Boyer...
proposed by Bob Boyer
Robert S. Boyer
Robert Stephen Boyer, aka Bob Boyer, is a retired professor of computer science, mathematics, and philosophy at The University of Texas at Austin. He and J Strother Moore invented the Boyer–Moore string search algorithm, a particularly efficient string searching algorithm, in 1977. He and Moore...
around 1993. Mizar is proprietary.
History
The system was created beginning in 1973 by Andrzej TrybulecAndrzej Trybulec
Andrzej Trybulec, is a Polish mathematician and computer scientist, at the University of Bialystok, in Białystok, Poland, noted for development of the Mizar system....
and is maintained at Białystok University, Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
, the University of Alberta
University of Alberta
The University of Alberta is a public research university located in Edmonton, Alberta, Canada. Founded in 1908 by Alexander Cameron Rutherford, the first premier of Alberta and Henry Marshall Tory, its first president, it is widely recognized as one of the best universities in Canada...
, Canada
Canada
Canada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...
, and Shinshu University
Shinshu University
is a national university in Nagano Prefecture, Japan. The University has five campuses in Matsumoto, Nishi-nagano , Wakasato , Ueda and Minami-Minowa, and 8 faculties with a total of around 10,000 students....
, Japan
Japan
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...
.
Mizar articles are written in ordinary ASCII
ASCII
The American Standard Code for Information Interchange is a character-encoding scheme based on the ordering of the English alphabet. ASCII codes represent text in computers, communications equipment, and other devices that use text...
. The Mizar language is close enough to the mathematical vernacular that mathematicians can read and understand Mizar articles almost immediately; it is formal enough that proofs can be checked automatically. All steps in a proof have to be justified, and it has been estimated that a Mizar article is about four times as long as an equivalent mathematical paper written in ordinary style.
The proof checker uses classical logic, is written in Pascal, and can be downloaded and freely used for non-commercial purposes. It runs on PC platform
IBM PC compatible
IBM PC compatible computers are those generally similar to the original IBM PC, XT, and AT. Such computers used to be referred to as PC clones, or IBM clones since they almost exactly duplicated all the significant features of the PC architecture, facilitated by various manufacturers' ability to...
s, Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...
, Solaris, FreeBSD
FreeBSD
FreeBSD is a free Unix-like operating system descended from AT&T UNIX via BSD UNIX. Although for legal reasons FreeBSD cannot be called “UNIX”, as the direct descendant of BSD UNIX , FreeBSD’s internals and system APIs are UNIX-compliant...
and Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...
, and Mac OS X
Mac OS X
Mac OS X is a series of Unix-based operating systems and graphical user interfaces developed, marketed, and sold by Apple Inc. Since 2002, has been included with all new Macintosh computer systems...
/Darwin
Darwin (operating system)
Darwin is an open source POSIX-compliant computer operating system released by Apple Inc. in 2000. It is composed of code developed by Apple, as well as code derived from NeXTSTEP, BSD, and other free software projects....
. The source code is available only to members of the Association of Mizar Users.
The Mizar distribution includes the Mizar Mathematical Library (MML) consisting of many definitions and theorems which can be referred to in newly written articles. These new articles, after having been reviewed and checked automatically, can be published in the associated Journal of Formalized Mathematics and then become part of the MML.
The MML is built on the axioms of the Tarski-Grothendieck set theory
Tarski-Grothendieck set theory
Tarski–Grothendieck set theory is an axiomatic set theory that was introduced as part of the Mizar system for formal verification of proofs. Its characteristic axiom is Tarski's axiom . The theory is a non-conservative extension of Zermelo–Fraenkel set theory...
. As of February 2010, it contained about 9500 definitions and 49,500 theorems. Examples are the Hahn–Banach theorem
Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
, König's lemma
König's lemma
König's lemma or König's infinity lemma is a theorem in graph theory due to Dénes Kőnig . It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic,...
, Brouwer fixed point theorem
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
, Gödel's completeness theorem
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....
, and facts about the Cantor set
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
.
Even though semantically all objects MML talks about are sets, the language nevertheless allows one to define and use syntactical types: a variable may for example be declared of type Nat if it stands for a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
, or of type Group if it denotes a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. This makes the notation more convenient and closer to the way mathematicians think of symbols.
Browsable abstracts of MML articles are available as the Journal of Formalized Mathematics and MML Query implements a search engine for MML.
External links
- Main Mizar site, contains links to the MML, to the Journal of Formalized Mathematics, and a bibliography section linking to several introductions to the system
- MML Query, is a search engine for MML.
- Freek Wiedijk's Mizar site, contains slides of a conference talk about the system, as well as a 40 page introductory article
- Association of Mizar Users
- A paper about Mizar history (ps)
- A comparison between Mizar and another (Automath-like) formal language for mathematics, Weak Type Theory.
- Mizar wiki