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ΩMEGA: Computer supported mathematics
 IN: PROCEEDINGS OF THE 27TH GERMAN CONFERENCE ON ARTIFICIAL INTELLIGENCE (KI 2004)
, 2004
"... The year 2004 marks the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number” (with Martin Davis’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated dedu ..."
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The year 2004 marks the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number” (with Martin Davis’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated deduction used machine oriented calculi to find the proof for a theorem by automatic means, the Automath project of N.G. de Bruijn – more modest in its aims with respect to automation – showed in the late 1960s and early 70s that a complete mathematical textbook could be coded and proofchecked by a computer. Classical theorem proving procedures of today are based on ingenious search techniques to find a proof for a given theorem in very large search spaces – often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited. The shift
Formal and computational aspects of dependency grammar: History and development of DG
 ESSLLI Course Notes, FoLLI, the Association of Logic, Language and Information
, 2002
"... • Goal: To provide an overview of the historical development of dependency grammar, set within the context of theoretical linguistics. • History in overview ..."
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• Goal: To provide an overview of the historical development of dependency grammar, set within the context of theoretical linguistics. • History in overview
Doing Mathematics on the ENIAC. Von Neumann’s and Lehmer’s different visions. ∗
"... In this paper we will study the impact of the computer on mathematics and its practice from a historical point of view. We will look at what kind of mathematical problems were implemented on early electronic computing machines and how these implementations were perceived. By doing so, we want to str ..."
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In this paper we will study the impact of the computer on mathematics and its practice from a historical point of view. We will look at what kind of mathematical problems were implemented on early electronic computing machines and how these implementations were perceived. By doing so, we want to stress that the computer was in fact, from its very beginning, conceived as a mathematical instrument per se, thus situating the contemporary usage of the computer in mathematics in its proper historical background. We will focus on the work by two computer pioneers: Derrick H. Lehmer and John von Neumann. They were both involved with the ENIAC and had strong opinions about how these new machines might influence (theoretical and applied) mathematics. 1
Computer Supported Formal Work: Towards a Digital Mathematical Assistant
 STUDIES IN LOGIC, GRAMMAR AND RHETORIC
, 2007
"... The year 2004 marked the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number ” (with Martin Davis ’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated ..."
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The year 2004 marked the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number ” (with Martin Davis ’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated deduction used machine oriented calculi to find the proof for a theorem by automatic means, the Automath project of N.G. de Bruijn – more modest in its aims with respect to automation – showed in the late 1960s and early 70s that a complete mathematical textbook could be coded and proofchecked by a computer. Roughly at the same time in 1973, the Mizar project started as an attempt to reconstruct mathematics based on computers. Since 1989, the most important activity in the Mizar project has been the development of a database for mathematics. International cooperation resulted in creating a database which includes more than 7000 definitions of mathematical concepts and more than 42000 theorems. The work by
Tag systems and Collatzlike functions
"... Tag systems were invented by Emil Leon Post and proven recursively unsolvable by Marvin Minsky. These production systems have shown very useful in constructing small universal (Turing complete) systems for several different classes of computational systems, including Turing machines, and are thus im ..."
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Tag systems were invented by Emil Leon Post and proven recursively unsolvable by Marvin Minsky. These production systems have shown very useful in constructing small universal (Turing complete) systems for several different classes of computational systems, including Turing machines, and are thus important instruments for studying limits or boundaries of solvability and unsolvability. Although there are some results on tag systems and their limits of solvability and unsolvability, there are hardly any that consider both the shift number v, as well as the number of symbols µ. This paper aims to contribute to research on limits of solvability and unsolvability for tag systems, taking into account these two parameters. The main result is the reduction of the 3n + 1problem to a surprisingly small tag system. It indicates that the present unsolvability line – defined in terms of µ and v – for tag systems might be significantly decreased. Key words: Tag Systems, limits of solvability and unsolvability, universality,
Comparative Analysis of Hypercomputational Systems Submitted in partial fulfilment
"... In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosoph ..."
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In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosophers and scientists are suggesting models of hypothetical computing devices which can outperform the Turing machine, performing some calculations the latter is unable to. The ChurchTuring Thesis, which the Turing machine model embodies, has raised discussions on whether it could be possible to solve undecidable problems which Turing’s model is unable to. Models which could solve these problems, have gone further to claim abilities relating to quantum computing, relativity theory, even the modeling of natural biological laws themselves. These so called ‘hypermachines ’ use hypercomputational abilities to make the impossible possible. Various models belonging to different disciplines of physics, mathematics and philosophy, have been suggested for these theories. My (primarily researchoriented) project is based on the study and review of these different hypercomputational models and attempts to compare the different models in terms of computational power. The project focuses on the ability to compare these models of different disciplines on similar grounds and
Formal Languages for Linguists: Classical and Nonclassical Models
, 2001
"... The basics of classical formal language theory are introduced, as well as a wide coverage is given of some new nonstandard devices motivated in molecular biology, which are challenging traditional conceptions, are making the theory revived and could have some linguistic relevance. Only definitions a ..."
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The basics of classical formal language theory are introduced, as well as a wide coverage is given of some new nonstandard devices motivated in molecular biology, which are challenging traditional conceptions, are making the theory revived and could have some linguistic relevance. Only definitions and a few results are presented, without including any proof. The chapter can be profitably read without any special previous mathematical background. A long list of references completes the chapter, which intends to give a flavour of the field and to encourage young researchers to go deeper into it.
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
Undecidability of FirstOrder Logic
"... 2.1 A signature for lines............................... 6 ..."