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37
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 & computational aspects of dependency grammar : Heads, dependents, and dependency structures. http://www.coli.unisb.de/g̃j/Lectures/DG.ESSLLI/index.phtml
, 2002
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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,
Knowledge Science and Technology: Operationalizing the Enlightenment
 In
, 2000
"... Abstract: The aspirations and achievements of research and applications in knowledgebased systems are reviewed and placed in the context of the evolution of information technology, and our understanding of human expertise and knowledge processes. Future developments are seen as a continuation of a ..."
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Abstract: The aspirations and achievements of research and applications in knowledgebased systems are reviewed and placed in the context of the evolution of information technology, and our understanding of human expertise and knowledge processes. Future developments are seen as a continuation of a longterm process of operationalizing the rational stance to human knowledge processes adopted in the enlightenment, involving further diffusion of artificial intelligence technologies into mainstream computer applications, and incorporation of deeper models of human psychological and social processes. 1
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.
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.
Mining Programming Language Vocabularies from Source Code
"... Abstract. We can learn much from the artifacts produced as the byproducts of software development and stored in software repositories. Of all such potential data sources, one of the most important from the perspective of program comprehension is the source code itself. While other data sources give ..."
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Abstract. We can learn much from the artifacts produced as the byproducts of software development and stored in software repositories. Of all such potential data sources, one of the most important from the perspective of program comprehension is the source code itself. While other data sources give insight into what developers intend a program to do, the source code is the most accurate humanaccessible description of what it will do. However, the ability of an individual developer to comprehend a particular source file depends directly on his or her familiarity with the specific features of the programming language being used in the file. This is not unlike the difficulties secondlanguage learners may encounter when attempting to read a text written in a new language. We propose that by applying the techniques used by corpus linguists in the study of natural language texts to a corpus of programming language texts (i.e., source code repositories), we can gain new insights into the communication medium that is programming language. In this paper we lay the foundation for applying corpus linguistic methods to programming language by 1) defining the term “word ” for programming language, 2) developing data collection tools and a data storage schema for the Java programming language, and 3) presenting an initial analysis of an example linguistic corpus based on version 1.5 of the Java Developers Kit. 1
Positional Value and Linguistic Recursion
 Springer Netherlands
, 2007
"... Computation and Natural Language The confluence of linguistic and mathematical thought in ancient India provides a unique view of how modern mathematics and computation rely on linguistic and cognitive skills. The linchpin of the analysis is the use of positional notation as a counting method for ..."
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Computation and Natural Language The confluence of linguistic and mathematical thought in ancient India provides a unique view of how modern mathematics and computation rely on linguistic and cognitive skills. The linchpin of the analysis is the use of positional notation as a counting method for ancient and modern arithmetical procedures. Positional notation is a primary contribution from India to the development of modern mathematics, and in ancient India bridges mathematics to Indian linguistics. Pān: ini’s grammar, while not thought of as mathematical, uses techniques essential to modern logic and the theory of computation, and is the most thoroughgoing historical example of algorithmic and formal methods until the nineteenth century. Taken together, modern logic and ancient algorithmics show how computation of all kinds is constructed from language pattern and use. To set the stage we start with the contemporary idea that all kinds of mathematics can be thought of as sets of formulas or sentences expressed in some formal language. Such sets are called theories, and are often thought of as being algorithmically generated by some precise rules of proof, such as the rules of predicate logic applied to domainspecific, or ‘‘nonlogical,’ ’ axioms with specially defined terms. So there are theories of arithmetic based on axioms for addition and multiplication; set theories based on axioms for set membership and formation; theories of the real numbers; various kinds of geometry, algebra, and so on. Today such proof systems can also be thought of as computations, which mainly means spelling out the details by which an
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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Cited by 1 (1 self)
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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
Universality in Two Dimensions
, 2012
"... ! daehl jilr d`ad jxcd zivgn z`xwl—mixd ogehe xwer,epzinre epxag,oepx`l Turing, in his immortal 1936 paper, observed that “[human] computing is normally done by writing... symbols on [twodimensional] paper”, but noted that use of a second dimension “is always avoidable ” and that “the twodimension ..."
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! daehl jilr d`ad jxcd zivgn z`xwl—mixd ogehe xwer,epzinre epxag,oepx`l Turing, in his immortal 1936 paper, observed that “[human] computing is normally done by writing... symbols on [twodimensional] paper”, but noted that use of a second dimension “is always avoidable ” and that “the twodimensional character of paper is no essential of computation”. We propose to exploit the naturalness of twodimensional representations of data by promoting twodimensional models of computation. In particular, programs for a twodimensional Turing machine can be recorded most naturally on its own twodimensional inputoutput grid, in such a transparent fashion that schoolchildren would have no difficulty comprehending their behavior. This twodimensional rendering allows, furthermore, for a most perspicacious rendering of Turing’s universal machine. 1