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OMDoc an open markup format for mathematical documents (version 1.2
 Number 4180 in LNAI
, 2006
"... This Document is an online version of the OMDoc 1.2 Specification published by ..."
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This Document is an online version of the OMDoc 1.2 Specification published by
The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations
, 1991
"... ..."
Hierarchical Contextual Reasoning
, 2003
"... VII Zusammenfassung IX Extended Abstract XI Acknowledgements XIII I ..."
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Cited by 18 (9 self)
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VII Zusammenfassung IX Extended Abstract XI Acknowledgements XIII I
Equality and Extensionality in Automated HigherOrder Theorem Proving
, 1999
"... Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Hintikka Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Primitive Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Model Existenc ..."
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Cited by 14 (11 self)
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Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Hintikka Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Primitive Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Model Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Extensional HigherOrder Resolution: ER 42 4.1 A Review of HORES and ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Lifting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.5 Theorem Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5 Extensional HigherOrder Paramodulation: EP 57 5.1 A Naive and Incomplete Adaptation...
Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λcalculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced
Revisiting the Notion of Function
"... Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both pro ..."
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Cited by 6 (5 self)
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Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both processes were implemented in Frege's Begriffschrift [17], Russell's Ramified Type Theory [42] and the lambdacalculus (originally introduced by Church [12, 13]) showing that the lambdacalculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the lambdacalculus and we show that various logical constructs (e.g., let expressions and definitions and the use of parameters in mathematics), can be seen as forms of the missing part of functionalisation. Our study of the function concept leads...
Reviewing the classical and the de Bruijn notation for λcalculus and pure type systems
 Logic and Computation
, 2001
"... This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentat ..."
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Cited by 3 (0 self)
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This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λcalculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λcalculus is introduced and some of its advantages are outlined.
An Abstract Syntax for a Formal Language of Mathematics
, 2001
"... This paper provides an abstract syntax for a formal language of mathematics. We call our language Weak Type Theory (abbreviated WTT ). WTT will be as faithful as possible to the mathematician 's language yet will be formal and will not allow ambiguities. WTT can be used as an intermediary be ..."
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This paper provides an abstract syntax for a formal language of mathematics. We call our language Weak Type Theory (abbreviated WTT ). WTT will be as faithful as possible to the mathematician 's language yet will be formal and will not allow ambiguities. WTT can be used as an intermediary between the natural language of the mathematician and the formal language of the logician. As far as we know, this is the rst extensive formalization of an abstract syntax of a formal language of mathematics. We compare our work with existing formalizations of languages of mathematics. 1
On the White Box Integration of Computer Algebra Algorithms into a Deduction System
 Master’s thesis, Universität des Saarlandes
, 2005
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ΩMKRP: A Proof Development Environment
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semigroups and automata [3] wi ..."
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In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semigroups and automata [3] with the firstorder theorem prover mkrp [11]. An important finding was that although current automated theorem provers have evidently reached the power to solve nontrivial problems, they do not provide sufficient assistance for proving the theorems contained in such a textbook. On account of this, we believe that significantly more support for proof development can be provided by a system with the following two features:  The system must provide a comfortable humanoriented problemsolving environment. In particular, a human user should be able to specify the problem to be solved in a natural way and communicate on proof