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Abstract Computerizing Mathematical Text with
"... Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which c ..."
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Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., LATEX and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerizing mathematical texts and knowledge which is flexible enough to connect the different approaches to computerization, which allows various degrees of formalization, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Three Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), and Robert Lamar (since 2006)) and over a dozen master’s degree and undergraduate students have worked on MathLang. The project’s progress and design choices are driven by the needs for computerizing real representative mathematical texts chosen from various
Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]:
"... The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction wh ..."
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The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction while it provides for an abbreviation mechanism at the level of terms. λδ enjoys some important desirable properties such as the confluence of reduction, the correctness of types, the uniqueness of types up to conversion, the subject reduction of the type assignment, the strong normalization of the typed terms and, as a corollary, the decidability of type inference problem.
Computerising Mathematical Text with MathLang
"... Mathematical texts can be computerised in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which c ..."
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Mathematical texts can be computerised in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., LATEX and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerising mathematical texts which is flexible enough to connect the different approaches to computerisation, which allows various degrees of formalisation, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Four Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), Robert Lamar (since 2006)), and Christoph Zengler (since 2008) and over a dozen master’s degree and undergraduate
The evolution of types and logic in the 20th century: A journey through Frege, Russell and . . .
 ILLC ALUMNI EVENT, AMSTERDAM 2004
, 2004
"... ..."
The Triumph of Types: Principia Mathematica’s Impact on Computer Science
"... Types now play an essential role in computer science; their ascent originates from Principia Mathematica. Type checking and type inference algorithms are used to prevent semantic errors in programs, and type theories are the native language of several major interactive theorem provers. Some of these ..."
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Types now play an essential role in computer science; their ascent originates from Principia Mathematica. Type checking and type inference algorithms are used to prevent semantic errors in programs, and type theories are the native language of several major interactive theorem provers. Some of these trace key features back to Principia. This lecture examines the influence of Principia Mathematica on modern type theories implemented in software systems known as interactive proof assistants. These proof assistants advance daily the goal for which Principia was designed: to provide a comprehensive formalization of mathematics. For instance, the definitive formal proof of the Four Color Theorem was done in type theory. Type theory is considered seriously now more than ever as an adequate foundation for both classical and constructive mathematics as well as for computer science. Moreover, the seminal work in the history of formalized mathematics is the Automath project of N.G. de Bruijn whose formalism is type theory. In addition we explain how type theories have enabled the use of formalized mathematics as a practical programming language, a connection entirely unanticipated at the time of Principia Mathematica’s creation.
The Formal System λδ
, 2008
"... The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction wh ..."
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The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction while it provides for an abbreviation mechanism at the level of terms. λδ enjoys some important desirable properties such as the confluence of reduction, the correctness of types, the uniqueness of types up to conversion, the subject reduction of the type assignment, the strong normalization of the typed terms and, as a corollary, the decidability of type inference problem.
The Formal System λδ
, 2008
"... The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction wh ..."
Abstract
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The formal system λδ is a typed λcalculus that pursues the unification of terms, types, environments and contexts as the main goal. λδ takes some features from the Automathrelated λcalculi and some from the pure type systems, but differs from both in that it does not include the Π construction while it provides for an abbreviation mechanism at the level of terms. λδ enjoys some important desirable properties such as the confluence of reduction, the correctness of types, the uniqueness of types up to conversion, the subject reduction of the type assignment, the strong normalization of the typed terms and, as a corollary, the decidability of type inference problem.
λTypes on the λCalculus with Abbreviations
, 2007
"... In this paper the author presents λδ, a λtyped λcalculus with a single λ binder and abbreviations. This calculus pursues the reuse of the term constructions both at the level of types and at the level of contexts as the main goal. Up to conversion λδ shares with Church λ → the subset of typable te ..."
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In this paper the author presents λδ, a λtyped λcalculus with a single λ binder and abbreviations. This calculus pursues the reuse of the term constructions both at the level of types and at the level of contexts as the main goal. Up to conversion λδ shares with Church λ → the subset of typable terms but in the “propositions as types ” perspective it can encode the implicative fragment of predicative logic without quantifiers because dependent types are allowed. λδ enjoys the properties of Church λ → (mainly subject conversion, strong normalization and decidability of type inference) and, in addition, it satisfies the correctness of types and the uniqueness of types up to conversion. We stress that λδ differs from the Automathrelated λcalculi in that they do not provide for an abbreviation construction at the level of terms. Moreover, unlike many λcalculi, λδ features a type hierarchy with an infinite number of levels both above and below any reference point.