Semantics of OpenMath and MathML3

Even though OpenMath has been around for more than 10 years, there is still confusion about the “semantics of OpenMath”. As the recent MathML3 recommendation semantically bases Content MathML on OpenMath Objects, this question becomes more pressing. One source of confusions about OpenMath semantics is that it is given on two levels: a very weak algebraic semantics for expression trees, which is extended by considering mathematical properties in content dictionaries that interpret the meaning of (constant) symbols. While this two-leveled way to interpret objects is well-understood in logic, it has not been spelt out rigorously for OpenMath. We present two denotational semantics for OpenMath: a construction-oriented semantics that achieves full coverage of all legal OpenMath expressions at the cost of great conceptual complexity, and a symbol-oriented one for a subset of OpenMath expressions. This subset is given by a variant of the OpenMath 2 role system, which—we claim—does not exclude any representations of meaningful mathematical objects.

[1]  John C. Reynolds,et al.  Towards a theory of type structure , 1974, Symposium on Programming.

[2]  Michael Kohlhase,et al.  A Better Role System for OpenMath , 2009 .

[3]  James H. Davenport,et al.  Unifying Math Ontologies: A Tale of Two Standards , 2009, Calculemus/MKM.

[4]  E. Zermelo.  Untersuchungen über die Grundlagen der Mengenlehre. I , 1908 .

[5]  Furio Honsell,et al.  A framework for defining logics , 1993, JACM.

[6]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[7]  J. Girard.  Une Extension De ĽInterpretation De Gödel a ĽAnalyse, Et Son Application a ĽElimination Des Coupures Dans ĽAnalyse Et La Theorie Des Types , 1971 .

[8]  James H. Davenport,et al.  A small OpenMath type system , 2000, SIGS.

[9]  Andreas Strotmann.  The Categorial Type of OpenMath Objects , 2004, MKM.

[10]  Till Mossakowski,et al.  Project Abstract: Logic Atlas and Integrator (LATIN) , 2011, Calculemus/MKM.

[11]  Olga Caprotti,et al.  First draft on the OpenMath standard (ESPRIT OpenMath Report) , 1998 .

[12]  A. M. Cohen,et al.  A Type System for OpenMath , 1999 .

[13]  J. Neumann.  Eine Axiomatisierung der Mengenlehre. , 1925 .

[14]  Michael Kohlhase,et al.  A scalable module system , 2011, Inf. Comput..

[15]  Hendrik Pieter Barendregt,et al.  Introduction to generalized type systems , 1991, Journal of Functional Programming.

[16]  D. Hilbert.  Über das Unendliche , 1926 .

[17]  Jamie Tappenden.  Recent Work in Philosophy of Mathematics , 2001 .

[18]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[19]  Terrence A. Brooks,et al.  World Wide Web Consortium (W3C) , 2010 .

[20]  Alfred Tarski,et al.  On the Application of Symbolic Logic to Algebra , 1953 .

[21]  Michael Kohlhase,et al.  An Exchange Format for Modular Knowledge , 2008, LPAR Workshops.

[22]  Claudio Sacerdoti Coen,et al.  A Foundational View on Integration Problems , 2011, Calculemus/MKM.

[23]  Michael Kohlhase,et al.  MathDox : mathematical documents on the web , 2006 .

[24]  B. Russell.  Mathematical Logic as Based on the Theory of Types , 1908 .

[25]  A. Tarski,et al.  Arithmetical extensions of relational systems , 1958 .

[26]  Bertrand Russell,et al.  Principia Mathematica Vol. Iii , 1913 .

[27]  Stephen M. Watt,et al.  Intelligent Computer Mathematics , 2014, Lecture Notes in Computer Science.

[28]  Stephen M. Watt,et al.  Mathematical Markup Language (MathML) Version 3.0 , 2001, WWW 2001.

[29]  Gérard P. Huet.  The Calculus of Constructions: State of the Art , 1987, FSTTCS.

[30]  J. Abbott,,et al.  Open Math: Communicating Mathematical Information Between Co-operating Agents in a Knowledge Network , 1998 .

[31]  Christoph Benzmüller,et al.  Higher-order semantics and extensionality , 2004, Journal of Symbolic Logic.

[32]  A. Tarski,et al.  Über unerreichbare Kardinalzahlen , 1938 .

[33]  Michael Kohlhase.  OpenMath Content Dictionaries , 2006 .

[34]  Josef Urban,et al.  Intelligent Computer Mathematics - 18th Symposium, Calculemus 2011, and 10th International Conference, MKM 2011, Bertinoro, Italy, July 18-23, 2011. Proceedings , 2011, Calculemus/MKM.

[35]  Florian Rabe,et al.  Representing logics and logic translations , 2008 .

[36]  K. Gödel.  Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I , 1931 .

[37]  P. Martin-Löf.  An Intuitionistic Theory of Types: Predicative Part , 1975 .

[38]  A. Pitts.  INTRODUCTION TO HIGHER ORDER CATEGORICAL LOGIC (Cambridge Studies in Advanced Mathematics 7) , 1987 .

[39]  G. Cantor,et al.  Grundlagen einer allgemeinen Mannichfaltigkeitslehre : ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen , 1883 .

[40]  G. Rota.  Introduction to higher order categorical logic , 1988 .

[41]  Mirina Grosz,et al.  World Wide Web Consortium , 2010 .

[42]  Michael Kohlhase,et al.  OMDoc - An Open Markup Format for Mathematical Documents [version 1.2] , 2006, Lecture Notes in Computer Science.