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The OpenMath Project Mike Dewar NAG Ltd, UK

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22/11/00Semantic Web Technologies What is OpenMath? A technology for encoding the semantics of mathematical objects so that they can be exchanged between applications and users IST Project International Society ( )

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22/11/00Semantic Web Technologies Understanding Mathematical Notation Explicit Context: –let p be any prime number; then a , a p a mod p Implicit Context: –a x +b x +c x +d x+e –f =f =1 ; f = f + f –ei–ei Conventions: –sin (x) vs f (x) –sin (x) vs f (x) –sin (x) vs f (x) Syntax: –sin(x) or Sin[x] or \sin{(x)} or... – R f(x)dx or R f(y)dy

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22/11/00Semantic Web Technologies OpenMath A standard mechanism for representing mathematical objects electronically –Unambiguous semantics –Machine-processable –System-independent –Extensible –Lightweight –Scalable

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22/11/00Semantic Web Technologies MathML Recommendation from W3C Mainly presentation Small content part (high school mathematics) Can embed OpenMath using semantics tag Can transform between OpenMath and MathML using e.g. XSL

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22/11/00Semantic Web Technologies OpenMath Objects Abstract model for a concrete mathematical entity

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22/11/00Semantic Web Technologies Content Dictionaries Specify semantics associated with symbols Extensible Standardised by OpenMath Society –Persistent –http://www.openmath.org/cd Private sin The sine function as described in Abramowitz and Stegun, section 4.3.

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22/11/00Semantic Web Technologies Content Dictionaries Definition of each symbol can include: –informal descriptions –formal mathematical properties –examples of use Associated information can include: –default presentation information –associated type information –other mathematical properties

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22/11/00Semantic Web Technologies Organising CDs CDs contain groups of related symbols –e.g. transc1 contains transcendental functions Related CDs collected into CD Groups –e.g. Linear Algebra CD Group linalg1: simple operations such as determinant linalg2: row-oriented constructors linalg3: column-oriented constructors linalg4: advanced operations such as eigenvalue linalg5: constructors for structured matrices

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22/11/00Semantic Web Technologies Some more CD Groups MathML –Symbols corresponding to content part of MathML 2 –Basic calculus, sets, linear algebra, arithmetic, numbers, transcendental functions... Polynomial Special Functions Number Theory Groups...

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22/11/00Semantic Web Technologies Encodings Provides concrete representation for an OpenMath object. Desirable features: –linear –compact XML encoding. Binary encoding....

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22/11/00Semantic Web Technologies The OpenMath Standard Formal specifications of: –OpenMath objects –XML encoding –Binary encoding –Syntax for Content Dictionaries –Basic symbols for Content Dictionaries

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22/11/00Semantic Web Technologies Associated Documents Type systems: –Simple Type System (STS) –Extended Calculus of Constructions (ECC) Content Dictionaries Also: –Guidelines for writing CDs –CD standardisation procedure –Correspondence between MathML(C) and CD Group

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Building Applications With OpenMath

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22/11/00Semantic Web Technologies Mathematical Components Interface described in terms of CDs “recognised” Use standard tools for translating between OpenMath and proprietary representations –Software layer which does this is called a Phrasebook Use any convenient mechanism to link components together –sockets, RPC, Active-X,...

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22/11/00Semantic Web Technologies Interactive Documents Mathematical objects encoded as OpenMath Manipulations/Calculations via plug-ins and applets –renderers, editors,... Translators for existing content in LaTeX Standard frameworks to handle interaction: –XML/DOM, CORBA,... –OMDoc

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22/11/00Semantic Web Technologies Mathematical Databases OpenMath API to existing databases Databases of OpenMath objects “Smart Databases”: –read CDs dynamically to extract mathematical properties –use properties in resolving queries

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22/11/00Semantic Web Technologies Computing with OpenMath Objects OpenMath does not mandate a default action. –Typesetter –Algebra system –Numerical software OpenMath consists of nouns not verbs. Onus is on phrasebook writer to implement “correct” behaviour.

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22/11/00Semantic Web Technologies Summary OpenMath provides a standard representation for semantically-rich mathematical data. OpenMath is a complementary technology to MathML.

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