Defining Geometric Algebra Semantics Samantha Zambo Naval Research Laboratory Stennis Space Center, MS 39529-5004, USA 228-688-4895
[email protected]
ABSTRACT This paper discusses the development of Open Math Content Dictionaries (CDs) for Geometric Algebra (GA) in support of efforts towards developing a Physics Markup Language (PML). This paper introduces the underlying principles of a PML. The background on the various bodies of knowledge that are being leveraged including markup languages, GA, and OpenMath are introduced. The approach for the creation of CDs specific to GA is presented with an amplifying example. Four GA CDs were constructed proving the capabilities for standardizing GA semantics and PML development.
Categories and Subject Descriptors I.7.2 [DOCUMENT AND TEXT PROCESSING]: Document Preparation – Markup languages, Format and notation, Desktop publishing.
General Terms Documentation, Standardization, Languages.
Keywords Physics Markup Language, Geometric Algebra, OpenMath, Content Dictionary.
1. INTRODUCTION The Naval Research Laboratory is currently engaged in the development of a Physics Markup Language (PML) for atmospheric and ocean models. Physics based models are used to drive other models. Since the physics behind them varies, the process of ‘joining’ these models is labor intensive. For example, the winds from atmospheric models are used as an input to ocean models for predicting wave heights. The physics used by the oceanographers and meteorologist are different and attempting to incorporate these different mathematics between the two groups of scientist can be difficult. Physicists are presented with the issue of successfully enabling machine to machine interoperability for ocean and atmospheric models. By developing a PML as a systematic process for expressing the semantics of the physics based models, we can alleviate this problem. A PML must
identify and define semantics essential for specifying physics based models, compose a standard notation, and construct a structured format to support computer applications and interoperability [1]. Therefore, a PML will improve communication of physics based models’ semantics, ultimately improving interoperability. The author’s contribution to this effort was standardizing the semantics for Geometric Algebra (GA) through OpenMath by developing Content Dictionaries (CDs). In section 2, we will discuss the background information regarding this research and in section 3, we will review OpenMath CD development for GA. Afterwards, section 4 will conclude with research results and anticipated future work.
2. BACKGROUND In the following, we will review relevant concepts that are pertinent to this work including: markup languages, GA, and OpenMath.
2.1 MARKUP LANGUAGES A markup language is a system for annotating a document for processing, defining, and presenting text in a syntactical way [2]. Hyper Text Markup Language (HTML) is a widely used webpage markup language with predefined structural markers for communicating presentational semantics [3]. These HTML tags markup the document in order to denote the presentation specifications for text and other data. An example of an HTML tag is ‘’ to indicate the heading beginning for a document and ‘’ to indicate the end of the heading. Extensible Markup Language (XML) has become as important as HTML for structuring, storing, and transporting information, extensively used in representing arbitrary data structures [4]. XML provides strong support with simplicity, while exclusively regarding the data’s meaning, instead of how the data is displayed. Similarly to HTML, XML is a set of standardized rules for encoding documents with structural markers, known as XML tags. Unlike HTML, XML tags do not have predefined semantics, permitting the author to establish unique and specific XML tags and document structure. Commonly applied together, HTML formats and displays data, while XML stores and transports. Markup languages relate to this research since we will be using XML to encapsulate the meaning of mathematical constructs involved in physical models.
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2.2 GEOMETRIC ALGEBRA
Copyright © 2010 ACM 978-1-4503-0064-3/10/04... $10.00
This research supports the development of a PML by exploring GA for physicists, an intuitive way of expressing geometric
The next section will describe GA, which was encoded in XML.
notions algebraically [5]. For example, an element of GA is the ‘inner product’, denoted ‘’. GA defines the algebra for the geometric concept ‘inner product’, which is the projection of one n-dimensional vector ‘A’ onto another n-dimensional vector ‘B’, expressed algebraically as A% _$__%_FRV *$ D W\SH RI Clifford Algebra, can represent the spatial part of physical variables for most classical and modern physics [6]. GA attempts to consolidate all of the mathematics that applies to physics. Hence, we have chosen GA to describe the underlying mathematics for the PML. Efforts to express semantics of GA were based on OpenMath whose background is given in the next section.
2.3 OPENMATH OpenMath is an XML based language for standardizing mathematical semantics for grades K-12 [7]. As an example OpenMath defines ‘minus’ as A-B = A + (-B) as shown in Fig. 1.
This research utilized the standards OpenMath provided to extend it to GA and the higher level concepts it encompasses. The next section entails the development of OpenMath CDs for GA.
3. OPENMATH CONTENT DICTIONARY DEVELOPMENT FOR GEOMETRIC ALGEBRA OpenMath standardizes mathematical semantics by developing CDs. These CDs consist of two parts. The first part is the heading containing the CD’s unique details including: CD name, description, revision date, review date, version number, CD status, CD base, CDURL, and CDComment. The second part contains a collection of related symbols each including a symbol name and description. Each symbol may also include these optional components: Commented Mathematical Properties (CMP) in plain text, Formal Mathematical Properties (FMP) encoded in OpenMath, examples encoded in OpenMath, and supplementary information such as type information with the use of any type system for instance Simple Type System (STS) for simple signatures. GA elements on the other hand are defined with a symbol, description, and properties. As previously stated, our objective is to leverage GA, which provides the rigorous mathematical definitions needed to describe physical model concepts, and to use the OpenMath CD’s structure to format these definitions in a way that is compatible with a markup language. Consequently, a mapping from GA’s syntax to CDs had to be developed. Additionally, it was decided to add images for each GA element to enhance the understanding of the meaning by the scientists. As an example, the GA shown in figure 2 represents the element ‘inner product’, which is an extraction from [5].
Figure 2. GA: 'Inner Product'
Figure 1. OpenMath CD: 'Minus'
By contrast, the equivalent CD is shown in figure 3.
of the GA element ‘inner product’ from these resources, a concise and basic definition was composed. This comprehension allowed the development of the CMP by expressing the given property for ‘inner product’ ArBs = (r-s) in plain text, correctly communicating the property’s meaning. The FMP was constructed by representing the given property, previously expressed within the CMP, in MathML encoding. Similarly to attaining the description, the image presented in the example for the ‘inner product’ was selected after reviewing several resources, and chosen for its proper communication of the ‘inner product’s’ semantics in a simple and clear manner, to even further enhance the understanding of the meaning by the scientists. Lastly, a STS signature file was developed and linked for the ‘inner product’, showing data types within the GA CDs assigning semi-formal signatures to the OpenMath object, ‘inner product’. Figure 4 shows the STS of ‘inner product’.
Figure 4. 'Inner Product' STS
Figure 3. OpenMath CD: 'Inner Product' Notice how the elements of each do not directly align. The ‘inner product’ CD components were not a straightforward integration from the given GA description. A systematic methodology needed to be constructed to allow GA elements to be transformed into CDs. Once this transformation is complete the GA elements will be provided with a way for standardizing their semantics. By ascertaining an extensive knowledge of GA elements and CD components, a one-to-one mapping was established enabling the required transformations. The symbol name developed for the ‘inner product’ was logically formed by simply substituting ‘_’ for the space. Supplementary information regarding the ‘inner product’s’ role was given as ‘application’ to further define how the GA element acts. ‘Application’ was decided since the inner product is applied upon two n-dimensional vectors. The description for ‘inner product’ was obtained through various resources including, but not limited to, [5], [8], and [9]. By gathering a comprehensive understanding
The ‘inner product’ is expressed as a specialized form of the ‘geometric product’, ArBs = (r-s), above. Currently OpenMath does not contain the GA element ‘geometric product’, standardizing the semantics. Due to this fact and that CDs must be self-contained, the ‘geometric product’ was defined within the same CD as ‘inner product’ and denoted ‘
’. This only further exemplifies and supports the desire for standardizing GA semantics in an effort to avoid such notational differences, as seen above in expressing the ‘geometric product’. Only through a collection of various resources lending to a comprehensive understanding of GA element components and CD components, was a correlation made identifying the required mapping from GA elements to CDs. This mapping, which was previously explored for the ‘inner product’ in the example above, becomes the generalized translation approach for all GA elements into OpenMath CDs. This approach was done for several types of GA elements, of which four CDs were produced for the areas of GA Basics, GA Products, GA Spaces, and GA Multivectors. A total of twenty-nine terms were defined within the CDs, all with enabled links and embedded images, verifying that the defined OpenMath mapping between CDs and GA elements were
reasonably effective in standardizing GA semantics, even though human reasoning was necessary.
4. CONCLUSION The objective of this research was to develop GA OpenMath CDs in an endeavor to standardize the mathematical semantics essential in designing a PML for physical models. This was accomplished by transforming the GA elements into OpenMath CDs developing a specific mapping of GA components to CD components, a valid method in standardizing mathematical semantics. Moreover, the four resulting CDs comprised of multiple GA elements, yielded positive results for the necessary mathematical semantics needed for a PML. Further research will be employed to continue the extension and development of GA, ultimately unifying mathematics and sciences, providing great advancements in communicating research and results.
5. ACKNOWLEDGMENTS This work was supported by the STEP program of the Naval Research Laboratory. The author would like to thank her academic mentor, Dr. Wendy Zhang of the Computer Science & IT Department, Southeastern Louisiana University, and research mentors, Dr. Fred Petry and Dr. Brian S. Bourgeois with the Naval Research Laboratory at Stennis Space Center, MS.
6. REFERENCES [1] J.B. Collins, “A Physics Markup Language for Ocean and Atmospheric Environmental Models”, FY09 6.1 External Review, July 2009.
[2] http://en.wikipedia.org/wiki/Markup_language Markup Language [3] http://en.wikipedia.org/wiki/HTML HTML [4] http://www.w3schools.com/XML/xml_whatis.asp Introduction to XML [5] D. Hestenes and G. Sobczyk, “Clifford Algebra to Geometric Calculus A Unified Language for Mathematics and Physics”. Dordrecht: Kluwer Academic Publishers, 1987. [6] J. B. Collins, “A Mathematical Type for Physical Variables”, Lecture Notes in Artificial Intelligence vol. 5144: Intelligent Computer Mathematics, pp.370-381, (Springer-Verlag, 2008) ISBN 978-3-540--85109-7. [Online]. Available: http://www.ait.nrl.navy.mil/MKM08_paperOffCopy.pdf [7] OpenMath Consortium OpenMath v2.0. (2004) “OpenMath Standard”, [Online]. Available: http://www.openmath.org/standard/om20-2004-0630/omstd20.pdf [8] L. Dorst, D. Fontijne, and S. Mann. “Geometric Algebra for Computer Science”. Elsevier, 2007 [9] C. Doran, and A. Lasenby. “Geometric Algebra for Physicists”. Cambridge University Press, 2003.
