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Title: Mathematics on the Net Stateoftheart and challenges


1
Mathematics on the Net State-of-the-art and
challenges
  • Dana Petcu
  • Computer Science Department,
  • Western University of Timisoara
  • and Research Institute e-Austria
    Timisoara,
  • Romania
  • http//web.info.uvt.ro/petcu

2
Overview
  • Motivation
  • Standards for Web of Mathematics
  • Web-based mathematical services
  • Grid-based mathematical computations
  • Mathematical knowledge management
  • Conclusions? What the future will bring

3
Motto
  • "It is reasonable to expect that in the year
    2010,
  • the predominant way of doing math
  • will no longer be by pen and paper,
  • but in an integrated
  • web-based math-development sys.
  • that supports the mathematician
  • in all aspects of mathematics. "
  • Michael Kohlhase,
  • MathWeb project (http//www.mathweb.org/)

4
Investigation
How far we are from this web-based math vision
fulfillment?
5
On-line Maths
  • On-line
  • mathematical computation
  • - mathematical information

6
E-Mathematics
  • Resources
  • bibliographic data (lots, with meta-data)
  • papers (lots, HTML)
  • software (some, user interfaces)
  • Services
  • citation indexes (very used)
  • computations (seldom used)

7
Mathematical problems
  • Solvable
  • Solvable manually or with the computing tools
  • Solvable by new constructions
  • Solvable only with the computers
  • Solvable, but missing computing power
  • Unsolved yet

8
Unsolved problems
9
Try to do this with the pen!
  • Find Groebner basis of the set of polynomials (91
    variables) a12, (-1 a1)(-1 b1), (-1
    b1)b1, (-1 a10)(-1 b10), (-1 b10)b10,
    (-1 a11)(-1 b11), (-1 b11)b11, (-1
    a12)(-1 b12), (-1 b12)b12, (-1 a13)(-1
    b13), (-1 b13)b13, (-1 a14)(-1 b14),
    (-1 b14)b14, (-1 a2)(-1 b2), (-1
    b2)b2, (-1 a3)(-1 b3), (-1 b3)b3, (-1
    a4)(-1 b4), (-1 b4)b4, (-1 a5)(-1
    b5), (-1 b5)b5, a11a5 b11b5, (-1
    a6)(-1 b6), (-1 b6)b6, (-1 a7)(-1
    b7), (-1 b7)b7, (-1 a8)(-1 b8), (-1
    b8)b8, (-1 a9)(-1 b9), (-1 b9)b9, c1,
    c10, b12/2 c12, c14, -1 p12 q12, -1
    p22 q22, -1 p32 q32, (-1/4 (p1p2p3
    - p3q1q2 - p2q1q3 - p1q2q3)2)(-1/4
    (p1p2p3 p3q1q2 p2q1q3 -
    p1q2q3)2)(-1/4 (p1p2p3 p3q1q2 -
    p2q1q3 p1q2q3)2) (-1/4 (p1p2p3 -
    p3q1q2 p2q1q3 p1q2q3)2), c11 a11u1,
    c3 a3u1, c4 a4u1, d22 - u12, d42 -
    u12, b10 c10 a10u2, b11 c11 a11u2, b5
    c5 a5u2, b8 c8 a8u2, b9 c9 a9u2,
    -1 d62 - (-u1 u2)2, x11, -u1 x6, x8,
    -(b13x1) a13y1, c12 a12x1 b12y1, c13
    (a13x1)/2 (b13y1)/2, c14 a14x1 b14y1,
    d12 - x12 - y12, -d12 - d22 2d1d2p1
    (-u1 x1)2 y12, c14 a14x10 b14y10, c3
    a3x10 b3y10, b6(u2 - x11) a6(-1 y11),
    c5 a5x11 b5y11, c6 (a6(u2 x11))/2
    (b6(1 y11))/2, c10 a10x12 b10y12,
    b7(x11 - x12) a7(-y11 y12), c7 (a7(x11
    x12))/2 (b7(y11 y12))/2, c5 a5x13
    b5y13, c6 a6x13 b6y13, b7(-x13 x14)
     a7(y13 - y14), c9 a9x14 b9y14, c7
    (a7(x13 x14))/2 (b7(y13 y14))/2, c4
    a4x15 b4y15, c8 a8x15 b8y15, c1
    a1x16 b1y16, c9 a9x16 b9y16, b2(u1 -
    x2) a2y2, c12 a12x2 b12y2, c2 (a2(u1
    x2))/2 (b2y2)/2, c3 a3x2 b3y2, d32 -
    (u1 - x2)2 - y22, -d32 - d42 2d3d4p2
    x22 y22, b2(u1 - x3) a2(-1 y3), c11
    a11x3 b11y3, c2 (a2(u1 x3))/2 (b2(1
    y3))/2, -(b13x4) a13(-1 y4), c10 a10x4
    b10y4, c13 (a13x4)/2 (b13(1 y4))/2, d52
    - (u2 - x5)2 - (1 - y5)2, b7(u2 - x5) a7(-1
    y5), c6 a6x5 b6y5, c8 a8x5 b8y5,
    -d52 - d62 2d5d6p3 (-u1 x5)2 y52,
    c7 (a7(u2 x5))/2  (b7(1 y5))/2, c12
    a12x6 b12y6, b2(-x6 x7) a2(y6 - y7), c4
    a4x7 b4y7, c2 (a2(x6 x7))/2 (b2(y6
    y7))/2, c12 a12x8 b12y8, b13(-x8 x9)
    a13(y8 - y9), c1 a1x9 b1y9, c13
    (a13(x8 x9))/2 (b13(y8 y9))/2, (-1
    ((x10 - x15)2 - (x10 - x16)2 (y10 - y15)2 -
    (y10 - y16)2)?1)(-1 ((x10 - x15)2 - (x15 -
    x16)2 (y10 - y15)2 - (y15 - y16)2)?2)

10
Powerful computer tools CAS
  • Computer Algebra Systems
  • Aim to manipulate a formula symbolically using
    the computer
  • A CAS provide algorithms for symbolic computation
  • Symbolic computation
  • a technology transfer method
  • takes mathematical ideas, techniques and
    theorems,
  • turn them into algorithms and computation
    tools
  • http//www.symbolicnet.org

11
Subfields of symbolic computing
  • computer algebra,
  • automated theorem proving,
  • computational combinatorics,
  • computational geometry,
  • automated programming,
  • functional or logic programming.

12
Symbolic methods - applications
  • computer aided design
  • software development
  • VLSI design
  • geometric modelling
  • reasoning
  • robot programming
  • human genome
  • etc
  • Investigating non-routine questions using CAS
    encourages diverse mathematical thinking and
    independent work

13
Problems behind CAS
  • Lagging relative to numerical computing,
  • mainly due to the inadequacy of available
    computational resources
  • computer memory
  • processor power.
  • Solution parallel distributed CA
  • Solving larger problems
  • Build new algorithms
  • Build new systems

14
Mathematical software
  • Thousands of packages of all kinds performing all
    kinds of mathematical computations
  • Maple, Mathematica, MuPAD, Maxima/Macsyma,
    Reduce, Axiom,
  • Theorist, Magma, Singular, Macaulay, Gb/RS,
    Cocoa, GAP, Cayley, Lie, PARI/GP,
  • Bernina, SYM, ACE, Hartmath, Jacal, Yacas, Giac,
    Ginac, AG libraries, LAPACK,
  • NAG libraries, IMSL, Matlab, Scilab, LAMEX,
    FRIDAY, The On-Line Encyclopedia of Integer
    Sequences
  • And counting

15
Mathematical software problem
  • Users may not be aware of existing tools to solve
    their math problems
  • Users may not be able to make best choice
  • Not realistic to install all packages locally
  • Know specifics of all software
  • Maintain up-to-date licenses of all software
    ()
  • Even for rarely used ones!

16
Needs
  • Need to normalize, categorize, and discover
    operations performed by mathematical packages.
  • Need for a standard taxonomy.
  • different packages perform the same operation
    under different names.
  • Semantic interface to abstract packages
    peculiarities
  • "Integrate a system of first-order ODEs"
  • instead of
  • "call NAG's D02BGF routine"

17
Divergence of Code Developers and Users
  • In the early days of numerical simulation Codes
    were used by the application experts themselves.
  • This is no longer true!
  • End user requirements
  • Software environment for the solution of a wide
    problem class (not just one special application)
  • Easy way to specify the problem (close to his
    language, no low-level coding)
  • Support by the software if choices are to be
    made, or the problem specification is
    inconsistent (recommender system)
  • Comfortable presentation of results
    (visualization)

18
Others
  • Real-World problems require very complex software
    solutions, model problem show cases not
    sufficient
  • Software development very expensive
  • Very slow productivity increase through modern
    programming techniques (OO, Java, Tool-support,
    )

19
Ways to Escape the Software Problem
  • In particular for Computer Science and
    Engineering
  • Lots of well-developed application codes and
    libraries available
  • As an application programmer
  • Avoid re-writing code that is already available
  • Challenge find comfortable ways to re-use them!

20
What we need?
  • Standard representation of mathematical objects
  • solved recently
  • Standard way to invoke, compose and discover
    mathematical packages
  • in progress

21
Standards for Web of Mathematics
22
A Web of Mathematical Resources and Services
  • Resources formal mathematical entities
  • Axiomatized theories
  • Definitions
  • Defined objects
  • Theorems
  • Proofs
  • Services mathematical problem solvers
  • Evaluators
  • Simpliers
  • Solvers
  • Provers
  • The Web of Mathematics is a collection of
    mathematical services
  • operating on formal resources.

23
Mathematical communication
  • Data formats for portable mathematical objects
  • OpenMath
  • MathML
  • OMDoc
  • Protocols and APIs
  • IAMC
  • MathWeb
  • JavaMath API

24
Mathematical Representation OpenMath
  • Standard developed by an European research
    consortium.
  • Abstract syntax model for mathematical objects
  • variable, symbol
  • quantier (variable, object), application(object,
    object), annotation(object, object)
  • Concrete syntax representations
  • Content dictionaries (CDs)
  • Collections of constant (function/predicate)
    symbols
  • Standard set of CDs plus extensions
  • Idea application (phrasebook") understands
    particular set of CDs.

25
OpenMath
  • Maple int(sin(x),x1..10)
  • OpenMath
  • ltOMOBJgt
  • ltOMAgt
  • ltOMS cd"calulus1" name"defint"/gt
  • ltOMAgt
  • ltOMS cd"interval1" name"interval/gt
  • ltOMIgt 1 lt/OMIgt
  • ltOMIgt 10 lt/OMIgt
  • lt/OMAgt
  • ltOMBINDgt
  • ltOMBVARgt
  • ltOMV name"x"/gt
  • lt/OMBVARgt
  • ltOMAgt
  • ltOMS cd"transc1" name"sin"/gt
  • ltOMV name"x"/gt
  • lt/OMAgt

26
Mathematical Representation MathML
  • Standard W3C for mathematical syntax
    (www.w3.org/Math/)
  • Presentation rather than representation standard
  • Presentation markup plus content markup.
  • Comparatively widely supported
  • Rendered by Web browsers
  • Computer algebra syntax (input/output format)
  • Driven by needs of electronic publishing.

27
Example MathML
  • x24x40
  • ltmrowgt
  • ltmrowgt
  • ltmsupgt ltmigtxlt/migt ltmngt2lt/mngt lt/msupgt
    ltmogtlt/mogt
  • ltmrowgt
  • ltmngt4lt/mngt
  • ltmogtInvisibleTimeslt/mogt
  • ltmigtxlt/migt
  • lt/mrowgt
  • ltmogtlt/mogt
  • ltmngt4lt/mngt
  • lt/mrowgt
  • ltmogtlt/mogt
  • ltmngt0lt/mngt
  • lt/mrowgt

28
OMDoc
  • Standard for mathematical documents
  • (Open Mathematical Documents http//www.mathweb.
    org/omdoc)
  • OM syntax for representation of mathematical
    objects.
  • Markup and formalization of mathematical
    concepts.
  • Theories, definitions, theorems, proofs,
    examples, . . .
  • Cross references across entities.
  • Input to MBase database.
  • Extraction of formal contents from mathematical
    documents.

29
Other standards to ensure interoperability
  • MSDL Mathematical Service Description Language
  • MPDL Mathematical Problem Description Language
  • MQL Mathematical Query Language
  • MEL Mathematical Explanation Language
  • MPL Mathematical Planning Language

30
IAMC
  • Internet-Accessible Mathematical Computation
  • HTTP-like protocol for server-client
    communication
  • Service referenced by URL
  • Computation and control requests from client.
  • Responses (also questions) from server
  • Informal description of service provided.
  • Abstract protocol for service access
    (machine-readable).
  • Requires insight to be used (not
    machine-understandable).
  • Target humans accessing service by web clients.

31
MathWeb (http//www.mathweb.org/)
  • Software bus combining mathematical agents
    (services)
  • Theorem provers, computer algebra systems.
  • Broker providing access object for service by
    name.
  • Abstraction from service locations.
  • Abstraction from object encodings.
  • No way to interact with previously unknown
    services.

32
Other activities
  • Digital mathematical libraries
  • More interested in library meta-data than service
    meta-data.
  • Computer algebra systems Maple, Mathematica
  • Read the docu" (sometimes read the code")
  • Axiomatic specifications OBJ, Larch, . . . ,
    CASL
  • Extensive collections of axiomatized theories.
  • Powerful modularization and structuring
    mechanisms.
  • Used only in theorem proving community.
  • Representation problem addressed

33
Web-enabled CASs
  • MapleNet
  • http//www.maplesoft.com/maplenet/
  • software platform to enhance mathematics and
    related courses over the web
  • the client machine runs a Java applet
  • the server manages concurrent Maple instances
    launched to serve client requests for
    mathematical computations and display services
  • a publisher machine is responsible for creating
    and editing content of web pages
  • webMathematica
  • http//www.wolfram.com/products/webmathematic
    a/
  • access to Mathematica applications through a web
    browser
  • deliver HTML pages that incorporate Mathematica
    commands and results

34
WIMS WWW Interactive Mathematics Server
35
Current Situation
  • Resources and services are mainly useable for
    humans only
  • Most resources have no formalized
    representation
  • Mathematical contents of papers, books,
    hypertext.
  • I/O interfaces of mathematical software systems.
  • Most resources/services do not provide
    meta-data
  • Bibliographic data, cross references.
  • Functional specifications (I/O conditions).
  • Non-functional specifications (time and memory
    complexity).
  • Situation need human-like insight into resources
    and services.

36
Web-based mathematical services
37
Mathematical Services
  • Originate from various communities
  • computer algebra (Maple, Mathematica, GAP,
    Magma, KANT, . . .)
  • numerical computation (FORTRAN libraries,
    NetSolve, . . .)
  • theorem proving (PVS, COQ, HOL, Isabelle, . . .)
  • visualization JavaView, KnotPlot, . . .
  • . . .

38
Mathematical Users
  • Consumers of resources and clients of services.
  • Humans
  • Databases
  • Software
  • Other resources and services
  • Vision Many future users on the web will be
    machines

39
Internetprojects
  • http//distributedcomputing.info/ap-math.html
  • finding large prime numbers,
  • factoring large numbers,
  • computing digits of Pi,
  • finding collisions on known encryption algorithms
  • etc.

40
Black boxes
  • Mathematics on the Web is mainly intended for
    human consumption!
  • Challenge make resources and services usable as
    black bloxes, no human insight needed i.e.
  • mathematical resource or service meant to be
    processed automatically
  • (machine-readable machine-understandable)

41
Machines as Users
  • Machine-readable
  • Formalization/standardization of representation.
  • Simplify communication.
  • Syntactic issue.
  • Machine-understandable
  • Meta-information on properties of the resource.
  • Simplify interpretation.
  • Semantic issue.

42
Web Service - definition
  • An application identified by URI
  • Designed to support interoperable
    machine-to-machine interaction over the network
  • Interfaces and bindings are defined/discovered as
    XML artifacts

43
Mathematical Web service
  • Service whose data and results are encoded using
    markup for mathematical content
  • Implements evaluators, solvers, simplifiers or
    provers
  • Mathematical resource a formal mathematical
    entity, e.g. definition, theorem, proof, object
  • Resource consumer human, database, piece of
    software, other resource or service

44
Initiatives for Math.Web services
  • MONET demonstrate the applicability of the
    semantic Web to the world of mathematical
    software (discover services dynamically)
  • MathWeb-SB access via broker by name
  • MathBroker Web registry to publish/discover
  • The way in which services are discovered is not
    standardize! Grids can help!

45
MONET Mathematics on the Net
  • Framework for web-based math services
  • http//monet.nag.co.uk
  • Demonstrate the applicability of the semantic web
    to the world of mathematical software
  • Ability to discover services dynamically based on
    published descriptions which express both their
    mathematical and non-mathematical attributes
  • A symbolic solver wrapper was designed to provide
    an environment that encapsulates CASs and expose
    their functionalities through symbolic services
  • Maple and Axiom used as computing engines

46
MONET-2
47
Grid-based Mathematical Services
48
What is Grid?
  • The short answer is that,
  • whereas the Web is a service for sharing
    information over the Internet,
  • the Grid is a service for sharing computer power
    and data storage capacity over the Internet.

49
Why Grid?
  • High potential as discovery accelerator
  • Way to categorize, explore, discover, invoke and
    compose thousand of software packages

50
Mathematics on Grids
  • GridSolve/NetSolve client/server to solve
    remotely
  • GENSS follows MONET, research on advertisement
    and discovery, ontology
  • gridMathematica HPC-Grid Maple parallel
    computing
  • GEMLCA deploy a legacy code
  • Maple-to-Grid
  • D.Petcu, D.Tepeneu, M.Paprzycki, T.Ida, Symbolic
    Computations on Grids, Chapter 6 in the book
    "Engineering the Grid status and perspective",
    eds. Beniamino di Martino, et al ASP, 2006, pp.
    91-107

51
GENSS project
  • Grid Enabled Numerical and Symbolic Services,
    initiated in 2004
  • http//genss.cs.bath.ac.uk/index.htm
  • follows the ideas formulated in the Monet project
  • intends to combine Grid computing and
    mathematical Web services
  • research was focused in two areas
  • matchmaking techniques for advertisement and
    discovery of mathematical services,
  • design and implementation of an ontology for
    symbolic problems

52
GENSS search service
53
Geodise
  • Implemented within Matlab environment
  • An engineering portal providing Grid access to
    computational fluid dynamics and design
    optimization tools
  • Two different mechanisms used to submit jobs to
    computing resources
  • use a web service interface to Condor
  • collection of Matlab functions
  • submission of jobs to Globus-enabled resources
    via Java CoG tools
  • functions allow users to run and control jobs on
    the grid, or to archive, query, and retrieve
    data, notify the (mobile) user about the status
    of the job.

54
Maple2g
  • Example Find Woodall primes
  • gt with(m2g) m2g_MGProxy_start()
  • m2g_connect, m2g_getservice, m2g_jobstop,
    m2g_jobsubmit, m2g_maple, m2g_MGProxy_end,
    m2g_MGProxy_start, m2g_rank, m2g_recv,
    m2g_results, m2g_send,m2g_size
  • Grid connection established
  • gt p4 a1 b2000 m2g_maple(p)
  • Connect kernel 1 successful
  • Connect kernel 2 successful
  • Connect kernel 3 successful
  • Connect kernel 4 successful
  • gt m2g_send("all",1,cat("sNULLa",a,"b",b,
  • " for i from am2g_rank to b by m2g_size do,
  • " if isprime(i2i-1) then ss,i fi od
    s")
  • gt m2g_recv("all",1)
  • 81,249,2,6,30,362,462,822,3,75,115,1
    23,751,384,512
  • gt m2g_MGProxy_end()
  • Grid connection closed

55
SCIEnce EU project (06-11)
  • goal improve integration between key
    world-leading developers and application experts
    in Symbolic Computation software systems.
  • Develop versions of the GAP, Maple, KANT and
    MuPAD systems which can inter-communicate via a
    common standard Web services interface
  • Develop common standards and middleware to allow
    the production of Grid-enabled systems for
    Symbolic Computation

56
SCIEnce partners
  • University of St Andrews, School of Computer
    Science, St Andrews, UK
  • Universtitaet Linz, Research Institute for
    Symbolic Computation, Linz, Austria
  • Centre National de la Recherche Scientifique,
    Laboratoire dInformatique UMR, Palaiseau, France
  • Universitaet Paderborn, Institute for Mathematics
    - AutoMATH, Paderborn, Germany
  • Technische Universiteit Eindhoven, Department of
    Mathematics and Computer Science, Eindhoven,
    Netherlands
  • Technische Universitat Berlin, Institut für
    Mathematik - KANT Group, Berlin, Germany
  • Institute e-Austria Timisoara, Timisoara, Romania
  • Waterloo Maple Inc., Dep. of Research and
    Development, Waterloo, Ontario, Canada
  • Heriot Watt University, School of Mathematical
    and Computer Sciences, Edinburgh, UK

57
Other RO projects at Timisoara
  • CFD on Grids, http//nanosim.ieat.ro
  • Web-PS Web simulator for membrane computing
    (simulating the behaviour of living cells),
    http//psystems.ieat.ro
  • MedioGrid Grid services for National, Agency
    of Meteorology for floods prevention,
    http//mediogrid.utcluj.ro
  • GridMOSI genetic algs.on and for Grids,
    http//gridmosi.info.uvt.ro
  • VISP Virtual Internet Service Provider (EU
    project, Web services) http//www.visp-project.org

58
Mathematical knowledge management
59
On-line publications
  • I think we are very shortly going to cross a
    sort of critical mass boundary where those
    publications that are not instantly available in
    full-text will become kind of second-rate in a
    sense, not because their quality is low, but just
    because people will prefer the accessibility of
    things they can get right away.
  • Clifford Lynch, 1997, Director of the Coalition
    for Networked Information

60
Mathematical Knowledge Management (MKM)
  • An emerging interdisciplinary field of research
    in the intersection of mathematics, computer
    science, library science, and scientific
    publishing.
  • Objective to develop new and better ways of
    managing mathematical knowledge using
    sophisticated software tools.
  • Challenge to create a universal digital
    mathematics library accessible via the World Wide
    Web.
  • http//www.mkm-ig.org/
  • Mathematical Knowledge Management Network
    (EU-funded Network September 2002 - November 2003
  • MoWGLI - Mathematics on the Web Get It by Logics
    and Interfaces (EU-funded project IST-2001-33562
    MOWGLI)
  • NA-MKM The North American Chapter of The MKM
    Consortium

61
MOWGLI (2002-2004)
  • Mathematics on the Web Get It by Logics and
    Interfaces
  • WWW - the largest resource of mathematical
    knowledge,
  • Almost all mathematical web documents are marked
    up only for presentation, severely crippling the
    potentialities for automation, interoperability,
    sophisticated searching mechanisms, intelligent
    applications, transformation and processing.
  • Goal overcome these limitations, passing from a
    machine-readable to a machine-understandable
    representation of the information, and developing
    the technological infrastructure for its
    exploitation.

62
SystheMathEx (2005-2007)
  • EU project
  • Aims to make a contribution to the field of MKM
    by developing knowledge bases through systematic
    exploration in two important case studies
    considered
  • theory of tuples and
  • theory of Groebner bases

63
Conclusions evolution
We are already seeing this
64
Conclusions because
  • Analogy
  • Mathematics (intellectual) Moving
    (physical)
  • mental calculation walking
  • paper pencil calculation cycling
  • automated calculation driving a car

65
Conclusions and the trends
and the current trend is to create environments
with self- combining mathematical services
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