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