Algebraic Foundations of Simulation Science

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Algebraic Foundations of Simulation Science

• Lecture 5 Jan-Feb 04
• Dr. Eduardo Mendoza
• Physics Department
• Mathematics Department Center for
  NanoScience
• University of the Philippines
  Ludwig-Maximilians-University
• Diliman Munich, Germany
• eduardom_at_math.upd.edu.ph
  Eduardo.Mendoza_at_physik.uni-muenchen.de

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Topics to be covered

• Motivation for the theory
• Sequential Dynamical Systems (SDS)
• Morphisms of SDS
• State Spaces of SDS
• Products and the Decomposition Theorem
• Further graduate research areas

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1. Motivation for the theory

• References
• Finite Dynamical Systems 28.02.2001 (by
  Reinhard Laubenbacher and Bodo Pareigis)
• Decomposition and Simulation of Sequential
  Dynamical Systems 20.06.2002 (by Reinhard
  Laubenbacher and Bodo Pareigis) .
• Update Schedules of Sequential Dynamical Systems
  20.06.2002 (by Reinhard Laubenbacher and Bodo
  Pareigis)

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Need for a Science of Simulations

• On one hand
• Computer simulations becoming ubiquitous tool for
  complex systems study
• Variety (Zoo ?) of toolscontinuous, discrete,
  hybrid--available

• On the other
• Few scattered results on structure and dynamics
  of simulations
• Simulation is still largely an art form
• Comparison of different simulations (of a system)
  comparison of different implementations
  difficult

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Some questions...

• How to replace a large simulation with a
  equivalent smaller one?
• How in general to relate structures and dynamics
  (their state spaces) of simulations?
• Can a large simulation be decomposed into
  smaller ones? If yes, is this (in some
  appropriate sense) unique?

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The original SDS concept(Barrett-Reidys, 1999)
For k 0,1
is
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Generalization (Laubenbacher-Pareigis, 2001)

• in the following aspects
• allow state set k to be arbitrary
• no restrictions on the local functions fa with
  respect to symmetry
• more general update schedules
• use of only a subset of local functions for
  global function construction
• arbitrary repetitions allowed

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2. Sequential Dynamical Systems

• Review of graph theory concepts
• (loop-free, undirected) graph
• graph morphism
• subgraph
• connected components of a graph
• 1-neighborhood (or vicinity) of a graph vertex

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Notation
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Permutation SDS ( original SDS)
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Sequential Dynamic System (SDS)
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3. Morphisms of SDS

• To define a morphism of SDS, we have to consider
  the following data which may be changed by maps
• the graph F
• the local functions fa
• the word a
• the set of states ka

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Adjoint map on state spaces
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Morphism definition considerations
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Examples for motivation (1)
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Example on motivation (2)
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Example Motivation (3)
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Example Motivation (4)
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Morphism definition (1)
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Morphism definition (2)
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Further example
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Composition of morphisms
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4. State spaces of SDS

• Definitions
• 1.The directed graph (kn, Ef) for a function f
  kn ? kn and Ef (x,f(x))x ? kn is called
  the state space of f.
• 2.A commutative diagram is
  called a morphism of the functions f, g and
  defines a morphism of directed graphs

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Main Result
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5. Products and the Decomposition Theorem
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Projection morphisms
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SDS Indecomposability
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Morphism decompositions
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Simulation comparisons (1)
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Simulation comparisons (2)
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Theory Evaluation
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Theory Limitations
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Solution Approach
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Basic Extensions (1) Ugraphs
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Basic Extensions (2) Pographs
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SDS on pographs
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6. Coop with Bodo Pareigis (LMU)

• Minisymposium
• SIAM Conference on Discrete Mathematics (Session
  MS22)
• Sequential Dynamical Systems a Mathematical
  Foundation of Computer Simulation
• 11.-14. August 2002, San Diego, CA, USA
• Sequential dynamical systems (SDS) were
  constructed as abstractions of computer
  simulations. They are similar to cellular
  automata, but carry more structure, which allows
  a successful mathematical analysis.
• SDS theory was first developed at Los Alamos
  National Laboratory (LANL) under Chris Barrett in
  context with large-scale traffic simulations.
• SDS theory also proves to be a rich mathematical
  subject. In a categorical framework one studies
  approximations of dynamical systems by SDS. This
  allows the mathematical formulation of questions
  regarding the replacement of one simulation by
  another, their linearizations, and the
  relationship between structural features of a
  simulation and the resulting dynamics

• Finite Dynamical Systems 28.02.2001 (by
  Reinhard Laubenbacher and Bodo Pareigis)
• Decomposition and Simulation of Sequential
  Dynamical Systems 20.06.2002 (by Reinhard
  Laubenbacher and Bodo Pareigis) .
• Update Schedules of Sequential Dynamical Systems
  20.06.2002 (by Reinhard Laubenbacher and Bodo
  Pareigis)

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Coop with Prof. Laubenbacher (VBI/Virginia Tech)
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Planned Activities to Support

• Joint Seminar _at_ LMU with B. Pareigis Sequential
  Dynamic Systems (April-July 04)
• Lecture Series on Algebra of Simulations in
  August/September 2004
• Co-advise PhD/MS theses with B. Pareigis and
  R.Laubenbacher (starting Fall 04)

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Thanks for your attention !

• Questions?