Bisimulation Relation

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Bisimulation Relation

• A lecture over
• E. Hagherdi, P. Tabuada, G. J. Pappas
• Bisimulation relation for dynamical, control, and
  hybrid systems

Rafael Wisniewski Aalborg University Ph.D. course
November 2005
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• Please ask as much as possible. I would be happy
  for all relevant to the topic questions.

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Labeled Transition Systems
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Product and Pullback
Product of C1 and C2
Pullback
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Product of Transition Systems
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Strong Bisimulation
Open Maps
Whenever
commutes
commutes then
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BranL Open Maps
P-bisimilarity
BranL is a full subcategory of TL of all
synchrinization trees with a single finite branch.
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Generalization of P-open maps
We generalize P-open maps to the category Dyn of
dynamical systems and Hyb the category of hybrid
dynamical systems.
Morphism
The path category P as the full subcategory of
Dyn with objects P I ? TI, where P(t) (t, 1)
and I is an open interval of R containing the
origin.
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P-open Maps
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P-bisimilarity for dynamical systems
Pullback in the category of P-open surjective
submersions
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Bisimilarity of Dynamical Systems
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Example
Consider the vector field X on M R2 defined
Also consider the vector field Y on N R defined
by
Then
is a Dyn-morphism
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Hybrid Dynamical Systems
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Category Hyb
Recall a time transition system from Henzinger
The state space is
Transition relation like in Henzinger
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Path Category in Hyb
The path category P is the full subcategory of
Hyb
dx/dt 1
t0
t1
t2
tk-1
tk
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Example of a path
Consider a path
This path is represented by the path object P
which has states l0, l1, l2
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P-open Maps for Hyb
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Characterization of bisimulation in Hyb
is said to be a bisimulation relation iff for all

implies
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Bisimulation Characterization
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Future Work

• Extension of the bisimulation notion from the
  article from timed transition systems to time
  abstract transition systems. This can be done by
  identify a whole flow line with a point in the
  space of flow lines.
• The strong simulation is too strong equivalence
  relation on dynamical systems is too strong. Try
  to use weaker equivalence relation some form of
  topological equivalency.

On Friday 18th Nov. try to understand the
definitions and go through proofs in the section
dealing with the dynamical systems. If you
understand P-open maps and bisimulation in the
category of dynamical systems the generalization
to hybrid systems seems natural.