Ichiro Hasuo Tracing Anonymity with Coalgebras TexPoint fonts used in EMF. Read the TexPoint manual...
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Transcript of Ichiro Hasuo Tracing Anonymity with Coalgebras TexPoint fonts used in EMF. Read the TexPoint manual...
Ich
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The ultimate aim
Computer systems
• pervasive, important• fail easily • …• we don’t quite understand them!
Bett er mathematical understanding of computer systems
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Coalgebras
Our mathemati cal presentati on of systems
Good balance:
• more applications are found• further mathematical theory is developed
In this thesis:
mathematical simplicity
(potential) applicability
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Coalgebras
coalgebraically
system coalgebra
behavior-preserving map
morphism of coalgebras
behavior by final coalgebra
F X
X
F X F Z
Xc
beh(c) Z¯ nal
F XF f
F Y
Xc
f Yd
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Overview
Coalgebraic theory of traces and
simulations (Ch. 2-3)•via coalgebras in a Kleisli category•apply to both•non-determinism•probability•case study: probabilistic anonymity (Ch. 4)
Concurrency in coalgebras (Ch. 5)
•the microcosm principle appears
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In Sets: bisimilarity
system as coalgebra
behavior by final coalgebraF X F Z
Xc
beh(c) Z¯ nal
F X
X
NB
• what they mean exactly depends on which category they’re in
standard
• they are in the category Sets• “behavior” captures bisimilarity
X, FX, FZ, … sets
X Y function
category = “universe”Sets, Top, Stone,
Vect, CLat, …
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Bisimilarity vs. trace semantics
When do we decide or ?
Bisimilarity
Anyway we get or
Trace semanti
cs
a aa
b bc c
=Also captured by
fi nal coalgebra?
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Coalgebraic trace semanticsBehavior by
final coalgebra
F X F Z
Xc
beh(c) Z¯ nal
captures…bisimilarity (standard)
in Sets
trace semantics (Ch. 2)
in Kl(T)
“Kleisli category”
o a category where branching is implicito X Y : “branching function” from X to Y
o T : parameter for branching-type
=Generic Trace Semantics via Coinduction IH, Bart Jacobs & Ana Sokolova Logical Method in Comp. Sci. 3(4:11), 2007
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non-deterministic branching
Different “branching-types”
in Kl(T) captures trace semantics
F X F Z
Xc
beh(c) Z¯ nal
T : parameter for “branching-type”
T = P probabilistic branching
T = D
a
b c
a a
b c
a 13
23
1 1
trace semantics: a b a c
trace semantics: a b : 1/3 a c : 2/3
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Coalgebraic simulations (Ch. 3)
morphism of coalgebras
functional bisimulation(standard)
in Sets
??
in Kl(T
)F XF f
F Y
Xc
f Yd
observati on
lax morphism= forward simulation
oplax morphism= backward simulation
9 fwd/bwd simulation trace inclusiontheorem (soundness)
genericity again : both for• T = P (non-determinism)• T = D (probability)
Generic Forward and Backward Simulations IH Proc. CONCUR 2006 LNCS 4137
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Summary so far
in Sets in Kl(T)
coalgebra system system
morphism of coalgebrafunctional bisimilarity forward simulation (lax)
backward similation (oplax)
by final coalgebrabisimilarity trace semanticsF X F Z
Xc
beh(c) Z¯ nal
F XF f
F Y
Xc
f Yd
F X
X Ch. 3
Ch. 2
theory of bisimilarity
theory of traces and simulations
genericity : both for• T = P (non-determinism)• T = D (probability)
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Case study: probabilistic anonymity (Ch. 4)
Simulation-based proof method for non-deterministic anonymity
[KawabeMST06]
Simulation-based proof method for probabilistic anonymity
generic, coalgebraic theory of traces and simulati ons
[Ch. 2-3]
T = P
T = D
Probabilistic Anonymity via Coalgebraic Simulations IH & Yoshinobu Kawabe Proc. ESOP 2007 LNCS 4421
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Concurrency
2-dimensional , nested algebraic structure
C k Drunning C and D in parallel
“concurrency” , “behavior”
2
inner k
outer k
the microcosm principle
final coalgebra
category of coalgebras
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Concurrency and the microcosm principle (Ch. 5)
mathematics
science of computer systems
concurrency, compositionality,
behavior, …
formalization of microcosm principle in 2-categories
L
1
C+ X
Cat
generic compositionality
theorem
The Microcosm Principle and Concurrency in Coalgebra IH, Bart Jacobs & Ana Sokolova To appear in Proc. FoSSaCS 2008 LNCS
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Summary
Coalgebraic theory of traces and
simulations (Ch. 2-3)•via coalgebras in a Kleisli category•apply to both•non-determinism•probability•case study: probabilistic anonymity (Ch. 4)
Concurrency in coalgebras (Ch. 5)
•the microcosm principle appears