Ichiro Hasuo Tracing Anonymity with Coalgebras

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The ultimate aim

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Coalgebras

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Coalgebras

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

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In Sets: bisimilarity

F X F Z

Xc

beh(c) Z¯ nal

F X

X

X, FX, FZ, … sets

function

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Bisimilarity vs. trace semantics

a aa

b bc c

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Coalgebraic trace semantics

F X F Z

Xc

beh(c) Z¯ nal

captures…

“Kleisli categoryKleisli category”

o a category where branching is implicitbranching is implicit

o X Y : “branching function” from X to Y

o T : parameter for branching-typebranching-type

=Generic Trace Semantics via Coinduction IH, Bart Jacobs & Ana Sokolova Logical Method in Comp. Sci. 3(4:11), 2007

Generic Trace Semantics via Coinduction IH, Bart Jacobs & Ana Sokolova Logical Method in Comp. Sci. 3(4:11), 2007

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Different “branching-types”

in Kl(T) captures trace semanticstrace semantics

F X F Z

Xc

beh(c) Z¯ nal

T : parameter for “branching-typebranching-type”

T = P T = D

a

b c

a a

b c

a 13

23

1 1

trace semantics: a b a c

trace semantics: a b a c

trace semantics: a b : 1/3 a c : 2/3

trace semantics: a b : 1/3 a c : 2/3

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Coalgebraic simulations (Ch. 3)

F XF f

F Y

Xc

f Yd

lax morphism= forwardforward simulationsimulation

lax morphism= forwardforward simulationsimulation

oplax morphism= backward backward simulationsimulation

oplax morphism= backward backward simulationsimulation

genericity againgenericity again : both for• T = P (non-determinism)• T = D (probability)

genericity againgenericity again : both for• T = P (non-determinism)• T = D (probability)

Generic Forward and Backward Simulations IH Proc. CONCUR 2006 LNCS 4137

Generic Forward and Backward Simulations IH Proc. CONCUR 2006 LNCS 4137

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Summary so far

F X F Z

Xc

beh(c) Z¯ nal

F XF f

F Y

Xc

f Yd

F X

X Ch. 3Ch. 3

Ch. 2Ch. 2

theory of bisimilaritytheory of

bisimilaritytheory of traces theory of traces and simulationsand simulationstheory of traces theory of traces and simulationsand simulations

genericity genericity : both for• T = P (non-determinism)• T = D (probability)

genericity genericity : both for• T = P (non-determinism)• T = D (probability)

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Case study: probabilistic anonymity (Ch. 4)

Probabilistic Anonymity via Coalgebraic Simulations IH & Yoshinobu Kawabe Proc. ESOP 2007 LNCS 4421

Probabilistic Anonymity via Coalgebraic Simulations IH & Yoshinobu Kawabe Proc. ESOP 2007 LNCS 4421

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Concurrency

“concurrency” , “behavior”“concurrency” , “behavior”

final coalgebr

a

final coalgebr

a

category of coalgebrascategory of coalgebras

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Concurrency and the microcosm principle (Ch. 5)

mathematicsmathematics

science of science of computer computer systemssystems

concurrency, concurrency, compositionality, compositionality,

behavior, …behavior, …

formalization of microcosm formalization of microcosm principle in 2-categoriesprinciple in 2-categories

L

1

C+ X

Cat

generic generic compositionality compositionality

theoremtheorem

The Microcosm Principle and Concurrency in Coalgebra IH, Bart Jacobs & Ana Sokolova To appear in Proc. FoSSaCS 2008 LNCS

The Microcosm Principle and Concurrency in Coalgebra IH, Bart Jacobs & Ana Sokolova To appear in Proc. FoSSaCS 2008 LNCS

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Summary