Ichiro Hasuo Tracing Anonymity with Coalgebras

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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