Transcript Slide 1

Ichiro Hasuo
Tracing Anonymity with Coalgebras
Tracing Anonymity with Coalgebras
Ichiro Hasuo
The ultimate aim
•
•
•
•
pervasive, important
fail easily
…
we don’t quite understand them!
Tracing Anonymity with Coalgebras
Ichiro Hasuo
Coalgebras
Good balance:
mathematical
simplicity
(potential)
applicability
In this thesis:
• more applications are found
• further mathematical theory is developed
Tracing Anonymity with Coalgebras
Ichiro Hasuo
Coalgebras
coalgebraically
FX
system
coalgebra
X
behaviorpreserving map
behavior
morphism of
coalgebras
by final
coalgebra
FX
Ff
c
X
FX
c
X
FY
d
f
Y
FZ
¯ nal
Z
beh( c)
Tracing Anonymity with Coalgebras
Ichiro Hasuo
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
Tracing Anonymity with Coalgebras
Ichiro Hasuo
In Sets: bisimilarity
system as coalgebra
FX
behavior by final
coalgebra
FX
c
X
X
FZ
¯ nal
Z
beh( c)
NB
• what they mean exactly depends on which
category they’re in
X, FX, FZ, …
sets
X Y
function
standard
• they are in the category Sets
• “behavior” captures bisimilarity
Tracing Anonymity with Coalgebras
Ichiro Hasuo
Bisimilarity vs. trace semantics
a

=
a
c
b
When do we
decide
c
b
Trace
semantics
Bisimilarity
or
a
Anyway we get

?

or
Tracing Anonymity with Coalgebras
Ichiro Hasuo
Coalgebraic trace semantics
Behavior by
final coalgebra
FX
c
X
“Kleisli category”
o a category where
FZ
¯ nal
Z
beh( c)
captures…
in
Sets
Generic Trace Semantics via
Coinduction
IH, Bart Jacobs & Ana Sokolova
Logical Method in Comp. Sci.
3(4:11), 2007
branching is implicit
o X  Y : “branching function” from X to Y
o
T : parameter for branching-type
bisimilarity
(standard)

in
Kl(T) trace
semantics
(Ch. 2)
=
Tracing Anonymity with Coalgebras
Ichiro Hasuo
Different “branching-types”
FX
c
X
T=P
FZ
¯ nal
Z
beh( c)
trace semantics:
ab
a c
non-deterministic branching
a
b
a
in Kl(T) captures
trace semantics
T
T : parameter for
trace semantics:
“branching-type”
a  b : 1/3
= D a  c : 2/3
probabilistic branching
a
c
b 1
1
3
2
3
a
1 c
Tracing Anonymity with Coalgebras
Ichiro Hasuo
Coalgebraic simulations (Ch. 3)
morphism of
coalgebras
FX
Ff
c
X
FY
d
f
Y
in
Sets
functional
bisimulation
(standard)
lax morphism
= forward
simulation
Generic Forward and Backward
Simulations
IH
Proc. CONCUR 2006
LNCS 4137
in
Kl(T)
??
oplax morphism
genericity again
: both for
= backward
• T = P (non-determinism)
simulation
• T = D (probability)
9 fwd/bwd simulation  trace inclusion
Tracing Anonymity with Coalgebras
Ichiro Hasuo
genericity : both for
• T = P (non-determinism)
• T = D (probability)
Summary so far
coalgebra
FX
in Sets
in Kl(T)
system
system
Ch. 3
X
morphism of coalgebra
FX
Ff
c
X
FY
functional bisimilarity
d
f
forward simulation (lax)
backward similation (oplax)
Y
Ch. 2
by final coalgebra
FX
c
X
FZ
¯ nal
Z
beh( c)
bisimilarity
trace semantics
theory of
bisimilarity
theory of traces
and simulations
Tracing Anonymity with Coalgebras
Ichiro Hasuo
Probabilistic Anonymity via
Coalgebraic Simulations
IH & Yoshinobu Kawabe
Proc. ESOP 2007
LNCS 4421
Case study:
probabilistic anonymity (Ch. 4)
Tracing Anonymity with Coalgebras
Ichiro Hasuo
category of
coalgebras
final
coalgebra
Concurrency
“concurrency” , “behavior”
2-dimensional , nested algebraic structure
Tracing Anonymity with Coalgebras
Ichiro Hasuo
Concurrency and the
microcosm principle (Ch. 5)
science of
generic
computer
compositionality
systems
theorem
The Microcosm Principle and
Concurrency in Coalgebra
IH, Bart Jacobs & Ana Sokolova
To appear in Proc. FoSSaCS 2008
LNCS
concurrency,
compositionality,
behavior, …
formalization of microcosm
principle in 2-categories
1
+ X
L
Cat
C
mathematics
Tracing Anonymity with Coalgebras
Ichiro Hasuo
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