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Slide1
Coalgebras in a Kleisli categorygeneric theory of traces and simulations
Ichiro HasuoRIMS, Kyoto Univ.PRESTO “Sakigake” Program, JST
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A
A
A
ASlide2
Scope of the workshop
theorypractice
synergy
Given a problem
,
what solution can I derive from mathematical theories?
Given a theory
,
what is its
killer application
?Slide3
applications as
driving force of development of theoryone way to ensure healthy development
of theory
The role of applications:
one theoretician’s point of view
theory
application
Extended theory
Further application
…
Unifying picture/
understanding
of the essenceSlide4
Thanks again for coming all over!!WelcomeSlide5
Theory of coalgebras as mathematical theory of state-based systemssystem as
coalgebra behavior by coinduction... in which category?Sets standard, “behavior” =
bisimilarity
Stone spaces
better compatible with modal logic
[
Kupke
, Kurz, Venema]
nominal sets/presheaves models for name-passing calculi
[Fiore, Staton]a Kleisli category suited for traces
and simulationsWhat this talk is aboutSlide6
The theory we develop
in Setsin
Kl
(
T
)
coalgebrasystemsystem
morphism of coalgebra
functional bisimulationforward simulation (lax)backward similation (oplax)
by final coalgebra
bisimilarity
trace semantics
theory of
bisimilarity
theory of traces and simulations
genericity
: both for
T
=
P
(non-determinism)
T
=
D
(probability)Slide7
Coalgebras in the Kleisli category Kl(
T)T: a parameter, for type of branchingT
=
P
non-deterministic branching
T = D probabilistic branching
“What we can do in a non-det. setting, we can also do in a probabilistic branching”Just change from
T = P to T = D
Exploited in verifying probabilistic anonymity
Genericity
of the theorySlide8
CoalgebraDefinition Let C
: a category F : C C a functor
A
coalgebra
is a
morphism
in CSlide9
System as
coalgebrax
y
z
a
c
b
as
in
C
C
=
Sets
F =
Σ
x _
“action and continue”
state space
type of transition
dynamicsSlide10
Transition-type:non-determinism
xy
b
a
as
in
C
C
=
Sets
F =
P
(
Σ
x _)
“non-det. choice over
(output and continue)” Slide11
Theory of coalgebras
coalgebraically
system
coalgebra
behavior-preserving map
morphism of coalgebras
behavior by final coalgebra
“coinduction”Slide12
A categorical principle…Definition An object Z in C is
final iff for any object X in
C
, there is a unique arrow
FinalitySlide13
Coinduction:behavior by final coalgebra
xy
z
a
c
b
as
in
C
C
=
Sets
F =
Σ
x _
“output and continue”
final
F
-
coalgebra
:Slide14
Coinduction:behavior by final coalgebra
x
(
a
,
x’
)
(
a,
beh(x’))beh
(x)
commutativity
of the diagramSlide15
Coinduction:behavior by final coalgebra
conventional def. of behavior
categorical def.
works for a variety of transition-types
i.e. various “signature”
functor
FSlide16
Why “coinduction”?
algebra
coalgebra
induction
by initial alg.
coinduction
by final
coalg
.
e.g.
e.g.
Ans.
Categorical dual of “induction”
well-founded
non-well-foundedSlide17
Bisimilarity by coinduction
C = SetsF = P
fin
(
A
£
_)Theorem beh(x) = beh
(y) iff x and
y are bisimilar.
x
F
-
coalgebras
are
(finitely-branching)
LTSs
modulo
bisim
.Slide18
Bisimilarity
vs. trace semantics
a
a
a
b
b
c
c
=
Also captured by final
coalgebra
?Slide19
Definition LTS with is a coalgebra
LTS with explicit termination
Leading example:
t
race semantics for LTS with
Slide20
Complete
trace semantics, to be precise
Recursive definition:
Trace semantics for LTS with
x
y
b
a
Slide21
Observationbeh(x
) gives “trace semantics” for xi.e. beh(x) = beh(y)
iff
x
and
y
are trace-equivalentcf. in Sets: beh(x) = beh(
y) iff x and
y are bisimilarTrace semantics via coinduction in Kl(T)
F
: parameter 1
“transition-type”
T
: parameter 2
“branching-type”Slide22
LTS with as a coalgebra
Two parameters:
separating transition-type and branching-type
a category where
branching is implicit
internal branching is unfolded awaySlide23
Kleisli category Kl(P)
Objects same as SetsArrowsComposition
non-deterministic branching is implicit
x
z
y
z’
z’’
y’
x
z
z’
z’’
inner branching str. is unfolded awaySlide24
Monad for branchinga
monad is a functor T equipped withPpowerset monad
D
subdistribution
monad
intuition
“unit”
singleton
“Dirac distr.”trivial branching
(with one choice)“multiplication”union
throwing internal branching away by flattening
Trivial branching
(with only one choice)
Forgetting internal branching
(by flattening)Slide25
Trace semantics via coinduction
in Kl(T)
Kleisli
category
branching is implicit
internal branching is unfolded
Commutativity of the diagram
amounts to the (conventional) definition of trace semantics such as
What is the final coalgebra in a Kleisli category?Slide26
Final coalgebra in a Kleisli categoryTheorem A final
coalgebra in Kl(P) is given by an initial algebra in Sets.Proof.
Generic Trace Semantics via
Coinduction
IH, Bart Jacobs & Ana
Sokolova
Logical Method in Comp. Sci.
3(4:11), 2007
for a polynomial/shapely
functor
F
,
9
distr. law
F
P
=>
P
F
,which lifts F to F
P : Kl(
P
) ->
Kl
(
P
)Slide27
Final coalgebra in a Kleisli categoryTheorem
T : a comm. monad s.t. Kl(T) is Cppo-enriched
A final
coalgebra
in
Kl
(
T) is given by an initial algebra in SetsProof.
Generic Trace Semantics via
Coinduction
IH, Bart Jacobs & Ana Sokolova
Logical Method in Comp. Sci.
3(4:11), 2007
for a polynomial/shapely
functor
F
,
9
distr. law
FT
=> TF,
which lifts
F
to
F
T
:
Kl
(
T
) ->
Kl
(
T
)Slide28
Initial algebra from the initial sequence…A closer look at the proofSlide29
whole diagram mapped by J: Sets -> Kl(T)
J (a left adjoint) preserves colimitsEach arrow here is an embedding
in a domain-theoretic sense…
A closer look at the proofSlide30
Take corresponding projectionscolimits are turned into limits (Smyth-Plotkin)cf. axiomatic domain theory, algebraic completeness/compactness, …
A closer look at the proofSlide31
The sequence turns out to be the final sequence Hence we’ve constructed a final coalgebraA closer look at the proofSlide32
non-deterministic branchingDifferent “branching-types”
in
Kl
(
T
) captures
trace semantics
T
: parameter for “
branching-type
”
T
=
P
probabilistic
branching
T
=
D
a
b
c
a
a
b
c
a
1
1
trace semantics:
ab
ac
trace semantics:
ab
: 1/3
ac
: 2/3Slide33
Coalgebraic simulations
observation
lax
morphism
=
forward
simulation
oplax
morphism
=
backward
simulation
9
fwd/
bwd
simulation
trace inclusion
theorem (soundness)
genericity
again
: both for
T
=
P
(non-determinism)
T
=
D
(probability)
Generic Forward and Backward Simulations
IH
Proc. CONCUR 2006
LNCS 4137Slide34
Summary so far
in Setsin
Kl
(
T
)
coalgebrasystem
systemmorphism
of coalgebrafunctional bisimilarityforward simulation (lax)backward
similation (oplax)by
final
coalgebra
bisimilarity
trace semantics
theory of
bisimilarity
theory of traces and simulations
genericity
: both for
T
=
P
(non-determinism)
T
=
D
(probability)Slide35
Case study: probabilistic anonymity
Simulation-based proof method for
non-deterministic
anonymity
[KawabeMST06]
Simulation-based proof method for
probabilistic
anonymity
generic,
coalgebraic
theory of traces and simulations
T
=
P
T
=
D
Probabilistic Anonymity via
Coalgebraic
Simulations
IH &
Yoshinobu
Kawabe
Proc. ESOP 2007
LNCS 4421Slide36
SummaryBisimilarity vs.
trace semanticsBisimilarity via
coinduction
in
Sets
Trace semantics
via
coinduction in a Kleisli categoryNon-deterministic branching (T=P)Probabilistic branching (T=D)Final
coalgebra in Kl(T) = initial algebra in
Sets(Monad + order) as essence of “branching” = ?
Thanks for your attention!
Ichiro Hasuo (RIMS, Kyoto-U)
http://www.kurims.kyoto-u.ac.jp/~ichiroSlide37
Serendipity: fool’s gold?