Post on 02-Feb-2016
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Coalgebraic Symbolic Semantics
Filippo Bonchi Ugo Montanari
Many formalisms modelling Interactive Systems
Algebras - SyntaxCoalgebras - Semantics
Bialgebras – Semantics of the composite system in terms of the semantics of the components
(compositionality of final semantics)CCS [Turi, Plotkin – LICS 97]
Pi-calculus [Fiore, Turi – LICS 01] [Ferrari, Montanari, Tuosto – TCS 05]
Fusion Calculus [Ferrari et al. – CALCO 05][Miculan – MFPS 08]
… in many interesting cases, this does not work…
Mobile Ambient [Hausmann, Mossakowski, Schröder – TCS 2006]
Formalisms with asynchronous message passingPetri Nets…
Plan of the Talk
• Compositionality• Saturated Semantics• Symbolic Semantics• Saturated Coalgebras• Normalized Coalgebras
As running example, we will use Petri nets
Bonchi, Montanari – FOSSACS 08
Petri Nets
p
q
B
c
d
P is a set of placesT is a set of transitionsPre:TP
Post:TP
l:T is a labelling
Given a set A, A is the set of all multisets over A,e.g., for A={a,b} ,then A={,{a},{b},{aa},{bb},{ab} ,{aab}…}
2
a marking is a multiset over P
The semantics is quite intuitive pc qcB
Open Petri NetsPetri net + interface
a b
$
interface
Input PlacesInput Places
Output Place
ClosedPlace
Interface=(Input Places, Output Places)
Petri Nets Contexts
Petri nets + Inner interfaces + Outer Interface
a
$
c
c
c c
c c
InnerInterface
OuterInterface
a b
$
a b
$
a b
$
x3
$
Bisimilarity is not a congruence
c d
$
5
c e
x3
$
cx exC$$$
e$$$
f
They are bisimilar
They are not
x3
$
e f
$
3
Plan of the Talk
• Compositionality• Saturated Semantics• Symbolic Semantics• Saturated Coalgebras• Normalized Coalgebras
As running example, we will use Petri nets
Saturated Bisimilarity
A relation R is a saturated bisimulationiff whenever pRq, then C[-]
• If C[p]→p’ then q’ s.t. C[q]→q’ and p’Rq’• If C[q]→q’ then p’ s.t. C[p]→p’ and p’Rq’
THM: it is always the largest bisimulation congruence
Saturated Transition System
p qC[-]
C[p] q
C[-] is a context is a label
Saturated Semantics for Open NetsAt any moment of their execution a token
can be inserted into an input place and one can be removed from an output place
b
$
a
$ $$ $$$
+$ +$ +$ +$
a
aa
+a
+a
-$ -$ -$
b
b$ +$ b$$
+$
a$ a$$
a$$$
+$ +$ +$ +$
+a +a +a$
a
Running Examples
a b
$
e f
$
3
g
i
h
c d
$
5
The activation is free.The service costs 1$.
The activation costs 5$.
The service is free.
The activation costs 3$. The service is free for 3
times and then it costs 1$.
THEY ARE ALL DIFFERENT
I have 1$ and
I need 1
I have 5$ and
I need 6
Running Examples
l
q
m
$
3
n
p
o
This behaves as a or e: either the activation is free and
the service costs 1$.Or the activation costs 3$ and then for 3 times the service is
free and then it costs 1$.
IS IT DIFFERENT
FROM ALL THE PREVIOUS???
a b
$
The activation is free.The service costs 1$.
$
$
a b
$
e f
$
3
g
i
h
Plan of the Talk
• Compositionality• Saturated Semantics• Symbolic Semantics• Saturated Coalgebras• Normalized Coalgebras
As running example, we will use Petri nets
Symbolic Transition System
p qC[-]
C[p] q
C[-] is a context is a label
intuitively C[-] is “the smallest context” that allows such transition
Symbolic Transition System
a b
$
c d
$
5
e f
$
3
g
i
h
a b $
c d5$
e
f
g
h
i
3$
$
Symbolic Semanticsa symbolic LTS + a set of deduction rules
In our running example
m nm$ n$
p qD[p] ’ E[q]
p,q
Inference relation
Given a symbolic transition system and a set of deduction rules, we can infer other transitions
p qC[-] p ’ q’C’[-]
Inference relation
a b
b$$$
$$$
b$n
$n
m n
m$ n$
a b
$
Bisimilarity over the Symbolic TS is too strict
l
q
m
$
3
n
p
o
l m n o
p
3$
$
q $
a b
$
a b $
Plan of the Talk
• Compositionality• Saturated Semantics• Symbolic Semantics• Saturated Coalgebras• Normalized Coalgebras
As running example, we will use Petri nets
Category of interfaces and contexts
• Objects are interfaces• Arrows are contexts
Functors from C to Set are algebras for Г(C)SetC AlgГ(C)
One object: {$}
Arrows: -$n: {$}{$}
for our nets
Saturated Transition System as a coalgebra
Ordinary LTS having as labels ||C|| and ΛF:SetSet F(X)=(||C||ΛX)
We lift F to F: AlgГ(C) AlgГ(C)
(saturated transition system as a bialgebra)
p qC[-]
Adding the Inference Relation
An F-Coalgebra is a pair (X, :XF(X))
The set of deduction rules induces an ordering on||C||ΛX
X
a b
b$$$
$$$
b$n
$n
Saturated Coalgebras
• A set in(||C||ΛX) is saturated in X if it is closed wrt
S: AlgГ(C) AlgГ(C)
the carrier set of S(X) is the set of all saturated sets of transitions
• E.g: the saturated transition system is always an S-coalgebra
X
Saturated CoalgebrasCoalgF
CoalgS
THM: CoalgS
is a covariety of CoalgF
THM: Saturated Coalgebras are not bialgebras
1F
1S
Redundant Transitions
… … … … … …
partial order ||C||ΛX,
X
Saturated Set
Given a set A in(||C||ΛX), a transition is redundant
if it is not minimal
Normalized Set
… … … … … …
partial order ||C||ΛX,
X
Saturated Set
A set in(||C||ΛX) is normalizedif it contains only NOT redundant
transitions
Normalized Set
SaturationNormalization
Normalized Coalgebras
N: AlgГ(C) AlgГ(C)
the carrier set of N(X) is the set of all normalized sets of transitions For h:XY, the definition of N(h) is peculiar
… … … … … … … …… …
||C||ΛX, X
||C||ΛY, y
This is redundant
Running Example
l m n o
p
3$
$
q $
a b $ b$
$$b$$b$
3$
lq m
$
3
n
p
o
a b
$
Isomorphism Theorem
Proof: Saturation
and Normalization
are two natural isomorphisms
between S and N
CoalgF
CoalgS
CoalgN
Saturation Normalization
Conclusions
• Bisimilarity of Normalized Colagebras coincides with Saturated Bisimilarity
• Minimal Symbolic Automata• Symbolic Minimization Algorithm
[Bonchi, Montanari - ESOP 09]
• Coalgebraic Semantics for several formalisms (asynchronous PC, Ambients, Open nets …)
• Normalized Coalgebras are not Bialgebras
Questions?