The Power of Convex Algebra Ana Sokolova CONCUR 17 Filippo Bonchi - - PowerPoint PPT Presentation
The Power of Convex Algebra Ana Sokolova CONCUR 17 Filippo Bonchi Alexandra Silva NII Shonan Meeting Enhanced Coinduction 15.11.17 probabilistic automata The true nature of PA as transformers of belief states Ana Sokolova Shonan
Ana Sokolova
NII Shonan Meeting “Enhanced Coinduction” 15.11.17
Alexandra Silva Filippo Bonchi
CONCUR ’17
probabilistic automata
Shonan 15-11-17 Ana Sokolova
Ana Sokolova
x1
a ✏
x2, x3 ✏
b / x3 b
g ˚
[Silva, Bonchi, Bonsangue, Rutten, FSTTCS’10]
Ana Sokolova
NFA X ➝ 2 x (P(X))A
x1
a
|
a
" x2 ✏ x3
b
k ˚
Shonan 15-11-17
X
c
Ñ FX Generative PTS X ➝ D (1 + A x X)
x1
a, 1
2
}
a, 1
2
! x2
b,1 ✏
x3
c,1
✏ x4
1 ✏
x5
1
✏ ˚ ˚
x1
a ✏ 1 2x2 ` 1 2x3 b
x
c
&
1 2x4
1 2 ✏
1 2x5
1 2
✏ ˚ ˚
[Jacobs, Silva, S. JCSS’15] [Silva, S. MFPS’11]
Ana Sokolova Shonan 15-11-17
X
c
Ñ FX PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
1 3x1 ` 2 3x2
+
a
u ⌘
a (
. . .
2 3x2 ` 1 6x3 ` 1 6x4
. . .
8 9x2 ` 1 9x3
belief-state transformer belief state
Ana Sokolova Shonan 15-11-17
X
c
Ñ FX PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
what is it? how does it emerge? foundation ?
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
Shonan 15-11-17
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
„
„ „d
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
1 3 ˆ2 3x2 ` 1 3x3 ˙ ` 2 3p1x2q Shonan 15-11-17
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
„
„ „d
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
1 3 ˆ1 2x3 ` 1 2x4 ˙ ` 2 3p1x2q
very infinite LTS on belief states
Shonan 15-11-17
Can be given different semantics:
strong bisimilarity probabilistic / combined bisimilarity belief-state bisimilarity
Ana Sokolova Shonan 15-11-17
”R
μ
a
৵
a
lifting of R to distributions assign the same probability to “R-classes”
R
bisimulation largest bisimulation
„
„
Ana Sokolova Shonan 15-11-17
”R
μ
a
৵
a
R
convex bisimulation largest convex bisimulation
„
„ „c
combined transition convex combination
Ana Sokolova Shonan 15-11-17
μ’
a
μ ৵
R
distribution bisimulation largest distribution bisimulation
„
„
transition in the belief-state transformer
„d
R
৵’
a
1 3 ˆ1 2x3 ` 1 2x4 ˙ ` 2 3p1x2q
is LTS bisimilarity on the belief-state transformer
„d
Ana Sokolova
[Hermanns, Krcal, Kretinsky CONCUR’13]
Shonan 15-11-17
X
c
Ñ FX PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
what is it? how does it emerge? foundation ?
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
Shonan 15-11-17
Uniform framework for dynamic transition systems, based on category theory.
X
c
Ñ FX
generic notion of behavioural equivalence
«
states
category C behaviour type functor on the base category C form a category too
CoAlgCpFq
Ana Sokolova Shonan 15-11-17
X
c
Ñ FX
all on
Sets
PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
Generative PTS X ➝ D (1 + A x X)
x1
a, 1
2
}
a, 1
2
! x2
b,1 ✏
x3
c,1
✏ x4
1 ✏
x5
1
✏ ˚ ˚
Ana Sokolova
NFA
x1
a
~} } }
a
A A A x2 ✏ x3
b
g
X ➝ 2 x (P(X))A
x1
a
|
a
" x2 ✏ x3
b
k ˚
Shonan 15-11-17
Sets
X ➝ (P D(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
„ “ «
X ➝ (C(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
„c “ «
convex algebras
X ➝ (Pc(X)+1)A
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
„d “ «
EMpDq
Ana Sokolova
and all convex combinations
…
Shonan 15-11-17
Mio FoSSaCS ’14
Ana Sokolova
pi P r0, 1s,
n
ÿ
i“1
pi “ 1 pA,
n
ÿ
i“1
pip´qiq
infinitely many finitary operations convex combinations
h ˜ n ÿ
i“1
piai ¸ “
n
ÿ
i“1
pihpaiq
n
ÿ
i“1
pi ˜ m ÿ
j“1
pi,jaj ¸ “
m
ÿ
j“1
˜ n ÿ
i“1
pipi,j ¸ aj
n
ÿ
i“1
piai “ ak, pk “ 1
satisfying
binary ones “suffice”
Shonan 15-11-17
Ana Sokolova
convex algebras abstractly satisfying
A
a η / DA a
✏ A
DDA
Da ✏ µ / DA a
✏ DA
a
/ A
EMpDq
DA
a
✏ A h DA
a
✏ A
DB
b
✏ B
DA
a ✏ Dh/ DB b
✏ A
h / B
Shonan 15-11-17
coalgebras on free convex algebras Minkowski sum constant exponent nonempty convex powerset termination
DS Ñ pPcpDSq ` 1qA
pA1 ` p1 ´ pqA2 “ tpa1 ` p1 ´ pqa2 | a1 P A1, a2 P A2u
free convex algebra
DDX
µ
✏ DX
DX “
convex combinations
Ana Sokolova Shonan 15-11-17
X
c
Ñ FX PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
what is it? how does it emerge? foundation ?
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
Shonan 15-11-17
X
c
Ñ FX PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
coalgebra over free convex algebra how does it emerge? foundation ?
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
Shonan 15-11-17
[Silva, Bonchi, Bonsangue, Rutten, FSTTCS’10]
CoAlgEMpT qpFq + CoAlgCpFTq 3 CoAlgCpFq
determinise forget needed: a lifting works for NFA not for generative PTS not for PA / belief-state transformer
Ana Sokolova Shonan 15-11-17
determinise forget needed: a lifting [Jacobs, Silva, S JCSS’15] [Silva, S. MFPS’11]
CoAlgEMpT qpGq + CoAlgCpTFq 3 CoAlgCpGq
works for generative PTS not for PA / belief-state transformer
Ana Sokolova Shonan 15-11-17
(Pc+1)A
is a quasi lifting and lax lifting of
C A
and
P A
Sets
determinise forget needed: a quasi lifting and a lax lifting works for PA /belief-state transformer
CoAlgEMpT qpHq + CoAlgCpFq 3 CoAlgCpGq
Ana Sokolova Shonan 15-11-17
X
c
Ñ FX PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
coalgebra over free convex algebra how does it emerge? foundation ?
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
Shonan 15-11-17
X
c
Ñ FX PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
via a generalised3 determinisation foundation ? coalgebra over free convex algebra
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
Shonan 15-11-17
X
c
Ñ FX PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
via a generalised3 determinisation are natural indeed coalgebra over free convex algebra
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
Shonan 15-11-17
R
bisimulation up-to convex hull
convex and equivalence closure of R to prove μ ~d ৵ it suffices to find a bisimulation up-to convex hull R with μ R ৵
„
„
μ’
a
μ ৵ ৵’
a
conv-conpRq there always exists a finite
Ana Sokolova Shonan 15-11-17
[S., Woracek JPAA’15] f.p. = f.g. for (positive) convex algebras
X
c
Ñ FX PA X ➝ (PD(X))A
x1
a
|
a
"
b
3
|
1 3" 1 2" 1 2
| x2
a
3 x3
b
k x4
b
k
a coalgebra over free convex algebra via a generalised3 determinisation are natural indeed sound proof method for distribution bisimilarity Thank You!
Ana Sokolova
1 3x1 ` 2 3x2
v ◆
a
) . . .
8 9x2 ` 1 9x3
. . .
2 3x2 ` 1 6x3 ` 1 6x4
Shonan 15-11-17