57
A Semiring-valued Temporal Logic Alberto Lluch Lafuente (based on joint-work with Ugo Montanari) Meeting, 25-26 September 2014, Aalborg
| Date post: | 27-Jun-2015 |
| Category: |
Science |
| Upload: | alberto-lluch-lafuente |
| View: | 257 times |
| Download: | 1 times |
A Semiring-valued Temporal Logic
Alberto Lluch Lafuente(based on joint-work with Ugo Montanari)
Meeting, 25-26 September 2014, Aalborg
NOTE: This presentation focuses on CTL and semiring multiplication as conjunction/universal. Our paper considers μ-calculus and operators based on the meet.
Disclaimers
This a 10-years aged work...
Disclaimers
This a 10-years aged work...
# doesn't mean I didn't work since then
Disclaimers
This a 10-years aged work...
# doesn't mean I didn't work since then
# I am not pretending it to be a milestone
Disclaimers
This a 10-years aged work...
# doesn't mean I didn't work since then
# I am not pretending it to be a milestone
# probably outdated
Semiring Temporal Logics
ok for multicriteria
but a bit illogical*
(*) Some standard results of CTLand μ-calculus do not lift.
Running Example
BA
AB
...possibly accessing the resource?
...possibly keep accessing the resource?Id of those {A,B}{A,B}
0$
1$1$
2$
...possibly accessing the resource?
...possibly keep accessing the resource?Price of 0 $
∞ $
0
11
0.5
...possibly accessing the resource?
...possibly keep accessing the resource?Certainty of1
1
DOES ?
DOES ?TO WHAT EXTENT
A
ABSORPTIVESEMIRINGS
Bistarelli, S., Montanari, U., & Rossi, F. (1997). Semiring-based constraintsatisfaction and optimization. Journal of ACM, 44, 201–236.
{A,B}
Ø
{B}{A}
Preferences
<{A,B},⊆ >
1
0
Preferences
<[1,0],≤>
0
∞
Preferences
(Nat,≥)
1
2
{A,B}
Ø
{B}{A}
Multi-Criteria
Ø
A
Ø
B (A,B)
(Ø,Ø)
(Ø,B)(A,Ø)X =
Ø
A
Ø
B (A,B)
(Ø,Ø)
(Ø,B)(A,Ø)X =
(A,Ø)⊔ (Ø,B)=(A,B)?
Ø
A
Ø
B (A,B)
(Ø,Ø)
(Ø,B)(A,Ø)X =(A,B)
(Ø,Ø)
(Ø,B)(A,Ø)
(Ø,Ø)
(Ø,B)(A,Ø)
(Ø,Ø)
(Ø,B)
(Ø,Ø)
(A,Ø)
(Ø,Ø)
(A,Ø)⊔ (Ø,B)=(A,B)?
Ø
A
Ø
B (A,B)
(Ø,Ø)
(Ø,B)(A,Ø)X =(A,B)
(Ø,B)(A,Ø)
(Ø,B)(A,Ø)
(Ø,Ø)
(A,Ø)⊔ (Ø,B)=(A,B)?
Ø
A
Ø
B (A,B)
(Ø,Ø)
(Ø,B)(A,Ø)X =(A,B)
(Ø,B)(A,Ø)
(Ø,B)(A,Ø)
(Ø,Ø)
(A,Ø)⊔ (Ø,B)=(A,B)?
{(A,Ø)}⊔ {(Ø,B)}={(A,Ø),(Ø,B)}
Semiring recipefor multi-criteria:Hoare Power Domain of Cartesian Product of individual criteria semiring
SEMIRING-VALUEDCTL
f(φ,...,φ)
S
S
S
x x x
BA
AB
...possibly accessing the resource?
...possibly keep accessing the resource?Id (φ) of those {A,B}
EFEGφ
EFφ
{A,B}
0$
1$1$
2$
...possibly accessing the resource?
...possibly keep accessing the resource?Price (φ) of 0 $
∞ $
EFEGφ
EFφ
0
11
0.5
...possibly accessing the resource?
...possibly keep accessing the resource?Certainty (φ) of1
EFEGφ
EFφ
1
(Ø,0$,0)
({B},1$,1)({A},1$,1)
({A,B},2$,0.5)
...possibly accessing the resource?
...possibly keep accessing the resource?QoS (φ) of
(Ø,0$,0) ({A},1$,1)({B},1$,1) ({A,B},2$,0.5)
({A},∞$,1) ({B},∞$,1)({A,B},∞$,0.5)EFEGφ
EFφ
SOMERESULTS
Minimal syntax?
Minimal syntax?
κ[⊥Rφ]f(φ,...,φ)
x
≥
x
x
≥
(1) For distributive semi-rings (x idempotent), doable via iterations (fixpoint semantics ok);
(2) For ECTL fragment via (old) graph problems, e.g. algebraic path problem, shortest paths, etc.;
(3) For the general case... I don't know!
What about model checking?
(1) For distributive semirings (x idempotent), doable via iterations (fixpoint semantics ok);
(2) For ECTL fragment via (old) graph problems, e.g. algebraic path problem, shortest paths, etc.;
(3) For the general case... I don't know!
What about model checking?
(1) For distributive semirings (x idempotent), doable via iterations (fixpoint semantics ok);
(2) For ECTL fragment via (old) graph problems, e.g. algebraic path problem, shortest paths, etc.;
(3) For the general case... I don't know!
What about model checking?
(1) For distributive semirings (x idempotent), doable via iterations (fixpoint semantics ok);
(2) For ECTL fragment via (old) graph problems, e.g. algebraic path problem, shortest paths, etc.;
(3) For the general case... we still don't know.
What about model checking?
What about bisimulation?
What about bisimulation?
11
[| AX 1 |] = 1+1 = 2 = 1 = [| AX 1 |]
1
NOTE: We can use the logic to compute the out-degree of nodes.
(1) Graph problems: e.g. reachability, (multi-criteria) path optimization, etc.
(2) (Quasi)-boolean model checking: e.g. “Multi-valued CTL” [Chechik et al,03].
(3) Quantitative model checking approaches: e,.g. “Fuzzy CTL” [de Alfaro et al.,03], “Discounted CTL [de Alfaro et al., 04]”.
What about generality?
CONCLUDINGREMARKS
(1) We lifted CTL & μ-calculus to absorptive Semirings.
(2) In the general case: no adequacy, fixpoint and path semantics disagree...
(3) We let some open parenthesis, e.g. model checking algorithms.
NOTE: This presentation focuses on CTL and semiring multiplication as conjunction/universal. Our paper considers μ-calculus and operators based on the meet.
Summary
(1) Consider cost/rewards in Stochastic Models?
(2) Study (bi)simulation metrics/distances?
Future Work
Semiring Temporal Logics
ok for multicriteria
but a bit illogical*
(*) Some standard results of CTLand μ-calculus do not lift.
THANKS!
Questions?
Meeting, 25-26 September 2014, Aalborg
