My presentation in the idea4cps.dk workshop held in Aalborg. Its about a semiring-valued temporal logic that me and Ugo Montanari developed some years ago. The logic is essentially a generalisation of CTL interpreted over absorptive semirings, an algebraic structure that is quite suitable to model quantitative aspects such as quality-of-service measures.

1. A Semiring-valued Temporal Logic
Alberto Lluch Lafuente
(based on joint-work with Ugo Montanari)
Meeting, 25-26 September 2014, Aalborg

3. NOTE: This presentation focuses on CTL and semiring multiplication as
conjunction/universal. Our paper considers μ-calculus and operators
based on the meet.

4. Disclaimers
This a 10-years aged work...

5. Disclaimers
This a 10-years aged work...
# doesn't mean I didn't work since then

6. 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

7. 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# probably outdated

8. Semiring Temporal Logics
ok for multicriteria
but a bit illogical*
(*) Some standard results of CTL
and μ-calculus do not lift.

9. Running Example

10. A B
AB
...possibly accessing the resource?
{A,B}
Id of those ...possibly keep accessing the resource? {A,B}

11. 0$
1$ 1$
2$
...possibly accessing the resource?
0 $
Price of ...possibly keep accessing the resource? ∞ $

12. 0
1 1
0.5
...possibly accessing the resource?
1
Certainty of ...possibly keep accessing the resource?
1

13. DOES ?

14. DOES ? TO WHAT EXTENT
A

15. ABSORPTIVE
SEMIRINGS
Bistarelli, S., Montanari, U., & Rossi, F. (1997). Semiring-based constraint
satisfaction and optimization. Journal of ACM, 44, 201–236.

17. Preferences
{A,B}
{A} {B}
Ø
<{A,B},⊆>

18. Preferences
1
0
<[1,0],≤>

19. Preferences
0
∞
(Nat,≥)
1
2

22. Multi-Criteria
{A,B}
{A} {B}
Ø

23. A
Ø
B (A,B)
X = (A,Ø) (Ø,B)
Ø
(Ø,Ø)

24. A
Ø
B (A,B)
X = (A,Ø) (Ø,B)
Ø
(Ø,Ø)
(A,Ø)⊔ (Ø,B)=(A,B)?

25. A
Ø
B (A,B)
X = (A,Ø) (Ø,B)
Ø
(Ø,Ø)
(A,B)
(A,Ø) (Ø,B)
(Ø,Ø)
(A,Ø) (Ø,B)
(Ø,Ø)
(Ø,Ø)
(Ø,B)
(Ø,Ø)
(A,Ø)
(Ø,Ø)
(A,Ø)⊔ (Ø,B)=(A,B)?

26. A
Ø
B (A,B)
X = (A,Ø) (Ø,B)
Ø
(Ø,Ø)
(A,B)
(A,Ø) (Ø,B)
(A,Ø) (Ø,B)
(Ø,Ø)
(A,Ø)⊔ (Ø,B)=(A,B)?

27. A
Ø
B (A,B)
X = (A,Ø) (Ø,B)
Ø
(Ø,Ø)
(A,B)
(A,Ø) (Ø,B)
(A,Ø) (Ø,B)
(Ø,Ø)
(A,Ø)⊔ (Ø,B)=(A,B)?
{(A,Ø)}⊔ {(Ø,B)}={(A,Ø),(Ø,B)}
Semiring recipe
for multi-criteria:
Hoare Power Domain of
Cartesian Product of
individual criteria semiring

28. SEMIRING-VALUED
CTL

30. f(φ,...,φ)

31. S

32. S
S
x x x

34. A B
AB
...possibly accessing the resource?
EFφ
{A,B}
Id (φ) of those ...possibly keep accessing the resource? {A,B}
EFEGφ

35. 0$
1$ 1$
2$
...possibly accessing the resource?
0 $
EFφ
Price (φ) of ...possibly keep accessing the resource?
∞ $
EFEGφ

36. 0
1 1
0.5
...possibly accessing the resource?
1
EFφ
Certainty (φ) of ...possibly keep accessing the resource?
EFEGφ
1

37. (Ø,0$,0)
({A},1$,1) ({B},1$,1)
({A,B},2$,
0.5)
(Ø,0$,0) ({A},1$,1)
({B},1$,1) ({A,B},2$,0.5)
...possibly accessing the resource?
EFφ
QoS (φ) of ...possibly keep accessing the resource?
({A},∞$,1) ({B},∞$,1)
EFEGφ ({A,B},∞$,0.5)

38. SOME
RESULTS

39. Minimal syntax?

40. Minimal syntax?
κ[⊥Rφ]
f(φ,...,φ)

42. x
≥

43. x

44. x
≥

45. What about model checking?
(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!

46. 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!

47. 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!

48. 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.

49. What about bisimulation?

50. What about bisimulation?
1 1
1
[| AX 1 |] = 1+1 = 2 = 1 = [| AX 1 |]
NOTE: We can use the logic to compute the out-degree of nodes.

51. What about generality?
(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]”.

52. CONCLUDING
REMARKS

53. Summary
(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.

54. Future Work
(1) Consider cost/rewards in
Stochastic Models?
(2) Study (bi)simulation
metrics/distances?

55. Semiring Temporal Logics
ok for multicriteria
but a bit illogical*
(*) Some standard results of CTL
and μ-calculus do not lift.

56. THANKS!

57. Questions?
albl@dtu.dk
albertolluch.com
Meeting, 25-26 September 2014, Aalborg