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A Semiring-valued Temporal Logic 
Alberto Lluch Lafuente 
(based on joint-work with Ugo Montanari) 
Meeting, 25-26 Septemb...
NOTE: This presentation focuses on CTL and semiring multiplication as 
conjunction/universal. Our paper considers μ-calcul...
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 miles...
Disclaimers 
This a 10-years aged work... 
# doesn't mean I didn't work since then 
# I am not pretending it to be a miles...
Semiring Temporal Logics 
ok for multicriteria 
but a bit illogical* 
(*) Some standard results of CTL 
and μ-calculus do ...
Running Example
A B 
AB 
...possibly accessing the resource? 
{A,B} 
Id of those ...possibly keep accessing the resource? {A,B}
0$ 
1$ 1$ 
2$ 
...possibly accessing the resource? 
0 $ 
Price of ...possibly keep accessing the resource? ∞ $
0 
1 1 
0.5 
...possibly accessing the resource? 
1 
Certainty of ...possibly keep accessing the resource? 
1
DOES ?
DOES ? TO WHAT EXTENT 
A
ABSORPTIVE 
SEMIRINGS 
Bistarelli, S., Montanari, U., & Rossi, F. (1997). Semiring-based constraint 
satisfaction and opti...
Preferences 
{A,B} 
{A} {B} 
Ø 
<{A,B},⊆>
Preferences 
1 
0 
<[1,0],≤>
Preferences 
0 
∞ 
(Nat,≥) 
1 
2
Multi-Criteria 
{A,B} 
{A} {B} 
Ø
A 
Ø 
B (A,B) 
X = (A,Ø) (Ø,B) 
Ø 
(Ø,Ø)
A 
Ø 
B (A,B) 
X = (A,Ø) (Ø,B) 
Ø 
(Ø,Ø) 
(A,Ø)⊔ (Ø,B)=(A,B)?
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 
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 
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)}={(...
SEMIRING-VALUED 
CTL
f(φ,...,φ)
S
S 
S 
x x x
A B 
AB 
...possibly accessing the resource? 
EFφ 
{A,B} 
Id (φ) of those ...possibly keep accessing the resource? {A,B} 
...
0$ 
1$ 1$ 
2$ 
...possibly accessing the resource? 
0 $ 
EFφ 
Price (φ) of ...possibly keep accessing the resource? 
∞ $ 
...
0 
1 1 
0.5 
...possibly accessing the resource? 
1 
EFφ 
Certainty (φ) of ...possibly keep accessing the resource? 
EFEGφ...
(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 t...
SOME 
RESULTS
Minimal syntax?
Minimal syntax? 
κ[⊥Rφ] 
f(φ,...,φ)
x 
≥
x
x 
≥
What about model checking? 
(1) For distributive semi-rings (x idempotent), 
doable via iterations (fixpoint semantics ok)...
What about model checking? 
(1) For distributive semirings (x idempotent), 
doable via iterations (fixpoint semantics ok);...
What about model checking? 
(1) For distributive semirings (x idempotent), 
doable via iterations (fixpoint semantics ok);...
What about model checking? 
(1) For distributive semirings (x idempotent), 
doable via iterations (fixpoint semantics ok);...
What about bisimulation?
What about bisimulation? 
1 1 
1 
[| AX 1 |] = 1+1 = 2 = 1 = [| AX 1 |] 
NOTE: We can use the logic to compute the out-deg...
What about generality? 
(1) Graph problems: e.g. reachability, 
(multi-criteria) path optimization, etc. 
(2) (Quasi)-bool...
CONCLUDING 
REMARKS
Summary 
(1) We lifted CTL & μ-calculus to absorptive 
Semirings. 
(2) In the general case: no adequacy, 
fixpoint and pat...
Future Work 
(1) Consider cost/rewards in 
Stochastic Models? 
(2) Study (bi)simulation 
metrics/distances?
Semiring Temporal Logics 
ok for multicriteria 
but a bit illogical* 
(*) Some standard results of CTL 
and μ-calculus do ...
THANKS!
Questions? 
albl@dtu.dk 
albertolluch.com 
Meeting, 25-26 September 2014, Aalborg
A Semiring-valued Temporal Logic
A Semiring-valued Temporal Logic
A Semiring-valued Temporal Logic
A Semiring-valued Temporal Logic
A Semiring-valued Temporal Logic
A Semiring-valued Temporal Logic
A Semiring-valued Temporal Logic
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A Semiring-valued Temporal Logic

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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.

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A Semiring-valued Temporal Logic

  1. 1. A Semiring-valued Temporal Logic Alberto Lluch Lafuente (based on joint-work with Ugo Montanari) Meeting, 25-26 September 2014, Aalborg
  2. 2. NOTE: This presentation focuses on CTL and semiring multiplication as conjunction/universal. Our paper considers μ-calculus and operators based on the meet.
  3. 3. Disclaimers This a 10-years aged work...
  4. 4. Disclaimers This a 10-years aged work... # doesn't mean I didn't work since then
  5. 5. 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
  6. 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 # probably outdated
  7. 7. Semiring Temporal Logics ok for multicriteria but a bit illogical* (*) Some standard results of CTL and μ-calculus do not lift.
  8. 8. Running Example
  9. 9. A B AB ...possibly accessing the resource? {A,B} Id of those ...possibly keep accessing the resource? {A,B}
  10. 10. 0$ 1$ 1$ 2$ ...possibly accessing the resource? 0 $ Price of ...possibly keep accessing the resource? ∞ $
  11. 11. 0 1 1 0.5 ...possibly accessing the resource? 1 Certainty of ...possibly keep accessing the resource? 1
  12. 12. DOES ?
  13. 13. DOES ? TO WHAT EXTENT A
  14. 14. ABSORPTIVE SEMIRINGS Bistarelli, S., Montanari, U., & Rossi, F. (1997). Semiring-based constraint satisfaction and optimization. Journal of ACM, 44, 201–236.
  15. 15. Preferences {A,B} {A} {B} Ø <{A,B},⊆>
  16. 16. Preferences 1 0 <[1,0],≤>
  17. 17. Preferences 0 ∞ (Nat,≥) 1 2
  18. 18. Multi-Criteria {A,B} {A} {B} Ø
  19. 19. A Ø B (A,B) X = (A,Ø) (Ø,B) Ø (Ø,Ø)
  20. 20. A Ø B (A,B) X = (A,Ø) (Ø,B) Ø (Ø,Ø) (A,Ø)⊔ (Ø,B)=(A,B)?
  21. 21. 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)?
  22. 22. 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)?
  23. 23. 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
  24. 24. SEMIRING-VALUED CTL
  25. 25. f(φ,...,φ)
  26. 26. S
  27. 27. S S x x x
  28. 28. A B AB ...possibly accessing the resource? EFφ {A,B} Id (φ) of those ...possibly keep accessing the resource? {A,B} EFEGφ
  29. 29. 0$ 1$ 1$ 2$ ...possibly accessing the resource? 0 $ EFφ Price (φ) of ...possibly keep accessing the resource? ∞ $ EFEGφ
  30. 30. 0 1 1 0.5 ...possibly accessing the resource? 1 EFφ Certainty (φ) of ...possibly keep accessing the resource? EFEGφ 1
  31. 31. (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)
  32. 32. SOME RESULTS
  33. 33. Minimal syntax?
  34. 34. Minimal syntax? κ[⊥Rφ] f(φ,...,φ)
  35. 35. x ≥
  36. 36. x
  37. 37. x ≥
  38. 38. 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!
  39. 39. 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!
  40. 40. 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!
  41. 41. 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.
  42. 42. What about bisimulation?
  43. 43. 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.
  44. 44. 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]”.
  45. 45. CONCLUDING REMARKS
  46. 46. 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.
  47. 47. Future Work (1) Consider cost/rewards in Stochastic Models? (2) Study (bi)simulation metrics/distances?
  48. 48. Semiring Temporal Logics ok for multicriteria but a bit illogical* (*) Some standard results of CTL and μ-calculus do not lift.
  49. 49. THANKS!
  50. 50. Questions? albl@dtu.dk albertolluch.com Meeting, 25-26 September 2014, Aalborg

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