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
2.
3. NOTE: This presentation focuses on CTL and semiring multiplication as
conjunction/universal. Our paper considers μ-calculus and operators
based on the meet.
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
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.
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]”.
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.