linear temporal logic for regular cost functions

Linear temporal logi for regular ost fun tionsDenis KuperbergLiafa/CNRS/Université Paris 7, Denis Diderot, Fran eSéminaire thésards23/02/2011

Introdu tion Cost fun tion : ounting extension of languages

Introdu tion Cost fun tion : ounting extension of languages LTL≤ : simple way to dene ost fun tions

Introdu tion Cost fun tion : ounting extension of languages LTL≤ : simple way to dene ost fun tions Translation from LTL≤ to automata

Introdu tion Cost fun tion : ounting extension of languages LTL≤ : simple way to dene ost fun tions Translation from LTL≤ to automata Algebrai hara terization of LTL≤-denable ost fun tions

OutlineIntrodu tionCounting events in wordsB-automataCost fun tionsQuantitative Linear temporal logi DenitionSemanti sAlgebrai hara terizationStabilization semigroups

B-automataAim : To represent fun tions A∗ −→ N ∪ ∞ with automata.B-automaton :

nondeterministi nite-state nite set of ounters, ranging over N, initial value 0 ea h transition perform a tions on ea h ounterAtomi a tions : in rement (i), reset (r), he k ( ).

Semanti s of B-automataCost fun tion asso iated with a B-automton A :[[A]]B(u) = infn/ there is a run with maximal he k value nwith inf ∅ = ∞.Example

[[A]]B = | · |a and [[A′]]B = minblo ka : u 7→ minn/an fa tor of u0a : i b : ε

1 2 3a, b : r b : ra : i b : ra, b : r

More on B-automataRemark : A standard automaton A omputing L an be viewed as aB-automaton without any ounter.Then [[A]]B = χL with χL(u) = 0 if u ∈ L∞ if u /∈ LTheorem ([Krob 94)The equivalen e of two B-automata is unde idable.Solution : Loosing some pre ision on the ounting, but keepinginformation about bounds.

Cost fun tionsIf f , g : A∗ → N ∪ ∞, thenf ≈ g if ∀X ⊆ A∗, f |X bounded ⇔ g |X bounded.i ∃ α : N → N su h as f ≤ α g and g ≤ α f (with α(∞) = ∞)Cost fun tion : equivalen e lass for ≈ relation.ExampleFor A = a, b, ,max(| · |a, | · |b) ≈ | · |a + | · |b but | · |a 6≈ maxblo ka

Known results on B-automataExtension of the notion of language via χL : L 6= L′ =⇒ χL 6≈ χL′ .Theorem (Col ombet 09)It is de idable whether two B-automata ompute the same ostfun tion (modulo ≈).Automata- omputable ost fun tions are alled regular.ExampleχL is a regular ost fun tion i L is a regular language.

Introdu tionCounting events in wordsB-automataCost fun tionsQuantitative Linear temporal logi DenitionSemanti sAlgebrai hara terizationStabilization semigroups

LTL on A des ribes regular languages :ϕ := a | Ω | ϕ ∧ ϕ | ϕ ∨ ϕ | Xϕ | ϕUϕwhere Ω marks the end of the word.


LTL≤ on A des ribes regular ost fun tions :ϕ := a | Ω | ϕ ∧ ϕ | ϕ ∨ ϕ | Xϕ | ϕUϕ | ϕU≤Nϕ



ϕU≤Nψ means that ψ is true somewhere in the future, and ϕis false at most N times until then.



ϕU≤Nψ means that ψ is true somewhere in the future, and ϕis false at most N times until then. The variable N is unique, and is shared by all o urren es ofU≤N operator.

Semanti s of LTL≤Satisability with integer parameter :We write (u, n) |= ϕ to signify that u ∈ A∗ satises the formula ϕ,with n ∈ N as value for all the o uren es of N in ϕ.Example

(aabb b a a b, 5) |= (aU≤Nb)UΩ


(aabb b a a b, 5) |= (aU≤Nb)UΩFrom formula to ost fun tion :[[ϕ]] is the ost fun tion asso iated to ϕ, dened by

[[ϕ]](u) = infn ∈ N, (u, n) |= ϕExampleFor all u ∈ a, b∗, we have [[bU≤NΩ]](u) = |u|a.

Introdu tionCounting events in wordsB-automataCost fun tionsQuantitative Linear temporal logi DenitionSemanti sAlgebrai hara terizationStabilization semigroups

Algebrai hara terization of regular ost fun tionsReminder : regular language ⇔ nite semigroup (Myhill) LTL⇔ star-free ⇔ aperiodi semigroup (S hützenberger) Nerode equivalen e for L : u ∼L v if

∀x , y ∈ A∗, xuy ∈ L ⇔ xvy ∈ L

Algebrai hara terization of regular ost fun tionsReminder : regular language ⇔ nite semigroup (Myhill) LTL⇔ star-free ⇔ aperiodi semigroup (S hützenberger) Nerode equivalen e for L : u ∼L v if

∀x , y ∈ A∗, xuy ∈ L ⇔ xvy ∈ LStabilization semigroup : S = 〈S , ·,≤, ♯〉, ordered semigroup with a

♯-operator : stabilization over idempotents (e = e · e).e♯ means "e repeated a lot of times".

Theorem (Col ombet 09)Stabilization semigroups re ognize exa tly the set of regular ostfun tions, and translations to or from B-automata are ee tive.

Theorem (Col ombet 09)Stabilization semigroups re ognize exa tly the set of regular ostfun tions, and translations to or from B-automata are ee tive.Theorem (Col ombet, Lombardy, K. 10)For all regular ost fun tion f , there is a quotient-wise minimalstabilization semigroup re ognizing f , and it an be omputed froma des ription of f (by automata or stabilization semigroup).

Theorem (Col ombet 09)Stabilization semigroups re ognize exa tly the set of regular ostfun tions, and translations to or from B-automata are ee tive.Theorem (Col ombet, Lombardy, K. 10)For all regular ost fun tion f , there is a quotient-wise minimalstabilization semigroup re ognizing f , and it an be omputed froma des ription of f (by automata or stabilization semigroup).TheoremWe an dene an analog to the Nerode equivalen e su h that for allregular ost fun tion f , the set of equivalen e lasses is isomorphi to the minimal stabilization semigroup re ognizing f .

Theorem (Col ombet 09)Stabilization semigroups re ognize exa tly the set of regular ostfun tions, and translations to or from B-automata are ee tive.Theorem (Col ombet, Lombardy, K. 10)For all regular ost fun tion f , there is a quotient-wise minimalstabilization semigroup re ognizing f , and it an be omputed froma des ription of f (by automata or stabilization semigroup).TheoremWe an dene an analog to the Nerode equivalen e su h that for allregular ost fun tion f , the set of equivalen e lasses is isomorphi to the minimal stabilization semigroup re ognizing f .TheoremAperiodi stabilization semigroups re ognize exa tly the set ofLTL≤-denable ost fun tions, hen e this lass is de idable.

Con lusionSummary Denition of LTL≤ to easily des ribe ost fun tions Translation from LTL≤ to B-automata Synta ti ongruen e for ost fun tions Algebrai hara terization and de idability of the lass ofLTL≤-denable ost fun tions.Future work Extension to innite words, trees,. . . Other hara terizations of this lass by rst-order logi ,star-free expressions,. . .

linear temporal logic for regular cost functions

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