Fast LTL to Büchi Automata Translation

Transcription

1 (LTL2BA) April 19, 2013

2 (LTL2BA) Table of contents 1 (LTL2BA) 2 Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA 3

3 (LTL2BA) (LTL2BA) (TGBA)

4 (LTL2BA) Very Weak Alternating Automata (VWAA) Definition 6: A co-büchi very weak alternating co-büchi automaton is a five tuple A = (Q, Σ, δ, I, F) - Q is the set of states - Let Q be the set of conjunctions of elements of Q. The empty conjunction is denoted by tt. We identity Q with 2 Q in the following - Σ is the alphabet, and we let Σ = 2 Σ - δ : Q 2 Σ xq - I Q is the set of initial states - F Q is the set of final states (co-büchi) - There exists a partial order on Q such that q Q, all the states appearing in δ(q) are lower or equal to q very weak

5 (LTL2BA) Figure: Automaton A θ. Some states (right) are unaccessible, they will be removed. I = {GFp F(q G r)}, δ(p) = {(Σ p, tt)} where Σ p = {a Σ p a}, δ(gfp) = {(Σ p, GFp), (Σ, GFp Fp)}.

6 (LTL2BA) A run σ of A on a word u 0 u 1... Σ ω is labeled DAG(V, E, λ) - V is partitioned in V i with E V i xv i+1 i=0 - λ: V Q is the labeling function i=0 - λ(v 0 ) I and x V i, (a, e) δ(λ(x)), u i a and e = λ(e(x)). A run σ is accepting if any (infinite) branch in σ has only a infinite number of nodes labeled in F (co-büchi acceptance condition).

7 (LTL2BA) Figure: Automaton A θ. Some states (right) are unaccessible, they will be removed. Figure: Example of an accepting run of the automaton A θ.

8 (LTL2BA) Definition 7: - For J 1 J 2 2 Σ xq : J 1 J2 = (a 1 a 2, e 1 e 2 ) (a 1, e 1 ) J 1 and (a 2, e 2 ) J 2 - Let ψ be an LTL formula in positive normal form. We define ψ by: ψ = {{ψ}} if ψ is a temporal formula ψ 1 ψ 2 = {e 1 e 2 e 1 ψ 1 and e 2 ψ 2 } and ψ 1 ψ 2 = ψ 1 ψ 2

9 (LTL2BA) Let ϕ be an LTL formula on a set Prop. Define the VWAA A ϕ : - Q is the set of temporal subformulae of ψ - Σ = 2 Prop, I = ψ - F is the set of until subformulae of ψ - δ is defined as follows ( extends δ to all subformulae of ϕ) δ(tt) = {(Σ, tt)} δ(p) = {(Σ p, tt)} where Σ p = {a Σ p a} δ( p) = {(Σ p, tt)} where Σ p = Σ\Σ p δ(x ψ) = {(Σ, e) e ψ} δ(ψ 1 ψ 2 ) = (ψ 2 ) ( (ψ 1 ) {(Σ, ψ 1 ψ 2 )}) δ(ψ 1 Rψ 2 ) = (ψ 2 ) ( (ψ 1 ) {(Σ, ψ 1 Rψ 2 )}) (ψ) = δ(ψ) if ψ is a temporal formula (ψ 1 ψ 2 ) = (ψ 1 ) (ψ 2 ) (ψ 1 ψ 2 ) = (ψ 1 ) (ψ 2 )

10 (LTL2BA) Example Let ϕ = (GFp G(q Fr)) - ϕ def = (ff R(ttUp)) (ttu(q (ff R r))) - Transition function (have 10 states totally) δ( r) = {(Σ r, tt)} δ(ff R r) = {(Σ r, ff R r)} δ(q (ff R r)) = (q) (ff R r) = {(Σ r Σ q, ff R r)} δ(ttup) = {(Σ p, tt), (Σ, ttup)}... - Initial state I = ϕ = GFp F(q G r) - Final states F is formula of type ϕ 1 Uϕ 2 = {F(q G r), Fp}

11 (LTL2BA) Transition Based Generalized Büchi Automata (TGBA) Definition 8: A generalized Büchi automaton is a five-tuple G = (Q, Σ, δ, I, T ) where: - Q is the set of states - Σ is the alphabet, and we let Σ 2 Σ - δ : Q 2 Σ xq is the transition function - I Q is the set of initial states - T = {T 1,..., T r } where T j QxΣ xq are the accepting transitions

12 (LTL2BA) Figure: Automaton G Aθ, before (left) and after (right). A run σ of G on word u 0 u 1... Σ ω is a sequence q 0, q 1,... of elements of Q such that q 0 I and i 0, a i Σ such that u i a i and (a i, q i+1 ) δ(q i ). A run σ is accepting if for each 1 j r it uses infinitely many transitions from T j. L(G) is the set of words on which there exists accepting run of G.

13 (LTL2BA) Let A = (Q, Σ, δ, I, F ) be a VWAA with co-büchi acceptance conditions. We define the GBA G A = (Q, Σ, δ, I, T ) where: - Q = 2 Q is identified with conjunctions of states - δ (q 1... q n ) = n δ(q i ) i=1 - δ is the set of -minimal transition of δ where the relation is defined by t t if t = (e,a,e ), t = (e,a,e ), a a, e e, and T T, t T t T - T = {T f f F } where T f = {(e, α, e ) f / e or (β, e ) δ(f ), α β and f / e e }

14 (LTL2BA) Example Find δ (GFp F(q G r)) - δ (GFp F(q G r)) (Σ p, GFp F(q G r)) : T Fp, (Σ p Σ q Σ r, GFp G r) : T Fp, T F(q G r), (Σ, GFp Fp F(q G r)) :, (Σ q Σ r, GFp Fp G r) : T F(q G r) - δ (GFp F(q G r) (Σ p, GFp F(q G r)), (Σ p Σ q Σ r, GFp G r), (Σ, GFp Fp F(q G r)), (Σ q Σ r, GFp Fp G r)

15 (LTL2BA) Example Find δ (GFp Fp G r) - δ (GFp Fp G r) (Σ p Σ r, GFp G r) : T Fp, T F(q G r), (Σ p Σ r, GFp Fp F(q G r)) : T F(q G r), (Σ r, GFp Fp F(q G r)) : T F(q G r) - δ (GFp Fp G r) (Σ p Σ r, GFp G r), (Σ r, GFp Fp F(q G r))

16 (LTL2BA) Büchi Automata (BA) Definition 9: A Büchi automaton is a five-tuple B = (Q, Σ, δ, I, F ) - Q is the set of states - Σ is the alphabet, and we let Σ 2 Σ - δ : Q 2 Σ xq is the transition function - I Q is the set of initial states - F Q is the set of repeated states (Büchi condition) Let G = (Q, Σ, δ, I, F ) be a GBA with T = {T 1,..., T 2 }. We define the BA B G = (Qx{0,..., r}, Σ, δ, Ix{0}, Qx{r}) δ ((q, j)) = {α, (q{, j )) (α, q ) δ(q) and j = next(j, (q, α, q ))}. max{j i r j < k i, t T k }if j r with next(j,t) = max{j < i r 0 < k i, t T k }if j = r

17 (LTL2BA) Figure: Automaton B GAθ after. A run σ of B on a word u 0 u 1... Σ ω is a sequence q 0, q 1,... of elements of Q such that q 0 I and i 0, a i Σ such that u i a i and (a i, q i+1 ) δ(q i ). A run σ is accepting if there exists infinitely many states in F. L(B) is the set of words on which there exists an accepting run of B.

18 (LTL2BA) Example T = {T F(q G r), T Fp } δ(a) = {(Σ p, A), (Σ, A), (Σ p Σ q Σ r, B), (Σ q Σ r, B)} next(0, (A, Σ p, A)) = 0 because T Fp next(0, (A, Σ, A)) = 0 next(0, (Σ p Σ q Σ r, B)) = 2 because T F(q G r), T Fp next(0, (Σ q Σ r, B)) = 1 because T F(q G r) δ (A, 0) = {(Σ p, (A, 0)), (Σ, (A, 0)), (Σ p Σ q Σ r, (B, 2)), (Σ q Σ r, (B, 1))}

19 (LTL2BA) - A state that is not accessible can be removed, - If a transition t 1 implies a transition t 2, then t 2 can be removed. t 1 = (q, a 1, q 1 ) implies t 2 = (q, a 2, q 2 ) if In VWAA: a 2 a 1 and q 1 q 2 In GBA: a 2 a 1, q 1 = q 2 and t T, t 2 T t 1 T In BA: a 2 a 1 and q 1 = q 2 - If two states q 1 and q 2 are equivalent, then they can be merged. q 1 and q 2 are equivalent if In VWAA: δ(q 1 ) = δ(q 2 ) and q 1 F q 2 F In GBA: δ(q 1 ) = δ(q 2 ) and (a, q ) δ(q 1 ), T T, (q 1, a, q ) T (q 2, a, q ) T In BA: δ(q 1 ) = δ(q 2 ) and q 1 F q 2 F

20 (LTL2BA) Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA

21 (LTL2BA) Alternating Formula Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Lemma 1. - Every pure eventuality formula µ satisfies the following: ω Σ ω, u Σ : ω = µ uω = µ (left-append closed languages) - Every pure universality formula ν satisfies the following: ω Σ ω, u Σ : uω = ν ω = ν (suffix closed languages) Lemma 2. - Every alternating formula ε satisfies the following: ω Σ ω, u Σ : ω = ε uω = ε (prefix-invariant languages)

22 (LTL2BA) Alternating Formula Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Definition 10: Let ϕ ranges over general LTL formula. Define classes pure eventuality formula µ, pure universality formula ν, and alternating formula ε - µ ::= Fϕ µ µ µ µ Xµ ϕuµ µrµ Gµ - ν ::= Gϕ ν ν ν µ Xµ νuν ϕrν Fν - ε ::= Gµ Fν ε ε ε ε Xε ϕuε ϕrε Fε Gε Reduction of LTL formula. Let ϕ, ψ range over LTL formula and γ ranges over alternating ones. XϕRXψ X(ϕRψ), ϕuγ γ, Fγ γ, Xγ γ, Xϕ Xψ X(ϕ ψ), ϕrγ γ, Gγ γ

23 (LTL2BA) Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Given an input LTL formula ϕ, an equivalent VWAA is constructed as A ϕ = (Q, Σ, δ, I, F ), where Q, Σ, F are defined as in the original construction, I = {ϕ}

24 (LTL2BA) Example Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Figure: VWAA for (GFa)Ub generate by (a) the translation of LTL2BA, (b) LTL3BA translation with suspension.

25 (LTL2BA) Example Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Let ϕ = (GFa)Ub def = (ff R(ttUa))Ub - δ(ttua) = {(Σ, ttua), (Σ a, tt)} - δ(ff R(ttUa)) = {(Σ a, GFa), (Σ, GFa Fa)} - In LTL2BA: δ((gfa)ub) = (b)u( (GFa) {(Σ, (GFa)Ub)}) = {(Σ b, tt), (Σ a, GFa ((GFa)Ub)), (Σ, GFa Fa ((GFa)Ub))} - In LTL3BA (we have GFa is alternating): δ((gfa)ub) = (b)u({(σ, GFa)} {(Σ, (GFa)Ub)}) = {(Σ b, tt), (Σ, (GFa ((GFa)Ub))}

26 (LTL2BA) Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Optimization of VWAA If O 1 O 2 then replace the label a 2 in t 2 by a 2 a 1. If O 1 = O 2, replace both transitions by the transition (q, a 1 a 2, O 1 ). Figure: VWAA after apply generalized optimization rule.

27 (LTL2BA) Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Definition 12: (Progress formula). Let M be the minimal set containing all VWAA states of the form ψrp and all subformula of their right operands p. The VWAA states outside M, called progress formula.

28 (LTL2BA) Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Let A ϕ = (Q, Σ, δ, I, F ) be a VWAA. Define G A = (Q, Σ, δ, I, T ) - δ (O) = n δ O (q i ), where i=1 {(Σ, {q i })}if O contains a progress non-alternating formula and q i is an alternating formula, δ O (q i ) = or O contains a progress formula and q is an alternating non-progress formula δ(q i ) otherwise - T = {T f f F } where T f = {(O, α, O ) f / O or ( β, O ) δ(f ), (γ, O ) f O f δ(f ) such that f / O, α = β γ, and O = O O )}.

29 (LTL2BA) Example Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Let ψ = GFa Fb tt:{3} a:{2, 3} b:{2, 3} start {1, 2} {1} Figure: A VWAA A ψ tt:{2} Figure: A TGBA G ψ corresponding to the VWAA of Figure 3.

30 (LTL2BA) Example Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA Find δ (GF a F b ) - We have F b is progress formula and non-alternating, GF a is an alternating formula. - δ (GF a F b ) = δ(gf a ) δ(f b ) - In LTL2BA: δ(gf a ) = {(Σ, GF a )} - In LTL3BA: δ(gf a ) = {(Σ a, GF a ), (Σ, GF a F b )} δ (GF a F b ) = {(Σ b, GF a ), (Σ, GF a F b )} - (GF a F b, Σ b, GF a ) T Fb, T Fa - (GF a F b, Σ, GF a F b ) T Fa

31 (LTL2BA) Optimization of BA Alternating Formula Improvements in LTL to VWAA Translation Improvements in VWAA to TGBA Translation Optimization of BA New rule states q 1 and q 2 can be merged if δ(q 1 )[q 1 /r] = δ(q 2 )[q 2 /r], where r is a fresh artificial state and δ(q)[q/r] is a δ(q) with all occurrences of q as a target node replaced by r.

32 (LTL2BA) - The translation proceeds in three basic steps: LTL formula is translated into a very weak alternating automaton (VWAA) VWAA is then translated into a transition-based generalized Büchi automaton (TGBA) TGBA is transformed into Büchi automaton (BA) - Each of the three automata is simplified during the translation - LTL2BA implementation is available at following address: - PAT is using LTL2BA, replace LTL2BA by LTL3BA

33 (LTL2BA) References Paul Gastin and Denis Oddoux, Fast LTL to Büchi Automata Translation, Tom Babiak, Mojmr Ketnsk, Vojtch ehk, Jan Strejek, LTL to Büchi Automata Translation: Fast and More Deterministic*, Kousha Etessami and Gerard J.Holzmann, Optimizing Büchi automata, O.Kupferman and M.Vardi, Weak alternating automata are not that weak, 1997.