Transparents

Fondements pour la v´erification
des syst`emes temps-r´eel et concurrents
Lecture 3
Alternation and LTL extensions
St´ephane Demri
October 8th, 2007

Summary from previous lecture
Bφ= (Σ,S,S0, ρ, F1,...,Fk)
◮Sis the set of maximally consistent sets wrt φ,
◮Σ = P(PROP),
◮S0={X ∈ S:φ∈ X },
◮Y ∈ ρ(X,a) iff
◮X ∩ PROP =a,
◮for Xψ∈cl(φ), Xψ∈ X iff ψ∈ Y,
◮If ψ1Uψ′
1,...,ψkUψ′
koccurs in φ, then
Fi
def
={X ∈ S:either ψiUψ′
i6∈ X or ψ′ i∈ X }
◮If U does not occur in φ, then k= 1 and F1=S.
St´ephane Demri Fondements pour la v´erification des syst`emes temps-r´eel et concurrents

Simple complexity properties
◮L(Bφ) = Models(φ).
◮Checking whether X ⊆ cl(φ) belongs to S[resp. S0,F1, . . . ,
Fk] can be done in polynomial-time in |φ|.
◮Checking whether Y ∈ ρ(X,a) can be done in
polynomial-time in |φ|.
◮|S|is in 2O(|φ|).
◮Elements in Scan be encoded in polynomial-space in |φ|.
St´ephane Demri Fondements pour la v´erification des syst`emes temps-r´eel et concurrents

NPSpace algorithm
1. Guess s0∈S0,s1∈F1, . . . , sk∈Fk;
2. i:= 0; s:= s0(current state);
3. While s6=s1and i<|S|do
3.1 Guess s′such that sa −→ s′for some a∈Σ;
3.2 i:= i+ 1; s:= s′.
4. If s6=s1, then abort otherwise
4.1 i:= 0; j:= 2;
4.2 While i:= 0 or (j6= 1 and i<|S| × k) do
4.2.1 Guess s′such that sa −→ s′for some a∈Σ;
4.2.2 i:= i+ 1; s:= s′.
4.2.3 if s′∈Fjthen nondeterministically choose either
j:= (j mod k) + 1 or skip;
4.3 If s=s1, then accept, otherwise abort.
St´ephane Demri Fondements pour la v´erification des syst`emes temps-r´eel et concurrents

Complexity
◮Bφis in exponential size in |φ|.
◮Testing on-the-fly the nonemptiness of Bφcan be done in
NPSpace.
◮By Savitch’s theorem: NPSpace =PSpace.
◮Satisfiability for LTL is in PSpace.
St´ephane Demri Fondements pour la v´erification des syst`emes temps-r´eel et concurrents