Méthodes logico-numériques pour la vérification des systèmes discrets et hybrides. (Logico-Numerical Verification Methods for Discrete and Hybrid Systems)

domains (§3.3) are often complete lattices, i.e., algebras over partially ordered sets. We recall in this section the fundamental notions of lattice theory, which are the basis for the soundness of over-approximations of reachable state sets computed by abstract interpretation methods. For further details and proofs we refer to textbooks on lattice theory [DP90] and static analysis [NNH05]. From partial orders to complete lattices Definition 3.3 (Partial order) A partial order xS,Ďy is a set S equipped with a reflexive, transitive and anti-symmetric binary relation Ď: S ˆ S. For example, the powerset of integers with the set inclusion x℘pZq,Ďy is a partial order. Partial orders can be visualized with the help of Hasse diagrams. Definition 3.4 (Upper and lower bounds) Let S1 Ď S: – Upper bound s P S: @s1 P S1 : s1 Ď s – Lower bound s P S: @s1 P S1 : s1 Ě s – Least upper bound (also called join or union operator) s “ Ů S1: for all upper bounds s1 P S1: s Ď s1 – Greatest lower bound (also called meet or intersection operator) s “ Ű S1: for all lower bounds s1 P S1: s Ě s1 For the powerset of integers x℘pZq,Ďy, we can define these operations as the usual intersection X and union Y for sets. For example, take S1 “ tt1, 2, 3u, t2, 4, 5u, t2, 5uu, then Ů S1 “t1, 2, 3u \ t2, 4, 5u \ t2, 5u“t1, 2, 3, 4, 5u, and Ű S1 “t2u. Abstract domains are often complete lattices: Definition 3.5 (Complete lattice) A complete lattice is an algebraic structure xS,Ď ,\,[,K,Jy with a partially ordered set S such that – all subsets have least upper and greatest lower bounds, – the least element K “ Ů H “ Ű S, and – the greatest element J “ Ű H “ Ů S. The powerset of integers is a complete lattice with K “ H and J “ Z.