Abstract Interpretation for Probabilistic Termination of Biological Systems

ion of states. We adopt intervals of integers, I = {[m,n] | m ∈ N,n ∈ N∪{∞}∧m ≤ n}. Over intervals we consider the standard order ⊑I , such that I ⊑I J iff min(I),max(I)∈ J. Moreover, we use ⊔I for the corresponding l.u.b.. The abstract states are defined by replacing multiplicities with intervals of multiplicities. Therefore, an abstract state is a function M◦ : X → I . We also use M ◦ for the set of abstract states. Obviously, given a multiset M, there exists an abstract multiset M◦, which is its most precise approximation. Indeed, each multiplicity, such as n, can be replaced with the exact interval [n,n]; for simplicity, we may even use n as a shorthand of [n,n]. In the following, α(M) stands for the best abstraction of a multiset M. Moreover, we use M◦[I/X ] for denoting the abstract state where the abstract multiplicity of reagent X is replaced by the interval I ∈ I . We adopt abstract operations of sum and difference, such that ∀X ∈ X , M◦⊕◦N◦(X) = M◦(X)+ N◦(X), I + J = [min(I)+ min(J),max(I)+ max(J)] M◦⊖◦N◦(X) = M◦(X)−N◦(X), I− J = [min(I)−̂max(J),max(I)−̂min(J)] It is immediate to define the following approximation order over abstract states. Definition 4.1 (Order on States) Let M◦ 1 ,M ◦ 2 ∈ M ◦, we say that M◦ 1⊑ ◦M◦ 2 iff, for each reagent X ∈ X , M◦ 1(X) ⊑I M ◦ 2(X). 144 Abstract Interpretation for Probabilistic Termination of Biological Systems The relation between multisets and abstract states is formalized as a Galois connection [8]. The abstraction function α : P(M ) → M ◦ reports the best approximation for each set of multisets S; the l.u.b. (denoted by ⊔◦) of the best abstraction of each M ∈ S. Its counterpart is the concretization function γ : M ◦ →P(M ) which reports the set of multisets represented by an abstract state. We refer the reader to [5] for the properties of functions (α ,γ). Definition 4.2 We define α : P(M ) → M ◦ and γ : M ◦ → P(M ) such that, for each S ∈ P(M ) and M◦ ∈ M ◦: (i) α(S) = ⊔◦ M∈Sα(M); (ii) γ(M◦) = {M′ | α(M′)⊑◦M◦}. Abstract transitions. The semantics of [5] uses abstract transitions of the form M◦ 1 Θ,∆◦,r −−−→ ◦ M◦ 2 where Θ ∈ L̂ , ∆◦ ∈ Q̂◦ = I ∪ (I ×I ), with arity(Θ) = arity(∆◦). Similarly as in the concrete case, Θ reports the label (the labels) of the basic action (actions), ∆◦ reports consistent information about the possible multiplicities, while r is the rate. In the proposed approach, such a transition is intended to approximate all the concrete moves, corresponding to label Θ, for each multiset M1 approximated by the abstract state M◦ 1 . This means that there exists a concrete transition M1 Θ,∆,r −−−→ M2, where the multiplicity (multiplicities) ∆ is included in the interval (intervals) ∆◦, and M2 is approximated by the abstract state M◦ 2 . Let us consider the environment E commented in Example 3.4 and a very simple abstract state such as M◦ 0 = {([1,2],X),([1,2],Y )}. The abstract state M ◦ 0 describes a set of experiments; thus, the abstract semantics has to model the system described by E , w.r.t. different initial concentrations. For approximating the duplication of X , i.e. the synchronization between X and Y along channel a, we would obtain M 0 (λ ,μ),([1,2],[1,2]),r −−−−−−−−−−→ ◦ M 0 ′ with M◦ 0 ′ = {([2,3],X),([0,1],Y )}. In this way, however, a hybrid state M◦ 0 ′ is introduced. Actually, M◦ 0 ′ represents terminated multisets, where the concentration of reagent Y is zero, as well as non terminated multisets, where reagent Y is still available. It should be clear that the moves corresponding to (λ ,μ) could be better approximated by adopting two different abstract transitions, M◦ 0 (λ ,μ),([1,2],[2,2]),r −−−−−−−−−−→ ◦ M◦ 1 (a) M ◦ 0 (λ ,μ),([1,2],[1,1]),r −−−−−−−−−−→ ◦ M◦ 3 (b) where M◦ 1 = {([2,3],X),([1,1],Y )} and M ◦ 3 = {([2,3],X),([0,0],Y )}. In this representation the labels capture a relevant information because they express a conflict. Actually, each multiset represented by M◦ 0 , realizes a move corresponding to (λ ,μ) which is abstracted either by transition (a) or by transition (b). Table 3 presents the refined abstract transition rules (as usual, w.r.t. a given environment E). The rules are derived from the concrete ones, by replacing multiplicities with intervals of multiplicities. The following operators are applied both to the target state and to the intervals, appearing in the transition labels, in order to properly split the intervals, such as [0,n]. For X ∈ X , we define א(X) = {(X = 0),(X > 0)}. Then, given an abstract state M◦ ∈ M and ♯ ∈ א(X) we define ▽(M) =    M◦[[0,0]/X ] if ♯ = (X = 0),M◦(X) = [0,n],n > 0 M◦[[1,n]/X ] if ♯ = (X > 0),M◦(X) = [0,n],n > 0 M◦ otherwise Roberta Gori & Francesca Levi 145 (Delay-a) E.X .λ = τr .Q ♯ ∈ א(X) M◦ λ ,(M(X)),r −−−−−−−→ ◦ ▽♯((M◦⊖◦{(1,X)})⊕◦α([[Q]])) (Sync-a) E.X .λ = ar .Q1 E.Y.μ = ār .Q2 ♯1 ∈ א(X) ♯2 ∈ א(Y ) M◦ (λ ,μ),((M◦(X))♯1 ,(M◦(Y ))♯2 ),r −−−−−−−−−−−−−−−−−→ ◦ ▽♯1,♯2(((M◦⊖◦{(1,X)})⊖◦{(1,Y )})⊕α([[Q1]])⊕α([[Q2]])) Table 3: Abstract transition relation With an abuse of notation, we may write ▽♯1,♯2(M◦) in place of ▽♯1(▽♯2(M◦)). Similarly, for an interval I = [n,m] ∈ I and ♯ ∈ א(X), I =    [n,1] if ♯ = (X = 0),n ≤ 1, [2,m] if ♯ = (X > 0),n ≤ 1,m ≥ 2, I otherwise. In the following we use LT S ◦ to denote the set of abstract LTS. We also assume that all notations defined for LTS are adapted in the obvious way. Hence, we write LTS((E,M◦ 0)) = (S ,→◦, M◦ 0 ,E) for the abstract LTS, obtained for the initial abstract state M◦ 0 by transitive closure. For the sake of simplicity we have presented an approximation where the number of states may be infinite. Further approximations can be easily derived by means of widening operators (see [5]).