MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

66 Robert Goldblattmodalities 〈 i 〉 for i ∈ I. The box modality [ i ] is defined to be ¬〈 i 〉¬. The relationp |= α, meaning “process p satisfies formula α”, is defined inductively, withp |= 〈 i 〉α iff for some i-experiment 〈p, p ′ 〉, p ′ |= α.Two processes are regarded as equivalent if there is no observable action that eithercan perform to distinguish them. Informally this means that to each observableaction that one can perform there is an action that the other can perform whichleads to an equivalent outcome, so each process can “simulate” the other. Spellingthis out,p is equivalent to q if, and only if,1. for every result p ′ of an experiment on p, there is an equivalent resultq ′ of an experiment on q; and2. for every result q ′ of an experiment on q, there is an equivalent resultp ′ of an experiment on p[Milner, 1980, p. 41]. As a definition of equivalence this appears to be circular,since the word “equivalence” occurs on both sides of the “if and only if”. Toformalise the idea, a sequence of equivalence relations ∼ n for n ≥ 0 is defined onP . For each relation S ⊆ P × P , define a relation E(S) by putting 〈p, q〉 ∈ E(S)if for every i ∈ I,1. 〈p, p ′ 〉 ∈ R i implies, for some q ′ , 〈q, q ′ 〉 ∈ R i and 〈p ′ , q ′ 〉 ∈ S; and2. 〈q, q ′ 〉 ∈ R i implies, for some p ′ , 〈p, p ′ 〉 ∈ R i and 〈p ′ , q ′ 〉 ∈ S.Put p ∼ 0 q for all p, q ∈ P , and inductively p ∼ n+1 q if 〈p, q〉 ∈ E(∼ n ). Thenp and q are defined to be observationally equivalent, written p ∼ q, if p ∼ n q forevery n.Now a relation R ⊆ P × P is image-finite if the set {p ′ : 〈p, p ′ 〉 ∈ R} is finite foreach p ∈ P . Hennessy and Milner gave a logical characterisation of observationalequivalence by showing that if each R i is image-finite, two processes are equivalentiff they satisfy the same formulas:p ∼ q iff for all formulas α, p |= α iff q |= α. (∗)Note that the operator E on relations is monotonic: R ⊆ S implies E(R) ⊆ E(S).This property implies, by induction, that ∼ n+1 ⊆ ∼ n , and so iteration of Egenerates a decreasing chain of relations∼ 0 ⊇ ∼ 1 ⊇ ∼ 2 ⊇ · · · ⊇ ∼ n ⊇ · · · · · ·Let ∼ ω = ⋂ {∼ n : n ≥ 0} be the intersection of the chain. Then in the image-finitecase, ∼ ω is the largest fixed point of the operator E, i.e. putting S =∼ ω givesthe largest solution to the equation S = E(S) (see [Hennessy and Milner, 1985,