Events in computation

because functicns like that of example 8.2. 5 which are not sequential but still included. By induction on types Berry shows that the stable ordering is "hidden" in the fully abstract model of PCF and that the functions in it are stable with respect to it. As remarked above the fullyabstract model cannot contain all such functions. For first order types (of the form (@, ,. rn; ? $ where ci and 7 are ground types) he shows that the stable order is the image of the syntactic order and that the extensional order is the image of Plotkin's operational preorder 5 on terms. He conjectures that this state of affairs holds at all types in the fully-abstract model. The work of Berry and Curien ([Ber and Cur], [Cur]) on models of algorithms shows the stable ordering will be very important for a semantic construction of the full*y-abstract model. Some obvious approaches do not work however. The stable ordering alone does not support sequential functions; both parts of axioms Q for 5 can fail (see 8.2.10) and even coherence of _$ goes (consider 5 for example 82.5). This is why they have produced models of algorithms which are not extensional but do preserve the concreteness axioms up the types. Crudely put,an algorithm is built up from "events" whichThe decisions to output or decisions to test input. Chapter 9. Higher tvpe event structures In this chapter we show how event structures may be used to represent exponentiations and products of domains. In particular we produce a category of stable event structures which represent a Cartesian closed full subcategory of Berry's bidomains. We construct the categoryindependently of Berry's results though, of course, the basic intuitions come from Berry's work. Finally we link up configurations of the event structures with bidomains. In fact this is how it was done based on a few heuristic guidelines which we present in the first section. There are many gaps in our understanding. In particular we introduce a new ordering G . L , a sort of dual to Berry's stable ordering; how is it to be interpreted and is there a natural operational characterisation like the one Berry conjectures for the stable ordering? In the final section we indicate how the techniques might be refined to construct a fullyabstract model of PCF which depends on capturing its sequential evaluation. There are many issues raised and left open by this chapter: in this sense it is an introduction albeit a rather lengthy one. We refer to [Mat] for the basic category theory used. 9.1 Introducing higher type event structures We start with a simple example of a higher type event structure which illustrates what we mean by them and how they are to be used. Let us look at event structures of the form cE,i,%) satisfying the single axiom e _> e1 % e" => e ,,' e". % These were introduced in chapter 4 where we showed how such event structures represented coherent prime algebraic domains. We showed that such an event structure determined and was essentially determined by a coherent prime algebraic domain; the left closed consistent subsets of an event structure E ordered by inclusion formed the coherent prime algebraic domain k( ) E and conversely such a domain D determined an event structure E, with events the complete primes, so that L() flV E =D. Suppose (Eii$$'i) for i = 0,l are two such event structures. Can we also represent the function space [%(E& -> i(E,)] of all continuous functions ordered pointwise? After Scott [Sco] we know the step functions form a basis of isolated elements. A little work characterises the complete primes of [ i(EO) -> i(~,)] as precisely those step functions of the form / y.yz x 1 --> [e],L , abbreviated as &wl, where x is an isolated element of L(E*) and e an event of E,. In fact [i(Eo) -> fJE,H is coherent and prime algebraic. Define the event structure EO -> E, to consist of events (x,e) (standing for e[x,y]) ordered bv (x,e) 5 (x',e') iff x' GO x 4 e $e' with conflict relation (x,e)$ (xf,el) iff XT x' ,&. e x, e' where we have simply expressed the ordering and incompatability in the functions space. Then by the representation result of chapter 4 we have 73 EO -> E,) 5 [i(EO) -> %(E,)]; the isomorphism simply expresses a continuous function f as tLe configuration Nx,e) 1 e E f(x)]. We have represented the function space as an event structure. &en more simply, we can represent products of coherent prime algebraic domains. Let (Ei,C,qi) for i = 0,l be two event structures as above. Take E. @ E, to be their disjoint juxtaposition defined by the disjoint union (g) of their sets and relations: Then $,(Eo @ E,) Gi(EJX f&): the isomorphism expresses a pair as the configuratton which is a disjoint union of the pair's ar=guments. Of course we have ignored intuition about what the causality relation 5 on event structures means. In the above constructions it can no longer generally mean mmust occur before in time'*. Accordingly a finiteness restriction on the relation such as an event dominates only finitely many events will not generally hold in representing a function space(,This occurs for the construction E0 -> E, in the innocent circumstances of E. including an infinite conflict-free subset and E, being non-nul_1.) A chief virtue of event structures is supposed to be their operational nature; they have previously prescribed possible behaviours in time. Can event structures like I E. -> E, representing a function space be made to reflect behaviour in time? Gnat finiteness restrictions can be imposed which reflect this? We expect some extra structure is involved in order to distinguish the behaviour of the functional events (in E. -> E, say) from say basic input events. Suppose (Ei,~~,~i) for i = 0,l are event structures representing input and output domains. To reflect this, on both we impose the additional axiom 1 g’bl 1 < CO for events e.