RECURSION. 13.1 Students will learn about the technique of recursion.
RECURSION. The grammar permits recursion through the presence of relative clause (which ex- pand to noun phrases which may introduce yet other relative clauses, etc.). This leads to sentences such as (6) in which the grammatical phenomena noted in (a-c) may be extend- ed over a considerable distance.
RECURSION. In this section we will show how to calculate reactive contracts for a restricted class of recursive models, with the particular aim of substantiating Theorem 5.3 in Section 5. In Section A.1 we showed that generalised reactive designs form a complete lattice. Thus for any monotonic process construction F we can be sure there exists fixed-points µ F and νF . However, in order to reason about reactive contract generally, we need to calculate the pre, peri, and postconditions of such constructions. • • In general, we are most interested in the weakest fixed-point for reactive designs, µ F , as the strongest fixed-point yields miraculous behaviour for erroneous processes [8]. For example, (νX X ) = Miracle, whereas in reality an infinite loop is a programmer error that should yield Chaos, which µ X X does. In order to calculate the reactive design of a weakest fixed-point we employ two results: (1) Hoare and He’s proof that guarded processes yield unique fixed-points [29, theorem 8.1.13, page 206], and (2) Kleene’s fixed- point theorem [34]. The latter allows us to convert from a recursive construction with a strongest fixed-point to an iterative construction, using a replicated internal choice of power constructions. Since we can calculate the reactive design of replicated processes, we can therefore tackle recursion. Hoare and He’s theorem states, informally, that guarded reactive processes yield a unique fixed-point. That is, if for any X , F (X ) is guarded then µ F = νF . Guardedness is defined as follows.
RECURSION. Recursion shall continue to have the obligation to make all payments owed under written agreements entered into by Recursion with Third Parties as of the Effective Date that relate to any Product, including the Existing Third Party Agreement Payments.
