Universe for VDM-SL Sample Clauses

Universe for VDM-SL. ‌ A prerequisite of our semantics is a representation of the universe of constructible VDM- SL values, which we call Usl. We will use this in our CML-based semantics of VDM-RT to represent parameters that are passed from object to object for method calls. This will allow us to have generic infrastructure processes for CPUs and busses which both simplifies translation, and also reduces the number of processes that need specific static analysis. Since we are mechanising our semantics in Isabelle/HOL, the universe must be repre- sentable within the HOL logical system. A possible candidate universe is the xxx Xxx- xxxx universe Vω+ω [72]. The xxx Xxxxxxx universe Vi is inductively defined for some index i by repeated application of the power-set for ordinal indices β, and generalised union for limit ordinals λ. V0 =^ ∅ Vβ+1 =^ P(Vβ) Vλ =^ Vβ [ β<λ ∈ Each limit ordinal index corresponds to the union of all sets constructed up to that level. In HOL, every finite type is representable by some Vn (for n N>0), and every infinite type by some Vω+n. For example, N can be represented by Vω and R by Vω+1. Then, Vω+ω, which is the limit of these sets, contains all possible types that are constructible in HOL. It is, therefore, impossible to formalise Vω+ω in HOL and thus this cannot be our universe. Nevertheless, since it is the universe of ordinary mathematics, we will use it to 3Please see xxxxx://xxxxxx.xxx/isabelle-utp/utp-main/ characterise the base logic world W , Vω+ω, which could be either HOL or some other equipollent logic. Fortunately, VDM-SL does not require the full generality of the xxx Xxxxxxx universe, as its type constructors are mainly of a finite nature. In particular, the power set operator is a finite power set operator. The constructible VDM-SL types Tsl include the following types, for A, B, Ti ∈ Tsl:
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