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2 Data flow analysis 2.3 Theoretical properties and semantics
and n2 as above, those bounds exits; the was nothing wrong with the above argumentation. Same for the upper bound of 3, 4, etc. numbers. But what about the following: is there an upper bound of all even numbers, i.e., the set {0, 2, 4, 6, . . .} ⊆ N? Well, there is no upper bound, at least not inside N. The point here is that there is a qualitative step from asking for the existance of bounds for finite sets to the existence of bounds for infinite sets. For the particular setting of the total order (N, ≤), upper bounds for infinite sets are not guaranteed, for the lower bound, there would always be at least one lower bound, namely
- 0. For (Z, ≤), also lower bounds for arbitrary sets are not guaranteed, obviously.
Let’s keep on staying with total orders for a while, for concreteness sake just (N, ≤). We know that, at least when sticking to finite sets, upper and lower bounds do exist, though there are typically more than one lower resp. upper bound. At least for upper bounds, it’s guaranteed that there are infinitely many upper bounds for any finite set of elements. Let’s focus on upper bounds in the discussion, since for lower bounds it’s the same, just the other way around. Now, as said, there are typically many upper bounds, and of course, the ≤-relation orders also the bounds themselves. In particular, that we are still in a total
- rder, for any two different upper bounds, one is larger and one is smaller. Actually, one
can easily establish: given a set of upper bounds for two numbers n1 and n2, there is all smallest upper bound. One conventionally would write max(n1, n2) for that, and it’s either n1 or n2, depending on which is larger (or “both”, if n1 = n2). One could likewise establish that this applies not just to upper bounds of two resp. finitely many elements, it would apply to arbitrary sets of elements. (N, ≤), however, is not suitable to illustrate this particular point, simply because there is no infinite set of elements that actually has an upper bound. If we switch two (Z, ≤) instead, there are now infitite sets that do have upper bounds. If a given set A′ ⊆ Z has upper bounds, then it’s guaranteed that the is a smallest or least upper bound. When swhich to lower bounds instead of upper bounds,
- ne would speak of greatest lower bounds, actually, the greatest lower bound, because
again, it’s unique, if it exists. For numbers, one uses min and max to refer to those. The uniquens of least upper bounds is ultimately a consequence of antisymmetry of ≤. For linear orders, this is very straightforward to see (“assume that there are two least upper bounds . . . ” ). For data flow analysis, we are not working with linear or total orders, but with partial
- rders. What changes then? Now, there elements may be uncomparable. Additionally,
and as a consequence, there may be pairs of (uncomparable) elements, for which no upper bound exists (same for lower bounds). The partially ordered sets are intended as the domain for the “search space”. When having two uncomparable safe approximations, then that’s still ok, if there is an upper bound; one can take that a approximation which is looser than the two uncomparable ones but still safe. Even better is, if not only there is an upper bound, but there is a least upper bound. Like in the total order setting, one can still establish that least upper bounds are unique (if they exists). The argument is a tad less immediate than with total orders, but still very easy and a consequence again of antisymmetry. Since least upper bounds and greatest lower bounds are so importantant for us, and since partial orders themselves don’t guarantee their existance, we just postulate their existance and focus on partial orders with lubs and glbs. Those are called complete lattices.
