MMP.8:4.3 - State the required quantifiers and performance criterion
For a deterministic success condition G(a,w), these are different questions:
| Working question | Mathematical statement |
|---|---|
| Is there some successful combination? | There exist a and w with G(a,w). |
| Does every circumstance have a successful action, if it were known? | For every w, there exists a with G(a,w). |
| Can one action be selected now that succeeds throughout the admitted circumstances? | There exists a such that, for every w, G(a,w). |
| Can an observation-dependent rule succeed throughout those circumstances? | There exists an allowed policy pi such that, for every w, G(pi(h(w)),w). |
Choose the statement from the work’s requirement. The first two can expose possibilities or limits even when the latter two fail. A stronger claim can require a different action or information source.
When the receiving use permits failures with a stated probability, name the event and the randomness over which that probability is calculated. For unknown fixed w and random observation O, the question may require P_w(failure of pi(O)) <= epsilon for every admitted w. If w itself is modeled as random and the question concerns performance averaged over circumstances, use the joint law of w and O. An expected cost needs the same choice of what is averaged. Neither a probability nor an objective follows just from listing possible circumstances. Feasibility, expected performance, tail risk and worst-case performance are alternative questions with different consequences. Use the common choice and characterization methods to decide which matters to the work.
Preserve dependence within the circumstance set. Independently combining several ranges can create impossible circumstances and overstate a requirement. Conversely, excluding an inconvenient circumstance changes the range of the conclusion and needs a subject reason or an agreed narrower use.