NOT.3:5.1 - Turn a condition diagram into a reading procedure

A team designs a notation for combinations of conditions. Labels P, Q and R refer to conditions with supplied truth values. An ALL enclosure holds only when every enclosed condition holds. An ANY enclosure holds when at least one enclosed condition holds. The conditions are stable while the expression is read, and reading them does not change their values.

Consider:

ALL {
  P
  ANY { Q R }
}

The legend defines the operators, but a new reader still needs a way to apply them to a nested expression. Supply this reading procedure: resolve each condition label from the given inputs; evaluate the innermost enclosure; replace it by its Boolean result; continue outward.

With P = true, Q = false and R = true, the inner enclosure is true because R is true. The outer enclosure combines true with true, so the whole condition holds. This result concerns the supplied conditions. Establishing their truth in an actual project is separate work.

A second route examines only values that can still change the answer. For ALL, one false child settles the result as false; for ANY, one true child settles it as true. With the same inputs, the reader checks P, then Q and R, and obtains the same result. If P changes to false, the outer ALL is false without inspecting Q or R. The stable, side-effect-free condition premise makes this omitted reading legitimate.

Now change the inner operator from ANY to ALL while keeping the original inputs. The inner result becomes false and therefore the outer result becomes false. The reader has followed the changed expression through the same interpretation rules. Replacing a label without resolving its value would leave a different gap: the input is missing, not the nesting rule.

The two reading routes expose a choice about obtaining the answer. They agree on these Boolean results; one shows every intermediate value, while the other can avoid work. A user who also needs all intermediate values should retain the first route or extend the second to produce them.

With P false and the values of Q and R not supplied, the whole expression is still false. A request for every intermediate value still needs those missing inputs.