Generative Structural Theory v1.94 — Cut Probing without Enumerating Cuts

The probing rule of v1.93 decides some undecided stops of the forcing procedure of v1.92 for one-sided re-hangings of a transit leaf: at an undecided stop (A, M), with R = A ∖ M, the procedure is run again from (A, M ∪ {c}) for the candidates c of a cut W of r ∪ M, and if every run for some cut ends impossible no step exists. As the finite checks of v1.93 apply it, the rule lists every cut. This text gives the same rule without listing cuts. A pair c = (s, t) of R is a candidate of some cut exactly when t is not reached from s in r ∪ M (Lemma ϣ1). For any set I ⊆ R, some cut has all its candidates in I exactly when r ∪ M ∪ (R ∖ I) is not strongly connected (Proposition ϣ2 (a)); with I the candidates whose run ends impossible, the rule that runs the procedure once for each candidate and tests r ∪ M ∪ (R ∖ I) once has the outcome of the probing rule at every closed state (Proposition ϣ2 (b)), with at most |R|^3 + |R| + 1 tests of strong connectivity and |R|^3 |H| + |R| searches of reachability, a number of steps polynomial in the number of nodes (Proposition ϣ3). The rule decides what the probing rule decides, no more; it is sound, and its completeness, (Q27) of v1.93, is not settled. If (Q27) has the answer "no" on a class of inputs, the existence of a feasible set at an undecided stop is decided there in polynomial time; the construction of a feasible set is apart. These statements hold for every finite node set. Findings of finite cases, from the records of finite check 23 (no computation run again for this text): the two rules give the same outcome at every undecided stop on four and five nodes (every minimal relation and every labelled tree), on six nodes (each isomorphism-class representative h of the minimal relations with every labelled tree S, 63 · 6^4 = 81648 pairs, not classes of pairs up to isomorphism) and at the 1178 inputs of v1.93 on seven nodes. On eight nodes, among the extensions by one node of those 1178 inputs (1102 up to the names of the nodes; 431964 labelled pairs, 1728766 undecided stops), probing decides "impossible" at each of the 150056 undecided stops without a feasible set and leaves undecided 115304 stops, each with a feasible set; in a sample of 3000 labelled pairs (1978 undecided stops) it decides "impossible" at 5, by soundness, and leaves undecided 60, each with a feasible set found. No choice asked for by (Q27) is found; the cases are not exhaustive on eight nodes, and (Q27), (Q28), (Q22) of v1.88 and (Q1) of v1.76 are open.