Generative Structural Theory v1.83 — Prioritizing Ear Moves: Reachability Preservation and Residual Cases

(Q11) of v1.82 asks for conditions under which the procedure P_F of v1.82, or a procedure with further rules, ends at a relation with a leaf of the cases (i)-(iii) of leaf detachment. At six nodes, of the 21192 residual instances of the population checked in v1.82 (every minimal relation with one tree of each shape), P_F leaves 2758 and its variant with the rule F tried first leaves 2562. This text does not settle (Q1) of v1.76, the two-deletion chain question; it gives partial results. Proposition Ͽ1: for every procedure by turns and every list of terminal rules (rules for which the last relation of every candidate has a leaf of cases (i)-(iii)), trying the terminal rules first loses no success of trying them last, with no more turns, and the instances gained are described by the turns of the procedure without them. As the rule F of v1.82 is terminal, P_F^first, the rule F tried before the six rules of P, succeeds wherever P_F does, without (Q1). Lemma Ͽ2: an open ear of a minimal strongly connected relation h whose end node is a leaf l of the tree S, moved to enter (or to leave) at the tree neighbour of l, gives a minimal relation inside h ∪ P_S with at most two deletions in which l is of case (ii) (or (iii)), under the hypotheses of Lemma Ͽ2, including that the new end pairs belong to h ∪ P_S and that condition (F3) of v1.82 holds for the new ends. Theorem Ͽ3 and Corollary Ͽ4: with (Q1) on one node less, a chain from h to P_S inside h ∪ P_S with deletion bound two follows; the procedure P_FG^first, the rules F and G (Lemma Ͽ2) before those of P, gives such a chain whenever it ends at a leaf of the cases (i)-(iii), and succeeds wherever P_F^first does. So the sets of instances in which P_F, P_F^first and P_FG^first succeed are each contained in the next. These statements hold for every finite node set. Finite checks at six nodes find that each of the 196 instances gained by the rule F first is reached by one step of F at the initial relation, and that in each of the 2562 left the procedure P ends for want of a rule, F applying at none of its turns. A search over every candidate of the existing rules reaches a leaf of the cases (i)-(iii) in 30 of the 2562; general steps inside h ∪ P_S reach one in all 2562 (least numbers over all 2562: one step 2446, two 114, three 2). The rule G applies at the initial relation in 358 of the 2562, so that P_FG^first reaches such a leaf in 18988 of the 21192 and leaves 2204; with G among the rules of the search over the candidates, some sequence of candidates reaches such a leaf in 26 of the 2204 and none in 2178, while general steps do in all. These figures are findings of the six-node enumeration, not general statements.