Differential Semigroups of Automaton Perturbations and Incremental Syntactic Reconstruction
This paper investigates the enumeration of elements in three classes of signed partial transformation semigroups, namely the partial signed order-preserving semigroup (PSPO_n), the partial signed order-decreasing semigroup (PSPD_n), and the partial signed order-preserving-or-order-reversing semigroup (PSPOD_n). Previous computations provide the initial cardinality sequences [ |PSPO_n| = 1, 13, 133, 1281, \ldots, ] [ |PSPD_n| = 1, 9, 81, 819, \ldots, ] and [ |PSPOD_n| = 1, 16, 209, 2302, \ldots, ] for (1 \leq n \leq 4). A numerical inconsistency in the reported data for (PSPD_4) is examined: the component values (191) and (700) sum to (891), whereas the reported cardinality is (819). Consequently, if the stated total (819) is retained, the corresponding first component must be (119). To provide a systematic framework for extending these finite enumerations, we define the cardinality functions [ A_n = |PSPO_n|, \qquad B_n = |PSPD_n|, \qquad C_n = |PSPOD_n|, ] and introduce the polarity-based enumeration function [ \mathcal{P}_{S}(n) ================== N_{S}^{-}(n) + N_{S}^{\ast}(n), \qquad S \in {O,D,OD}. ] A refined domain-rank enumeration framework is proposed as [ \mathcal{P}_{S}(n) ================== \sum_{k=1}^{n} \binom{n}{k} \sum_{r=1}^{k} E_{S}(n,k,r), ] where (E_{S}(n,k,r)) represents the number of admissible signed partial transformations of domain size (k) and rank (r) satisfying the structural condition (S). Using Newton finite-difference interpolation on the known cardinalities, computational formulae are obtained as [ \widehat{A}_n ============= 1 * 12\binom{n-1}{1} * 108\binom{n-1}{2} * 920\binom{n-1}{3}, ] [ \widehat{B}_n ============= 1 * 8\binom{n-1}{1} * 64\binom{n-1}{2} * 602\binom{n-1}{3}, ] and [ \widehat{C}_n ============= 1 * 15\binom{n-1}{1} * 178\binom{n-1}{2} * 1722\binom{n-1}{3}. ] These expressions reproduce exactly the available cardinalities for (1 \leq n \leq 4) and provide a systematic procedure for generating conjectural values for larger (n). The interpolated expressions are distinguished from exact semigroup enumeration formulae and are therefore treated as computational conjectures pending verification from the defining transformation conditions. The resulting framework provides a basis for computational enumeration, recurrence analysis, conjecture testing, and the subsequent derivation of exact closed-form counting formulae for these finite signed partial transformation semigroups.