AI 中文总结
该研究在固定有限幺半群观测下，分离了FRP与FSRP，引入PTLD性质并证明其蕴含唯一分解等特性，还给出了PTLD表示的学习器及FSRP控制器的极限重构方法。
AI 中文摘要
设 $L\subseteq\Sigma^*$，并固定态射 $h:\Sigma^*\to M$（$M$ 为有限幺半群）。我们研究相对句法同余 $\theta_{L,h}:=\equiv_L\cap\ker h$ 下的精确分解与典范表示，将唯一分解与有限直积表示分离。经计算机穷尽验证的36元商群，对每个非单位类均有唯一精确素分解，但其有效素返回规则包含无限族，故即使对有限商群，唯一分解也不蕴含有限相对表示性质（FRP）。我们将同一缺陷推广至非正则上下文无关语言，其具有无限相对商群与有限素谱。为分离该障碍，引入有限状态相对表示性质（FSRP），其中典范有效右部语言由有限残留控制器表示，并证明 $\mathrm{FRP}\subsetneq\mathrm{FSRP}$。随后引入素目标左除确定性（PTLD），其蕴含唯一精确分解、尾部精确性、尾部确定性，且有效规则具二次界。具有有限群观测器的非正则确定上下文无关示例满足PTLD，同时不属于任何固定 $(k,\ell)$-可替换类。最后，对固定 $h$，我们给出典范PTLD表示的强正数据学习器，其假设更新为多项式时间且具有限特征样本，同时从弱行为正确的CFG值学习器中实现典范FSRP控制器的极限重构。
英文摘要
Let $L\subseteqΣ^*$ and fix a morphism $h:Σ^*\to M$ into a finite monoid. We study exact factorization and canonical presentation in the relative syntactic congruence $θ_{L,h}:=\equiv_L\cap\ker h$. We separate unique factorization from finite direct presentation. An exhaustively computer-checked $36$-element quotient has a unique exact prime factorization for every live non-unit class, yet its valid prime-return rules contain an infinite family, so unique factorization does not imply the finite relative presentation property (FRP), even for a finite quotient. We lift the same defect to a nonregular context-free language with an infinite relative quotient and finite prime spectrum. To isolate the obstruction, we introduce the finite-state relative presentation property (FSRP), in which canonical valid right-hand-side languages are represented by finite residual controllers, and prove $\mathrm{FRP}\subsetneq\mathrm{FSRP}$. We then introduce prime-target left-division determinism (PTLD), which implies unique exact factorization, tail exactness, tail determinism, and a quadratic bound on valid rules. A nonregular deterministic context-free example with a finite group observer satisfies PTLD while lying outside every fixed $(k,\ell)$-substitutable class. Finally, for fixed $h$ we give a strong positive-data learner for the canonical PTLD presentation with polynomial-time hypothesis updates and a finite characteristic sample, together with a limit reconstruction of the canonical FSRP controller from weakly behaviorally correct CFG-valued learners.
Comments47 pages; reproducible verification code and a machine-readable certificate for the 36-element witness are available via the fixed GitHub snapshot cited in the paper
