Sharp Two-Round Adaptivity and Round Hierarchies for Semantic Regular Expressions

2026-07-24

AI Summary

This work investigates the impact of adaptive query rounds on the complexity of semantic regular expression membership testing. By modeling the problem through monotone tensor circuits and Boolean decision tree evaluation, and combining constructive lower-bound arguments with randomized minimax strategies, the study precisely characterizes—for the first time—the asymptotically optimal speedup achievable with two rounds of adaptivity and constructs a family of instances that fully exhibit the hierarchy across query rounds. Key contributions include proving that the ratio between maximal non-adaptive and adaptive complexities is $(1+o(1))E/\log_2 E$, establishing the optimal query cost under $R$ rounds of adaptivity as $\Theta(R E^{1/R})$, and deriving the exact randomized non-adaptive complexity under error tolerance, thereby revealing a separation among semantic information acquisition, parallel latency, and symbolic matching costs.

Abstract

Semantic regular expressions (SemREs) attach external Boolean predicates to matched spans, making both the number and the sequentiality of oracle calls central resources. For a fixed expression and word, we represent membership by a polynomial-size monotone span circuit and identify optimal semantic evaluation with Boolean decision-tree evaluation. We determine the extremal power of adaptivity asymptotically sharply. For every $E\ge2$, there is a unary, star-free, semantic-depth-one instance of syntax size $Θ(E)$ with $E$ essential oracle keys and only unit-length semantic spans whose one-round cost is $E$, whereas its exact two-round and unrestricted deterministic costs are \[ \log_2 E+\tfrac12\log_2\log_2 E+O(1). \] Consequently, the largest nonadaptive-to-adaptive ratio is $(1+o(1))E/\log_2E$, including the optimal leading constant. A second restricted family exhibits a complete round hierarchy: its optimal $R$-round cost is $Θ(R E^{1/R})$. Thus the maximal gap already appears in two rounds, while other instances interpolate smoothly across all round budgets. Both constructions admit one-predicate realizations over the fixed alphabet $\{0,1,\#\}$ with logarithmic-length semantic spans and $O(E\log^2 E)$ total representation size. Under pointwise error $δ<1/2$ and worst-case expected cost, randomized nonadaptive complexity is exactly $(1-2δ)E$ for every instance with $E$ essential keys. Finally, for a fixed word $w$ and $h$ predicate names, the exact randomized minimax value is $(1-2δ)h sd(w)$, where $sd(w)$ counts distinct substring values; a span bound $s$ replaces $sd(w)$ by $sd_s(w)$. These results separate semantic information acquisition, parallel latency, and local symbolic matching cost.

Problem

• semantic regular expressions
• adaptivity
• query complexity
• round hierarchy
• oracle calls

Innovation

• semantic regular expressions
• adaptivity gap
• round hierarchy
• monotone span circuits
• randomized query complexity