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Deduction and static equivalence are central knowledge problems in the formal analysis of security protocols and are known to be undecidable for general equational theories. Several decidable classes have been identified through structural restrictions, including subterm convergent theories, shallow permutative theories, contracting theories and some are implemented in tools such as ProVerif and DeepSec. We identify two recurring themes: symbol preservation, where symbols are maintained across axioms, and symbol contraction, where symbols decrease in depth or number from left to right. For symbol-preserving systems, we introduce measure-invariant (MI) and separate measure-invariant (SMI) theories, generalizing permutative classes and providing new decidable fragments for deduction and static equivalence. Depth-sensitive refinements, including depth-preserving permutative (DPP) and depth-preserving variable-permuting (DPVP) theories, are case studies to understand whether the depth of occurrences of symbols matter. For symbol-contraction systems, we define depth-decreasing (DD) and variable-preserving function-decreasing (VBFD) theories, capturing some simple relaxations of term contraction; while deduction is undecidable in general, these restrictions highlight potential decidable fragments. Overall, our results show that controlling symbol dynamics in rewrite rules provides a unifying perspective on the decidability of knowledge problems, offering conceptual clarity on what makes these problems hard for different equational theories.