I'm looking for a construct that is to semantic-analysis what NFA/DFA are to scanning and DPDA is to parsing
Repeated substring string replacement is Turing complete.

To emulate a Turing machine you can use the following:

The current string encodes the tape with a new symbol $\$$ followed by an encoding of the current state inserted before the position where the current head is. The allowed replacements can be encoded by taking each transition $\delta(n, b) = (m, c, N|L|R)$ by creating the replacement rules to replace $a\$nb$ with $a\$mc$ $\$mac$ or $ac\$m$ for not moving the head or moving it left or right where a is every symbol in the alphabet.