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Is there name for class of languages exactly such that their words can be parsed in $O(n)$ by program in conventional Turing-complete language (SML)?
(i.e. without backtracking)
I do not know of a specific name for this class of languages.
However, there are many results on grammar types. For example, LR(1) and LL(1) grammars can be parsed in linear time, but using different parsing strategies.
Did you mean linear bounded automata?
Required, but never shown