It came to my mind that automata (say to start DFA) can be thought as a special kind of rewriting systems. So if one has a word w , one tries to reduce it to the $\epsilon$ word. In other words acceptance is just a particular case of equational reasoning in term rewriting systems.
My question is, has this approach been developed before? In particular, it would be interesting to see if Turing machines can be modelled in this language.
Someone has suggested regular grammars do model deterministic automata. Maybe it is useful to recall the difference between abstract reduction system and term rewriting system. From "Term rewriting and all that":
A rewrite rule is an identity $l \approx r$ such that l i not a variable and $Var(l) \supseteq Var(r)$. In this case we may write $l \to r$. A term rewriting system is a set of rewrite rules.
It seems that formal gramars fit better the definition of abstract reduction system:
An abstract reduction system is a pair $(A,\to)$ where the reduction $\to$ is a binary relation on the set $A$, i.e. $\to \subseteq A \times A$.
For the future I note: