Connections With Other Notions
A semi-Thue system is also a term-rewriting system—one that has monadic words (functions) ending in the same variable as left- and right-hand side terms, e.g. a term rule is equivalent with the string rule .
A semi-Thue system is also a special type of Post canonical system, but every Post canonical system can also be reduced to an SRS. Both formalism are Turing complete, and thus equivalent to Noam Chomsky's unrestricted grammars, which are sometimes called semi-Thue grammars. A formal grammar only differs from a semi-Thue system by the separation of the alphabet in terminals and non-terminals, and the fixation of a starting symbol amongst non-terminals. A minority of authors actually define a semi-Thue system as a triple, where is called the set of axioms. Under this "generative" definition of semi-Thue system, an unrestricted grammar is just a semi-Thue system with a single axiom in which one partitions the alphabet in terminals and non-terminals, and makes the axiom a nonterminal. The simple artifice of partitioning the alphabet in terminals and non-terminals is a powerful one; it allows the definition of the Chomsky hierarchy based on the what combination of terminals and non-terminals rules contain. This was a crucial development in the theory of formal languages.
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