Correspondence between automata and formal grammars?

Question

From Wikipedia

Since there is a one-to-one correspondence between linear-bounded automata and such grammars, no more tape than that occupied by the original string is necessary for the string to be recognized by the automaton.

Note: If I am correct, "such grammars" mean context-sensitive grammars.

I wonder if the quote means that:

a context-sensitive language can have multiple context-sensitive grammars that can generate it, and multiple linearly bounded machines that can have it as their recognized languages, and there exists a one-to-one correspondence between the linearly bounded machines and the context-sensitive grammars, for the context-sensitive language.

More generally, a recursively enumerable formal language can have multiple formal grammars that can generate it, and multiple Turing machines that can have it as their recognized languages. So I wonder if there exists a one-to-one correspondence between Turing machines and formal grammars, for a recursively enumerable formal language.

Given a formal grammar, how can we construct a Turing machine to have the language generated by the grammar as its recognized language?

Conversely, given a Turing machine, how can we construct a formal grammar which can generate the language which is the one recongized by the Turing machine?

Answer 1

The expression "one-to-one correspondence" seems a little too strong for me. It suggests that for every grammar there is a specific automaton. It should be read as: for every grammar an automaton can be constructed for the same language and vice-versa.

Context-sensitive languages are accepted by linear bounded automata and generated by context-sensitive grammars. Context-sensitive grammars have productions of the form $\beta A \gamma \to \beta\alpha\gamma$ where $A$ is a nonterminal and $\alpha$ is nonempty. They are equivalent to length-increasing (more properly noncontracting, or monotone) grammars, which have the form $\alpha\to \beta$ where $|\alpha| \le |\beta|$ (usually $\alpha$ is assumed to include at least one nonterminal).

The languages of Turing machines are generated by unrestricted, or type-0, grammars. See Chomsky Hierarchy.

Answer 2

Since there is a one-to-one correspondence between linear-bounded automata and such grammars, no more tape than that occupied by the original string is necessary for the string to be recognized by the automaton.

that wikipedia statement can be understood as that there is a 1-1 correspondence between unique LBAs and CSLs/CSGs. in other words every "unique" CSL has an LBA and vice versa but multiple LBAs can be equivalent (to the same CSL). in other words the same CSL can be "encoded" in different LBAs & CSGs. but note that determining equivalence of CSLs (as encoded eg as CSGs or LBAs) is undecidable.[1]

for your other question about TMs←→grammars conversions see [2] which also hints at both directions.

[1] Which properties of context sensitive languages are decidable?

[2] How can I convert the Turing machine the recognizes language $L$ into an unrestricted grammar?