Self doubt

Question

I know that for a recursively enumerable language there exists an unrestricted grammar and we have formally defined unrestricted grammar. I want to know whether for every recursive language, there is a grammar or not, and if there is, is there a formal definition of the same?

Comments

recursive languages doesn't have a specific grammar to them.

they are derived by Unrestricted grammar only, but there should be atleast one Total Turing Machine ( Haling turing Machine ) which accepted to the language generated by those unrestricted grammar.

 

Simply,

Unrestricted grammar have one-to-one correspondence with Turing Machines.

but Unrestricted grammar doesn't have one-to-one correspondence with Total Turing Machines.

 

in other words,

every Recursive language generated by Unrestricted grammar, but every language generated by Unrestricted grammar is not Recursive language.

 

Normally we say as, ( about RL, CFG or CSG )

every Regular language generated by Regular grammar, and every language generated by Regular grammar is Regular language.

that's why we are saying, Regular Language has a separate grammar which is called as Regular Grammar

 

every Context Free language generated by Context Free grammar, and every language generated by Context Free grammar is Regular language.

that's why we are saying, Context Free Language has a separate grammar which is called as Context Free Grammar

 

every Context Sensitive language generated by Context Sensitive grammar, and every language generated by Context Sensitive grammar is Context Sensitive language.

that's why we are saying, Context Sensitive Language has a separate grammar which is called as Context Sensitive Grammar

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Unrestricted grammar also generate recursive enumerable language which is recognize by Turing machine and partially decidable

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Thank you! That is what my understanding was too.