patternMinor
Decidable languages and unrestricted grammars?
Viewed 0 times
unrestrictedlanguagesgrammarsdecidableand
Problem
Turing machines and unrestricted grammars are two different formalisms that define the RE languages. Some RE languages are decidable, but not all are.
We can define the decidable languages with Turing machines by saying that a language is decidable iff there is a TM for the language that halts and accepts all strings in the language and halts and rejects all strings not in the language. My question is this: is there an analogous definition of decidable languages based on unrestricted grammars rather than Turing machines?
We can define the decidable languages with Turing machines by saying that a language is decidable iff there is a TM for the language that halts and accepts all strings in the language and halts and rejects all strings not in the language. My question is this: is there an analogous definition of decidable languages based on unrestricted grammars rather than Turing machines?
Solution
A language is decidable, iff it is semi-decidable and its complement is semi-decidable. Moreover, a language is recursive-enumerable iff it is semi-decidable and thus you can find an unrestricted Grammar. Therfore:
A language $L$ is decidable iff there is both an unrestricted Grammar $G$ with $L(G) = L$ and an unrestricted Grammar $\bar G$ with $L(\bar G) = \bar L$.
A language $L$ is decidable iff there is both an unrestricted Grammar $G$ with $L(G) = L$ and an unrestricted Grammar $\bar G$ with $L(\bar G) = \bar L$.
Context
StackExchange Computer Science Q#10124, answer score: 7
Revisions (0)
No revisions yet.