Unrestricted Grammars

Recall, a grammar is an abstract entity which attempts to describe the "strings" [aka sentences] of a language. As a formal extension of a context-free grammar:

Definition: an unrestricted grammar is a 4-tuple (T,N,P,S), consisting of:

1. T = set of terminals (the legal "tokens" of the language)
2. N = set of nonterminals (aka variables)
3. P = as set of productions, each of the form:
   v -> w (what does this mean?)
   where v and w are strings consisting of nonterminals and terminals.
4. S = a special nonterminal called the start symbol.

This type of grammar is also known as a Type 0 grammar. It is equivalent to a Turing Machine. [What does this mean? Given a Type 0 grammar, there exists a TM that can "recognize" the same language as that grammar. And, conversely, a Type 0 grammar can always be found for the language recognized or generated by any TM.]

It is also equivalent to a recursively enumerable language. [What does this mean?]

E.g.: G with set of productions:

    P = {
            S  -> ACaB
            Ca -> aaC
            CB -> DB
            CB -> E
            aD -> Da
            AD -> AC
            aE -> Ea
            AE -> ε
        }

L(G) (language generated or recognized by G) =

   {w | w = ai and i = 2k, k > 0}

Note: normal method of specifying repetition of strings.

Definition: context-sensitive grammar [aka Type 1 grammar]

All productions are of form

   v -> w  where |v| < |w|

Note: one special normal form [what does this mean?] of this kind of grammar requires productions of form:

   uAv -> uwv with w != epsilon,

I.e., A -> w but only in the context of u _ v.

A context-sensitive grammar is equivalent to a linear bounded automaton (LBA) and to a context-sensitive language.

Almost all languages are context-sensitive. [What does this mean exactly? - they may actually be simpler than context-sensitive.]

Definition: context-free grammar [aka Type 2 grammar]

All productions are of the form:

   A -> x  — where A is nonterminal, x is a string of nonterminals and terminals

A context-free grammar is equivalent to a pushdown automaton (PDA) and to context-free languages.

Definition: Regular grammar [aka Type 3 grammar, aka linear grammar]

All productions are of the form:

   A -> wB
   A -> w  where A, B are nonterminals,
                 w is a string of terminals

Regular grammars are equivalent to regular sets [what does equivalent to mean here?]

And they are also equivalent to finite automata.

   -------------------------------------------------------
   |                                                     |
   |         Type 0 - r.e. languages and TM's            |
   |                  unrestricted grammars              |
   |     -----------------------------------------       |
   |     |                                       |       |
   |     |    Type 1 - c.s. languages and LBA's  |       |
   |     |             c.s. grammars             |       |
   |     |                                       |       |
   |     |       ---------------------------     |       |
   |     |       | Type 2 - c.f. lang's    |     |       |
   |     |       |          PDA's          |     |       |
   |     |       |          c.f. grammars  |     |       |
   |     |       |   ------------------    |     |       |
   |     |       |   |                |    |     |       |
   |     |       |   | Type 3 -       |    |     |       |
   |     |       |   |   reg lang's   |    |     |       |
   |     |       |   |   DFA's NFA's  |    |     |       |
   |     |       |   |   reg grammars |    |     |       |
   |     |       |   |                |    |     |       |
   |     |       |   |                |    |     |       |
   |     |       |   |                |    |     |       |
   |     |       |   |                |    |     |       |
   |     |       |   ------------------    |     |       |
   |     |       |                         |     |       |
   |     |       ---------------------------     |       |
   |     |                                       |       |
   |     |                                       |       |
   |     |                                       |       |
   |     -----------------------------------------       |
   |                                                     |
   |                                                     |
   |                                                     |
   -------------------------------------------------------