Regular vs Context Free Grammars

问题:

I'm studying for my computing languages test, and there's one idea I'm having problems wrapping my head around.

I understood that regular grammars are simpler and cannot contain ambiguity, but can't do a lot of tasks that are required for programming languages. I also understood that context-free grammars allow ambiguity, but allow for some things necessary for programming languages (like palindromes).

What I'm having trouble with is understanding how I can derive all of the above by knowing that regular grammar nonterminals can map to a terminal or a nonterminal followed by a terminal or that a context-free nonterminal maps to any combination of terminals and nonterminals.

Can someone help me put all of this together?

回答1:

Regular grammar is either right or left linear, whereas context free grammar is basically any combination of terminals and non-terminals. Hence you can see that regular grammar is a subset of context-free grammar.

So for a palindrome for instance, is of the form,

S->ABA
A->something
B->something

You can clearly see that palindromes cannot be expressed in regular grammar since it needs to be either right or left linear and as such cannot have a non-terminal on both side.

Since regular grammars are non-ambiguous, there is only one production rule for a given non-terminal, whereas there can be more than one in the case of a context-free grammar.

回答2:

I think what you want to think about are the various pumping lemmata. A regular language can be recognized by a finite automaton. A context-free language requires a stack, and a context sensitive language requires two stacks (which is equivalent to saying it requires a full Turing machine.)

So, if we think about the pumping lemma for regular languages, what it says, essentially, is that any regular language can be broken down into three pieces, x, y, and z, where all instances of the language are in xy*z (where * is Kleene repetition, ie, 0 or more copies of y.) You basically have one "nonterminal" that can be expanded.

Now, what about context-free languages? There's an analogous pumping lemma for context-free languages that breaks the strings in the language into five parts, uvxyz, and where all instances of the language are in uvⁱxyⁱz, for i ≥ 0. Now, you have two "nonterminals" that can be replicated, or pumped, as long as you have the same number.

回答3:

The difference between regular and context free grammar: (N, Σ, P, S) : terminals, nonterminals, productions, starting state Terminal symbols

● elementary symbols of the language defined by a formal grammar

● abc

Nonterminal symbols (or syntactic variables)

● replaced by groups of terminal symbols according to the production rules

● ABC

regular grammar: right or left regular grammar right regular grammar, all rules obey the forms

1. B → a where B is a nonterminal in N and a is a terminal in Σ
2. B → aC where B and C are in N and a is in Σ
3. B → ε where B is in N and ε denotes the empty string, i.e. the string of length 0

left regular grammar, all rules obey the forms

1. A → a where A is a nonterminal in N and a is a terminal in Σ
2. A → Ba where A and B are in N and a is in Σ
3. A → ε where A is in N and ε is the empty string

context free grammar (CFG)

○ formal grammar in which every production rule is of the form V → w

○ V is a single nonterminal symbol

○ w is a string of terminals and/or nonterminals (w can be empty)

回答4:

Regular Expressions

• Basis of lexical analysis
• Represent regular languages

Context Free Grammars

• Basis of parsing
• Represent language constructs

回答5:

A grammar is context-free if all production rules have the form: A (that is, the left side of a rule can only be a single variable; the right side is unrestricted and can be any sequence of terminals and variables). We can define a grammar as a 4-tuple where V is a finite set (variables), _ is a finite set (terminals), S is the start variable, and R is a finite set of rules, each of which is a mapping V
regular grammar is either right or left linear, whereas context free grammar is basically any combination of terminals and non-terminals. hence we can say that regular grammar is a subset of context-free grammar. After these properties we can say that Context Free Languages set also contains Regular Languages set

回答6:

Regular grammar:- grammar containing production as follows is RG:

V->TV or VT
V->T

where V=variable and T=terminal

RG may be Left Linear Grammar or Right Liner Grammar, but not Middle linear Grammar.

As we know all RG are Linear Grammar but only Left Linear or Right Linear Grammar are RG.

A regular grammar can be ambiguous.

S->aA|aB
A->a
B->a

Ambiguous Grammar:- for a string x their exist more than one LMD or More than RMD or More than one Parse tree or One LMD and One RMD but both Produce different Parse tree.

                S                   S

              /   \               /   \
             a     A             a     B
                    \                   \
                     a                   a

this Grammar is ambiguous Grammar because two parse tree.

CFG:- A grammar said to be CFG if its Production is in form:

   V->@   where @ belongs to (V+T)*

DCFL:- as we know all DCFL are LL(1) Grammar and all LL(1) is LR(1) so it is Never be ambiguous. so DCFG is Never be ambiguous.

We also know all RL are DCFL so RL never be ambiguous. Note that RG may be ambiguous but RL not.

CFL: CFl May or may not ambiguous.

Note: RL never be Inherently ambiguous.

回答7:

Basically regular grammar is a subset of context free grammar,but we can not say Every Context free grammar is a regular grammar. Mainly context free grammar is ambiguous and regular grammar may be ambiguous.

回答8:

a regular grammer is never ambiguous because it is either left linear or right linear so we cant make two decision tree for regular grammer so it is always unambiguous.but othert than regular grammar all are may or may not be regular