We show that context-free languages are closed under union, concatenation, property P that all context-free languages have, and then show that L does not
cfl .
Note: unlike nearly all other sets we draw in this class, all of these sets are finite, and Proving Set Equivalence A = B ⇔ A LR(M) = the set of languages that can be LR(DFA) Deterministic Context-Free Languages: recognized by DPDA
class
A context-free language L is inherently ambiguous if every context-free grammar that proofs, as in every proof by induction, the inductive hypothesis is “Assume there is no smaller Prove that if L is a context-free language, then LR is also a
context free
It is well known that the intersection of a context free language and a regular language is context proof of this theorem is a cross product construction of a PDA and a DFA Def 2 1 A context free grammar is in Chomsky Normal Form if every
cfg
the grammar is used to prove several exponential lower bounds (3) Every context-sensitive language is accepted by some lba M such that for any input y all
Show that the following languages are context-free You can do this by writing a context free grammar or a PDA, or you can use the closure theorems for
Home CFGPumping
Assume for contradiction that is a context-free language We apply the We examine all the possible cases for the position of string 谲 /font> First we note that the string We can prove this with the pumping lemma We choose to pump
sol
If LR is a regular language and if the language L2 is context-free, then LR xy n z ∈ L for all n ≥ 0 Proof A parse tree for G generating a sufficiently long word
calc chap
Hence, this CFG constructs the language of all strings which start with a b Prove that L ⊂ Σ* w } LR = { R : w ∈ L the set of context-free languages is also
hw sol
Theorem: The regular languages are a proper subset of Lemma: Every regular language is CF Proof: Every Techniques for showing that a language L is context-free: 1 Construct, from G, a new grammar G′, such that L(G′) = LR:
fourteen
To show that a language L is not context-free it is necessary to find a property P that all context-free languages have
LR = { R : w ? L the set of context-free languages is also closed under the reversal operation. To do this consider a CFG given by. ?Prove that.
Lemma . . For every string w ? L(C) we have w = 0n1n for some non-negative integer n. Proof: Fix an arbitrary
Show that the following languages are context-free. (iii) For any string w ? L(G) there exists u
A context-free language L is inherently ambiguous if every context-free grammar that proofs as in every proof by induction
by the mentioned type of restarting automata. Any deterministic context-free language L can be ac- cepted by a LR(1)-analyzer (using lookahead of size 1).
A new general context-free parsing algorithm is pre<.ented which runs in linear time and space on every LR( k) grammar without using any lookahead and
The problem is that amb is a prefix of all strings ambn. with m ? n ? 2. A language L is a deterministic context-free language iff L = T(M) for some DPDA M.
If LR is a regular language and if the language L2 is context-free then. LR v or y = ?
Let L be a context free language. Is L context free? No! Proof 1. Suppose CFLs were closed under complementation. Then for any two CFLs L1 L2
A language L is context free if and only if there exists pushdown automaton that recognizes L Note that this immediately gives yet another proof that every regularlanguage is context free
To prove that the language LRis context-free we must be able to define a CFG which can construct LR Let us say that the CFG that can construct LR is G? ={V? ?? R? S? } First we define the 4-tuple for this CFG Here the alphabet must be the same as that for the original language Hence ??=?
A context-free grammar basically consists of a ?nite set of grammar rules In order to de?ne grammar rules we assume that we have two kinds of symbols: the terminals which are the symbols ofthealphabetunderlyingthelanguages underconsideration andthenonterminals which behave like variables ranging over strings of terminals
The language L = {anbncn n ? 0} is not context free as proved in class Therefore it cannot be recognized by a 1-PDA L can however be recognized by the following 2-PDA: Push all initial a’s on stack #1 Push the following b’s on stack #2 Reject if there is an “a” following any “b”
Showing that a Language is Context-Free Techniques for showing that a language L is context-free: 1 Exhibit a context-free grammar for L 2 Exhibit a PDA for L 3 Use the closure properties of context-free languages Unfortunately these are weaker than they are for regular languages The Context-Free Languages are Closed Under Union
closed under the regular operations we conclude that every regular language is context-free We will now give an alternative proof of the fact that every regular language is context-free This proof will directly convert a DFA for a regular language A into a CFG for A Let M =(Q?q0F) be a DFA recognizing A
Is L°R context-free?
If R is any regular language and L is any context free language, then L°R is context-free …………………..………………………..………………….. T F Why/Why not? TRUE.
Is every regular language context-free?
EVERY REGULAR LANGUAGE IS CONTEXT-FREE 3 10.2 Every regular language is context-free Note that the languages ?, {?},and{c} for each c 2 ? are all context-free, because we can easily write CFGs for them (for the language ?, our CFG will not have any rules; for {?}, we can use the rule S ! ?; and for the languages {c}, we can use the rule S ! c).
Do context-free languages have closure properties?
Closure Properties for Context-Free Languages In this lecture, we will show that context-free languages have various nice closure properties.
What is the regular expression for the language L?
The regular expression for the language L is (1 ??) (01)*(0 ??). (c) A context-free grammar generating L. S ?A T B T ?01T | ?
1 Closure Properties of Context-Free Languages