# Theory of Computation 5. Assignment by yaoyufang

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```									                              Theory of Computation                                              Problem 4      (Closure properties) [6 points]

5. Assignment                                                Suppose L is context-free and R is regular. Are the following languages necessarily context-
free? If yes, prove it; if not, give a counter-example.
Due April 8
(a) L ∩ R
(b) L − R
Rudolf Fleischer                                                                                   (c) R − L

Spring 2011
Problem 5      (Pumping Lemma) [2 points]

Show that L = {an bm an | n < m} is not context-free.
Problem 1       (PDA) [6 points]

If the stack is empty when the input word has been completely read we say the PDA accepts     Problem 6      (Self-embedding variables) [4 points]
by empty stack (i.e., we do not distinguish between accepting and non-accepting states).
Consider the following PDA. It has three states, the initial state s and states f and g. It      Let G = (Σ, V, R, S) be a CFG. A variable A of G is called self-embedding if and only if
has the following transitions:                                                                   A ⇒+ uAv for some u, v ∈ (Σ ∪ V )∗ .

(1) ((s, a, ), (s, a))                                                                          (a) Give an algorithm to test whether a speciﬁc variable A is self-embedding.
(2) ((s, b, ), (s, b))                                                                          (b) Show that if G has no self-embedding variable, then L(G) is regular.
(3) ((s, , ), (f, ))
(4) ((f, a, a), (f, ))                                                                        Challenge Problem E        (A game)    [1 bonus point]
(5) ((f, b, b), (f, ))
Consider the following game played on words over a ﬁnite alphabet Σ. If a word w is of
(6) ((f, , ), (g, ))                                                                             the form w = uck v, where c ∈ Σ, k ≥ 2, u, v ∈ Σ , u does not end with c, and v does not
start with c, then we can contract w to w = uv. In other words, we can remove a maximal
(a) [2 points]
subsequence of equal letters (of length at least two) from w. A word is solvable if and only
Trace the PDA on the following strings. Does it accept the input by empty stack?             if there is a sequence of contractions leading to the empty string.
1. abba                                                                                     Show that a word w is solvable if and only if w is generated by the context-free grammar
2. abb
(b) Which language is accepted by the PDA?                                                                                      S → | SS | aSa | aSaSa ,

for all a ∈ Σ.
Problem 2       (PDA II) [4 points]

(a) Construct a PDA accepting by empty stack the language generated by the CFG

S → aS, S → aSb, S →
(b) Which language is generated by this CFG?

Problem 3       (PDA III) [2 points]

Construct a PDA that accepts the language {an bm | n ≤ 2m} by empty stack.

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