UMBC CMSC 651, Automata Theory & Formal Languages,
Spring 2005, Section 0101
Homework Assignments
From Introduction to the Theory of Computation by Michael Sipser,
PWS Publishing unless otherwise noted.
 1.4 (e, h & l)
 1.5 (b, c)
 1.10
 For any language L, subset of Sigma*, define HALF(L) as follows:
HALF(L) = { x in Sigma*  there exists y in Sigma*, x = y and xy is in L }.
Show that if L is regular, then HALF(L) must also be regular.
 1.23 (b, c & d)
 1.30
 1.41
 Describe an algorithm that, given a DFA M, determines
whether L(M) contains an infinite number of strings.
Argue that your algorithm is correct.
 Let A and B be subsets Sigma*. Define TwoWay(A,B) as follows:
TwoWay(A,B) = { x in Sigma*  x is in A and x^{R} is in B }
where x^{R} denotes the reversal of x. Show that if A and B
are regular, then TwoWay(A,B) must also be regular.
 2.6 (b & d). Please comment your grammars.
 2.18 (b, c & d)
 2.26
 3.10
 3.12
 3.16
 4.18
 5.10
 5.14 & 5.15
 5.20

A set is cofinite if it is the complement
of a finite set. Let
COF = { < M >  M is a TM and L(M) is cofinite}
FIN = { < M >  M is a TM and L(M) is finite}
Prove that FIN ≤_{m} COF. That is FIN reduces to COF via
a manyone reduction (a.k.a. a "mapping" reduction).

Prove directly that A_{TM} ≤_{m}reduces to FIN and
the complement of A_{TM} ≤_{m}reduces to FIN.
 6.5
 6.15
 9.3
 9.20. Note: the definition of pad(A, f(n)) should be
pad (A, f(n)) = { pad(s, f(n))  where n is the length of s and s is in A }.
 9.21
 7.21
 7.27
 7.28
 7.29
 7.23
 Show that Dominating Set (DOM) defined below is NPcomplete.
DOM = { (G,k)  G = (V,E) is an undirected graph and there exists
V' subset of V such that V' ≤ k and for each vertex
u in V  V' there exists a v in V' such that
(u,v) in E }
 8.8
 8.20
 10.7
 10.11
 10.13
Hint: Think closure under polynomialtime manyone reductions.
 10.14
Hint: Show that an NP^{SAT} machine only needs
to make one query to the SAT oracle.
Last modified: 09 May 2005 11:43:16 EDT
by
Richard Chang,
chang@umbc.edu
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