UMBC CMSC 651, Automata Theory & Formal Languages,
Spring 2003, Section 0101
Homework Assignments
From Introduction to the Theory of Computation by Michael Sipser,
PWS Publishing unless otherwise noted.
 1.4 (c, f & l)
 1.5 (b, c)
 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 (c & d)
 1.31
 See handout (in PDF).
 2.6 (a & d)
 2.17 (a & b)
 2.18 (c & d)
 3.10
 3.12
 3.16
 5.10
 5.14 & 5.15
 Prove directly that the following language is undecidable
(i.e., don't use Rice's Theorem).
L_{17} = { i  L(M_{i}) contains exactly 17 strings },
where M_{i} is the ith Turing Machine in the canonical
list of Turing Machines.
Optional: Show that L_{17} is not Turing recognizable.
 5.20
 5.21

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).
 6.5
 6.17

Prove directly that A_{TM} ≤_{m}reduces to FIN and
the complement of A_{TM} ≤_{m}reduces to FIN, where:
FIN = { < M >  M is a TM and L(M) is finite}
 9.18 & 9.19
 9.20
 9.21
 7.21
 7.27
 7.30
 Prove that Dominating Set is NPcomplete by showing that
Vertex Cover reduces to Dominating Set via a polynomialtime
manyone reduction, where
VC = { < G, k >  G = (V,E) is an undirected graph and
there exists a subset V' of V with V' <= k such that for
each edge (u,v) in E either u or v is in V' }
DOM = { < G, k >  G = (V,E) is an undirected graph and
there exists a subset V' of V with V' <= k such that for
each vertex u in V  V' there exists v in V' such that (u,v) is
in E.
 Construct a logspace transducer that takes as input an undirected
graph encoded as a list of edges and outputs the same graph encoded
as an adjacency matrix.
 8.19
Last modified: 01 May 2003 10:19:28 EDT
by
Richard Chang,
chang@umbc.edu
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