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.


Homework 1, due Thursday 02/06

  1. 1.4 (c, f & l)
  2. 1.5 (b, c)
  3. 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.


Homework 2, due Thursday 02/13

  1. 1.23 (c & d)
  2. 1.31
  3. See handout (in PDF).


Homework 3, due Thursday 02/27

  1. 2.6 (a & d)
  2. 2.17 (a & b)
  3. 2.18 (c & d)


Homework 4, due Thursday 03/06

  1. 3.10
  2. 3.12
  3. 3.16


Homework 5, due Thursday 03/20

  1. 5.10
  2. 5.14 & 5.15
  3. Prove directly that the following language is undecidable (i.e., don't use Rice's Theorem).
    L17 = { i | L(Mi) contains exactly 17 strings },
    where Mi is the ith Turing Machine in the canonical list of Turing Machines.
    Optional: Show that L17 is not Turing recognizable.


Homework 6, due Thursday 04/03

  1. 5.20
  2. 5.21
  3. 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 many-one reduction (a.k.a. a "mapping" reduction).


Homework 7, due Thursday 04/10

  1. 6.5
  2. 6.17
  3. Prove directly that ATMm-reduces to FIN and the complement of ATMm-reduces to FIN, where:
    FIN = { < M > | M is a TM and L(M) is finite}


Homework 8, due Thursday 04/17

  1. 9.18 & 9.19
  2. 9.20
  3. 9.21


Homework 9, due Thursday 05/01

  1. 7.21
  2. 7.27
  3. 7.30


Homework 10, due Thursday 05/08

  1. Prove that Dominating Set is NP-complete by showing that Vertex Cover reduces to Dominating Set via a polynomial-time many-one 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.
  2. 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.
  3. 8.19


Last modified: 01 May 2003 10:19:28 EDT by Richard Chang, chang@umbc.edu
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