CSci 4011: Formal Languages and Automata Theory
Fall 2008, University of Minnesota
Homework 5

Posted: Oct 30, 2008
Due: Before class on Nov 13, 2008

Problem 1

(Exercise 4.7 from the book.)

Let T = {(i,j,k) | i,j,k are natural numbers}. Show that T is countable.

Problem 2

(Problem 4.10 from the book.)

Let

INF_PDA = {<M> | M is a PDA and L(M) is an infinite language}.

Show that INF_PDA is decidable.

Problem 3

(Problem 4.24 from the book.)

Let

PAL_DFA = {<M> | M is a DFA that accepts some palindrome}.

Show that PAL_DFA is decidable. As a hint, properties of context-free languages may be useful here. For concreteness, you may assume that the alphabet is simply {a,b}.

Problem 4

(Problem 4.26 from the book.)

Let

C = {<G,x> | G is a CFG that generates at least one string that has x as a substring}.

Show that C is decidable. (Suggestion: an elegant solution to this problem uses the decider for E_CFG.)

Problem 5

(Problem 4.28 from the book.) Let A be a Turing-recognizable language consisting of descriptions of Turing machines {<M₁>, <M₂>,...}, where every M_i is a decider. Prove that some decidable language D is not decided by any decider M_j whose description appears in A. (Hint: you may find it helpful to consider an enumerator for A.)

Problem 6

(Problem 4.17 from the book.)

Let C be a language. Prove that C is Turing-recognizable if and only if a decidable language D exists such that C = {x | ∃y <x,y> &isin D}. (Hint: For any fixed n, is it decidable if a Turing machine accepts a string in (at most) n steps?)

Last updated on Oct 30, 2008 by gopalan atsign cs dot umn dot edu.