CMPSCI 601: Theory of Computation

David Mix Barrington

Spring, 2003

Practice Exam #1

Posted Sunday 23 March 2003

Solutions to be posted Wednesday 26 March 2003

Actual (in-class) exam is Monday 31 March 2003

Directions: Each question is a statement that may be true or false. Please state whether it is true or false -- you will get five points for a correct boolean answer and there is no penalty for a wrong guess. Then justify your answer -- the remaining points will depend on the quality and validity of your justification. Problems with higher point values will tend to require longer justifications, but this will not always be the case.

Crib sheet: I will state some useful definitions after the questions -- these will also be available during the in-class exam.

Types of variables: Unless otherwise indicated, a variable A or B denotes a language, D denotes a DFA, G denotes a context-free grammar, M denotes a Turing machine, n denotes a number, and w denotes a string. Remember that N is the set of all numbers, which we may also think of as the set Sigma^* of all strings.

Note: Question 6 assumes knowledge of Lecture 15 (26 March 2003) or of Chapter 6 of [P]. (Please ignore my earlier reference to Question 1.) The other questions assume only knowledge of Lectures 1-13.

• Question 1 (10): If A is r.e., and A is reducible to its own complement, then A is recursive.

• Question 2 (20): The set {M: L(M) &ne N and L(M) &ne &emptyset} is neither r.e. nor co-r.e.

• Question 3 (10): The set {D: L(D) is empty} is recursive.

• Question 4 (10): If A and B are both r.e., then A is reducible to B (A &le B).

• Question 5 (15): The set {M: for all w: w in L(M) iff w^R in L(M)} is recursive.

• Question 6 (15): The set FO-VALID, which is {&phi: &phi is true in all models of first-order logic} is recursive.

• Question 7 (20): The formula "[forall x:(cube(x) implies small(x))] implies not[exists y:(cube(y) and medium(y))]" is true in all models of Tarski's World.

Crib Sheet:

• L(M) is the set of numbers n such that M halts on input n with output 1.
• L(D) is the set of strings w such that D running on w finishes in an accepting state.
• L(G) is the set of strings w such that w can be generated from G's start symbol according to G's rules.
• A language is r.e. if it is equal to L(M) for some Turing machine M.
• A language is recursive if it is equal to L(M) for some Turing machine M that always halts.
• A function f is total recursive if there is a Turing machine that always halts and gives output f(n) on any input n.
• A is reducible to B if there exists a total recursive function f such that for any n, n is in A iff f(n) is in B.
• A is r.e.-complete iff A is r.e. and for any language B, if B is r.e. then B is reducible to A.
• The language K = {n: n in L(M_n} is r.e.-complete. (definition corrected 30 March 2003)
• The Bloop programming language is defined in terms of basic operations and bounded loops. The Floop programming language also has free loops.
• If w is any string, w^R is w written backward.

Last modified 30 March 2003