CSCI (Math) 361 Theory of Computation

Homework Assignments Fall, 2002

Homework is generally assigned two class periods before it is due. Please note that homework is due at the beginning of class and no late homework will be accepted. The two homeworks to be dropped are to cover situations like illness or other circumstances that prevent you from attending class or doing homework.

Due Date To Turn In: To Do On Own: Solutions:
9/11 L&P: 1.4.1b Annotate chapters of Jones w/ errata from course web page. Solutions
9/13
1. A rational number is a number that can be represented by a fraction m/n where m and n are integers and n is different from 0. Show that the set of rational numbers is countable. (Don't forget that rational numbers can be represented by fractions in many ways. E.g., 1/2 = 2/4 = ... Address why that does not cause problems with your proof.)
A. Let g: A -> B be surjective (onto) and suppose A is countable. Show B is also countable.
L&P: 1.5.8
Solutions
9/16
1. A real number between 0 and 1 may be represented as the infinite string of integers to the right of the decimal point in its infinite decimal expansion. E.g. 1/3 is represented by < 3,3,3,...> while 1/2 is represented by < 5,0,0,0,...>. Use the diagonal method directly (i.e., don't use any of the theorems or propositions proved in class) to show the set of real numbers is uncountable. Warning:Worry about the fact that some real numbers have more than one infinite decimal expansion. E.g., .50000... = .49999... Make sure that your proof takes that into consideration or it will be wrong!. That is, make sure the counterexample you generate is really different from all of those in the list. (L&P: 1.5.11 is similar, but it is harder to do correctly in binary.)
L&P: 1.5.7, 1.5.9 Solutions
9/18 L-P: 1.5.2, 1.5.6 L&P: 1.5.1, 1.5.3, 1.5.5 Solutions
9/20 Jones: 2.1
Hint: use a stack -- and write and test the program in a higher-level language first! In the high-level language define a list data structure that supports operations cons, hd, and tail -- and no fair using recursion - that makes it too easy! When you are done convert your program to our while language -- though you can use abbreviations we developed in class like if-then-else to make your life simpler.
Write a program in the while language to compute the "and" of two boolean values. That is, your program should take as input (X.Y) where X and Y are elements of {true,false} (or rather their encodings) and it should output the encoding of true or false as appropriate. I don't care what the program returns if one or more of X and Y fails to be a boolean value. Solutions
9/23 Jones: 2.5
Hint: Let C stand for the while loop in the reverse program. Prove by induction on n, that for all k, if Y = (e1,...,ek) and X = (d1,...,dn) before executing C, then Y = (dn,...,d1,e1,...,ek) and Y = nil after C has finished executing. For example, the base case is when n = 0, and hence X = () = nil = false according to our abbreviations. Once you have finished proving the above claim, then it should be easy to finish the proof that reverse does what it is supposed to.
Jones: 2.3, 2.4,
A. On page 34 of Jones is a program to calculate the sum of two numbers. Write WHILE programs to calculate the product of two numbers, and another to calculate the first number raised to the second number power, i.e, AB. Feel free to use procedures computing already known functions (like addition).
Solutions
9/25 On page 41 of Jones, it is stated that no variable is allowed to occur twice in a pattern. Give an example of a case statement where one variable occurs twice in a pattern and show why the translation given in class (and in the text) is incorrect. That is give an example (for example one involving the pattern (X.X)). Give the translation of the case statement to the WHILE language (though you can use if-then-else, but not case). Finally, explain why that translation does not give the expected behavior. Translate the program in Example 2.3.2 on page 40 into the WHILE language (where you are allowed to use IF statements). Your translation should be exactly the translation that would be obtained using the translation described on pages 41 and 42 of the text. A freehand translation is not acceptable. Solutions
9/27 Define a compiling function from the WHILE language with if-then-else to the pure WHILE language. The definition of the WHILE language with if-then-else is obtained by adding an extra clause in Definition 2.1.3. Add to the end of the definition of Commands the line:
```   | if E then C else D
```
The compiling function should be defined on all Programs of this language. You will have to define it separately first for expressions (a trivial recursive function based on the clauses in the definition of expressions), then for commands (where the translation of commands different from if-then-else are trivial), and finally for programs. See the definition of concrete syntax for WHILE in Figure 3.1 for an example of a definition by recursion on expressions, commands, and programs.
Extra Credit: Design a compiler in the WHILE language with if-then-else and case statements that compiles this extended language to the pure while language. This program should operate on the concrete syntax for WHILE (exended as necessary to include concrete syntax for if-then-else). Solutions
9/30 Rewrite the program to add two numbers so that it only uses a single variable. Write the tlrep and consrep functions defined on page 63 of Jones. Solutions
10/2 Jones: 4.3. The handout distributed in class stated invariants for the evaluation of expressions and commands run in the interpreter. Show by induction that the invariant holds for all expressions. That is, show the following for all expressions E of the WHILE language:

Suppose p evaluates E in store [v1 -> d] to e and suppose initial values of Cd, St, V1 are (E.Cd'), S, and d before evaluation of E. Then eventually, new values will be Cd', e.S, and d.

Solutions
10/7 Jones: 5.1, parts 2, 4, and 5 only Show that there exists a fixed program, p0 such that determination of whether [[p0]](d) terminates for d in D is undecidable. I.e., find a program p0 such that {d in D | [[p0]](d) terminates} is undecidable. Solutions
10/9 Jones: 5.3 (Do NOT use Rice's Theorem. Prove it directly as we did with the problem involving acceptance on nil.) Prove that it is undecidable whether there exists an input to a given program for which it halts and writes nil. I.e., show the set {p in WHILE-programs | there is a d, such that [[p]](d) = nil} is undecidable. Do NOT use Rice's Theorem. Solutions
10/11 Jones: 5.13, 5.14 Note: The order referred to in 5.14 is the order given in Lemma 5.7.1 Jones: 5.8. Solutions
10/21 L+P: 1.7.2c, 1.7.4ab L+P: 1.7.3, 1.7.5, 1.7.6 Solutions
10/25 L+P: 1.8.3ab, 2.1.3ad, 2.1.4a(i,ii) L+P: 1.8.2abcd, 1.8.5abcd, 2.1.1, 2.1.2abc, 2.1.3bc Solutions
10/28 2.2.2b, 2.2.3ac 2.2.1ab, 2.2.2a, 2.2.4, create ndfa accepting (a ∪ b)*ab+(aaa ∪ aba)b* Solutions
10/30 2.2.6, create dfa accepting (a ∪ b)*ab+(aaa ∪ aba)b* 2.2.7, 2.2.9b, 2.2.10 Solutions
11/1 2.3.3, 2.3.6ag, 2.3.11a 2.3.1, 2.3.2, 2.3.5, 2.3.6f Solutions
11/4 2.3.7b 2.3.4b,2.3.7a Solutions
11/6 2.4.3ad, 2.4.4, 2.4.5a, 2.4.8ac 2.4.2,2.4.3bce,2.4.8b Solutions
11/8 2.5.2, Find min state dfa equiv to dfa here 2.5.1(i)(iii), both parts a and b Solutions
11/11 Describe an algorithm to determine, given M1 and M2, if L(M1) and L(M2) are disjoint.   Solutions
11/13 3.1.3ab, 3.1.5b, 3.1.7 3.1.2, 3.1.3c, 3.1.4, 3.1.5a, 3.1.9ad Solutions
11/15 No homework because Midterm due
11/18 3.2.2, 3.3.2b, 3.3.3 3.2.3, 3.2.4b, 3.3.1, 3.3.2acd Solutions
11/20 3.4.1   Solutions
11/22 3.5.1b, 3.5.2cd, show {aibjcidj | i,j >= 0} is not a cfl 3.5.1acd Solutions
11/25 3.5.3a, 3.7.5a 3.7.1a Solutions
12/2 1. Design a deterministic Turing machine to accept the language {wcw | w in {a,b}*}. You may use a tape language as large as you like. Hint: As you move back and forth comparing characters, remember that you can change characters on the tape.
2. Discuss how a non-deterministic Turing machine could accept the language {ww | w in {a,b}*}. Discuss how you could modify your deterministic Turing machine from problem 1 to accept {ww | w in {a,b}*} deterministically.
Design a deterministic Turing machine to accept the language {anbncn | n >= 0}. Hint: Use a multitape Turing machine. You may describe the computation in English rather than write the detailed transitions. Solutions
12/4 1. Design a 2 tape deterministic TM that copies the contents of the first tape onto the second tape. Originally the read head is to the left of the first non-blank character of the first tape. I.e., the initial configuration is (s,Bx,B) where we have shown the contents of both tapes. The final configuration should be (h,Bx,Bx). That is, both read heads should be at the blank before the input.

2. Design a deterministic Turing machine that will halt at the "." for input a term of D of the form (x.y) where x and y themselves are elements of D. Example: If the input is ((0.(0.0)).(0.0)), then the TM should halt at the 3rd ".". Hint: Use a scratch tape to keep track of any extra information needed. Recall that "0" is used to encode nil on the TM and the tape alphabet includes B, 0, "(", ")", and ".".

Solutions