DF CS 454: Theory of Computation

CS 454, Section 001	Sonoma State University	Spring, 2024

Theory of Computation
Instructor: Henry M. Walker Lecturer, Sonoma State University Professor Emeritus of Computer Science and Mathematics, Grinnell College

Although much of this course has been well developed in recent semesters, the SSU CS faculty recently have approved an updated course description. Also, the required SSU Signature Project for SSU's Upper Division GE Area B Requirement for CS Majors has been rethought for this course. Currently, the Web site is reasonably stable, but modest refinements are likely. Check these pages regularly for adjustments.

Contract Negotiations Have Yielded a Tentative Agreement

Since May, 2024, the California Faculty Association (CFA) – the labor union of professors, lecturers, librarians, counselors, and coaches across the 23 California State University campuses – has been in negotiations with the management of the California State University System. After a one-day strike on Monday, January 22, the two sides have reached a tentative agreement, and the strike has been called off. Effective Tuesday, January 23, SSU classes (including CS 454) will be held as scheduled.

Assignment on Proofs, DFAs, and NFAs

Prove: 1² + 2² + ... + n² = n(n+1)(2n+1)/6 for all positive integers n.
Definition: The height of a non-empty tree is defined as the number of nodes on the longest path from a leaf of a tree to its root; the height of an empty tree is defined to be 0.

Nodes in a Binary Tree: Suppose a binary tree T has height h. Prove that T contains at least h nodes and at most 2^h - 1 nodes.
For each of the following, construct a state diagram for the specified language over the alphabet Σ = {0, 1}.
1. All strings containing exactly two 0's (and any number of 1's).
2. All strings in which every 0 is followed by exactly two 1's. (Note the empty string is allowed here, as are strings with just 1's.)
3. All strings that are either all 0's or all 1's. (note the empty string is not allowed here.)
4. All strings that contain exactly three 0's and two 1's.
Non-negative binary integers divisible by 4 (decimal) [100 (binary)]: Consider the binary representation of non-negative integers that are divisible by 4. These numbers have the following properties:
- The number zero (0) is included.
- All non-zero integers must end with the binary digits 00.
- For this problem: All non-zero integers must start with the digit 1 (no leading 0's, except for the number zero itself).
Since the problem involves binary numbers, the relevant alphabet is Σ = {0, 1},
1. Write a DFA with no more than 12 states that accepts exactly these non-negative binary integers that are divisible by 4.
2. Write an NFA with no more than 8 states that accepts exactly these integers. (For this part, the NFA must not also be a DFA.)
For each automaton, be sure all transitions are clearly identified and all states are clearly specified as accepting or not.

Comments in C/C++: C/C++ allows two types of comments:

Any character sequence [on one or more lines] that starts with /*, ends with */ and does not contain */ except at its end.
Any part of a line that begins with // and concludes at the end of the line.

For the purposes of this problem, suppose all characters within a C/C++ program are from the set {/, *, a, b, N, E}, where N represents the end-of-line character and E represents the end-of-file character. (In a real C/C++ program, of course, the characters a, b would be extended to all letters, digits, punctuation, etc., but that seems too extensive for this exercise.)

Notes:

A comment beginning with /*, but having no concluding */, is considered to have a syntax error. A compiler will generate an error, and the program likely will not compile.
All programs must end with an end-of-file character E, but that character may appear only at the end.
An end-of-line character is not required before the end-of-file. Thus, a comment beginning with // may be completed with either N or E.
Comments in C/C++ do not nest.
In the discussion for this exercise, string literals (strings enclosed by double quotes ". . .") are ignored

Here are selected examples from Section 6.4.9 Comments in the 2011 C Standard.

// */;	// comment, not syntax error
f = g/**//h;	// equivalent to f = g / h;
//// l();	// equivalent to l();
m = n//**/o + p;	// equivalent to m = n + p;

Exercise: Consider the challenge of checking the comment structure of a program, ignoring non-comment code and considering statements involving only this limited alphabet.
Write an NFA with no more than 12 states that accepts exactly "programs" in which all comments are correctly specified in C/C++ and rejects any "program" that does not have the proper comment structure..

created Fall, 2023 revised Fall, 2023
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.