DF CS 454: Theory of Computation

CS 454, Section 001	Sonoma State University	Spring, 2026

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, some details may be adjusted from semester to semester. For example, the Signature Project for this course has satisfied SSU's Upper Division GE Area B Requirement for CS Majors for several years, and satisfying that requirement again this semester. However, details of this project likely vary from instructor to instructor and from semester to semester. Also,

This course syllabus can be expected to remain stable through the semester (except for clarifying statements or for correcting typographical errors, if any).
The course schedule, assignments, elements of the Signature Project, and other course materials may be adjusted as the semester progresses, until the week before student work is due. Be sure to check/reload Web pages for the course before starting required course work.

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.

Observation: When a conclusion has two parts, a proof might include one argument for the two parts together, or a proof might be in two separate/independent pieces (one for each part).
For each of the following, construct a state diagram (DFA) for the specified language over the alphabet Σ = {a, b}.
1. All strings containing exactly two b's (and any number of a's).
2. All strings in which every b is followed by exactly three a's. (Note the empty string is allowed here, as are strings with just a's.)
3. All strings that are either all a's or all b's. (note the empty string is not allowed here.)
4. All strings that contain exactly three a's and two b's.
Note: Any DFA must have transitions from each state for every character in the alphabet Σ. Just showing transitions for accepted strings is not adquate.
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. Draw a DFA (state diagram) with no more than 12 states that accepts exactly these non-negative binary integers that are divisible by 4.
2. Draw an NFA (state diagram) 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 Σ = {/, *, 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: Draw an NFA state diagram 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.

The NFA should read any sequence of characters from the above alphabet until the end-of-line character E is encountered.
Once the start of a comment is encountered, subsequent characters are ignored until the end of the comment is found.
When not within a comment, all characters should be ignored (except to identify the start of the next comment—elements of C/C++ systax other than comments should not be analyzed).

created Fall, 2023 revised Fall, 2023 revised October, 2025
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.

Copyright © 2011-2026 by Henry M. Walker.
Selected materials copyright by Marge Coahran, Samuel A. Rebelsky, John David Stone, and Henry Walker and used by permission.
This page and other materials developed for this course are under development.
This and all laboratory exercises for this course are licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.