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 Undecidability and Reducibility

Proof Summaries: Recall that the Problem 4 of the Assignment on Turing Machines introduced the notion of a proof summary. For this problem, each of the following summaries should be at least 1/2 page.
1. Write a proof summary that outlines the idea behind the proof that the the rational numbers are countable. (Discussion of this result may be found in the textbook.)
2. Write a proof summary that outlines the idea behind the proof of Corollary 4.18 in the text, "Some languages are not Turing-recognizable."
EQ_TM: Write a proof summary for Theorem 5.30. The summary should be at least 2/3 page for the "Turing-recognizable" part of the theorem and at least 2/3 page for the "co-Turing-recognizable" part of the theorem.
Exercise 5.28 in either the second or third edition of Sipser's book presents a careful statement of Rice's Theorem, and a proof of this theorem (more of a proof outline) is stated in the "Selected Solutions" section at the end of that chapter.
1. The formal statement of Rice's Theorem in Exercise 5.28 indicates two necessary conditions for the Theorem. Restate each of those conditions in your own words. (Condition 1 may be relatively straightforward to understand and restate, but Condition 2 may require additional thought to understand and clarify.)
  Note: Discussion of Rice's Theorem and its conditions is discussed extensively on the Web. For example, you might read the Wikipedia article on Rice's theorem.
2. Give a formal statement of Rice's Theorem in your own words, which draws upon your answer to part a of this problem.
3. Use Rice's Theorem to prove exercise 5.30, part b, in Sipser's book (either second or third edition).
Reducibility of a Decidable Language: Suppose A is a language over an alphabet Σ. Also, suppose L is the infinite language {0^*1^*} With this notation, Exercise 5.23 suggests that A is decidable if and only if A ≤_m L. For this problem, assume that Exercise 5.23 has been proven correct.

With this assumption, rather than consider an infinite language, let w be a given string over alphabet {0, 1}, and let W be the language consisting of the one string w; that is W = {w}.

Is it true that A is decidable if and only if A ≤_m W? Justify your answer.

created Fall, 2023 revised Fall, 2023, January14, 2024 revised September, 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.