HW1CSE105W24: Homework assignment 1

In this assignment,

You will practice reading and applying the definitions of alphabets, strings, languages, Kleene star, and regular expressions. You will use regular expressions and relate them to languages and finite automata. You will use precise notation to formally define the state diagram of finite automata, and you will use clear English to describe computations of finite automata informally.

Resources: To review the topics for this assignment, see the class material from Week 1. We will post frequently asked questions and our answers to them in a pinned Piazza post.

Reading and extra practice problems: Sipser Section 0, 1.3, 1.1. Chapter 1 exercises 1.1, 1.2, 1.3, 1.18, 1.23.

For all HW assignments: Weekly homework may be done individually or in groups of up to 3 students. You may switch HW partners for different HW assignments. Please ensure your name(s) and PID(s) are clearly visible on the first page of your homework submission and then upload the PDF to Gradescope. If working in a group, submit only one submission per group: one partner uploads the submission through their Gradescope account and then adds the other group member(s) to the Gradescope submission by selecting their name(s) in the “Add Group Members" dialog box. You will need to re-add your group member(s) every time you resubmit a new version of your assignment. Each homework question will be graded either for correctness (including clear and precise explanations and justifications of all answers) or fair effort completeness. For “graded for correctness” questions: collaboration is allowed only with CSE 105 students in your group; if your group has questions about a problem, you may ask in drop-in help hours or post a private post (visible only to the Instructors) on Piazza. For “graded for completeness” questions: collaboration is allowed with any CSE 105 students this quarter; if your group has questions about a problem, you may ask in drop-in help hours or post a public post on Piazza.

All submitted homework for this class must be typed. You can use a word processing editor if you like (Microsoft Word, Open Office, Notepad, Vim, Google Docs, etc.) but you might find it useful to take this opportunity to learn LaTeX. LaTeX is a markup language used widely in computer science and mathematics. The homework assignments are typed using LaTeX and you can use the source files as templates for typesetting your solutions. To generate state diagrams of machines, we recommend using Flap.js or JFLAP. Photographs of clearly hand-drawn diagrams may also be used. We recommend that you submit early drafts to Gradescope so that in case of any technical difficulties, at least some of your work is present. You may update your submission as many times as you’d like up to the deadline.

Integrity reminders

• Problems should be solved together, not divided up between the partners. The homework is designed to give you practice with the main concepts and techniques of the course, while getting to know and learn from your classmates.

• You may not collaborate on homework questions graded for correctness with anyone other than your group members. You may ask questions about the homework in office hours (of the instructor, TAs, and/or tutors) and on Piazza (as private notes viewable only to the Instructors). You cannot use any online resources about the course content other than the class material from this quarter – this is primarily to ensure that we all use consistent notation and definitions (aligned with the textbook) and also to protect the learning experience you will have when the ‘aha’ moments of solving the problem authentically happen.

• Do not share written solutions or partial solutions for homework with other students in the class who are not in your group. Doing so would dilute their learning experience and detract from their success in the class.

You will submit this assignment via Gradescope (https://www.gradescope.com) in the assignment called “hw1CSE105W24”.

Assigned questions

1. (Graded for completeness) ¹ Finding examples and edge cases (12 points):

   With \(\Sigma_1 = \{0,1\}\) and \(\Sigma_2 = \{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}\) and \(\Gamma = \{0,1,x,y,z\}\)

   1. Give an example of the shortest string over \(\Sigma_1\) that is meaningful to you in some way, and explain why it’s meaningful to you.

   2. List all examples of strings of length \(1\) over \(\Sigma_2\) and explain why your list is exhaustive.

   3. Calculate the number of distinct strings of length 3 over \(\Gamma\) and explain your calculation.

   4. With the ordering \(x < y < z < 0 < 1\), list the first ten strings over \(\Gamma\) in string order.

   5. Give an example of a finite set that is a language over \(\Sigma_1\) and over \(\Sigma_2\) and over \(\Gamma\), or explain why there is no such set.

   6. Give an example of an infinite set that is a language over \(\Sigma_1\) and over \(\Gamma\), or explain why there is no such set.

2. (Graded for completeness) Regular expressions (10 points):

   1. Give three regular expressions that all describe the set of all strings over \(\{a,b\}\) that have even length. Ungraded bonus challenge: Make the expressions as different as possible!

   2. A friend tells you that each regular expression that has a Kleene star (\(~^*\)) describes an infinite language. Are they right? Either help them justify their claim or give a counterexample to disprove it and then fix the formula.

3. (Graded for correctness) ² Functions over languages (15 points):

   For languages \(L_1, L_2\) over the alphabet \(\Sigma_1 = \{0,1\}\), we have the associated sets of strings \[SUBSTRING(L_1) = \{ w \in \Sigma_1^* ~|~ \text{there exist } a,b \in \Sigma_1^* \text{ such that } awb \in L_1\}\] and \[L_1 \circ L_2 = \{ w \in \Sigma_1^* ~|~ w = uv \text{ for some strings } u \in L_1 \text{ and } v \in L_2 \}\]

   1. Specify an example language \(A\) over \(\Sigma_1\) such that \(A \neq \emptyset\) and yet \(\textsc{Substring}(A) = \emptyset\), or explain why there is no such example. A complete solution will include either (1) a precise and clear description of your example language \(A\) and a precise and clear description of the result of computing \(\textsc{Substring}(A)\) using relevant definitions to justify this description and to justify the set equality with \(\emptyset\), or (2) a sufficiently general and correct argument why there is no such example, referring back to the relevant definitions.

   2. Specify example languages \(B, C\) over \(\Sigma_1\) such that \(B \neq \Sigma_1^*\) and \(C \neq \Sigma_1^*\) and yet \(B \circ C = \Sigma_1^*\), or explain why there are no such examples. A complete solution will include either (1) a precise and clear description of your example languages \(B,C\) and a precise and clear description of the result of computing \(B \circ C\) using relevant definitions to justify this description and to justify the set equality with \(\Sigma_1^*\), or (2) a sufficiently general and correct argument why there is no such example, referring back to the relevant definitions.

   3. Specify example finite languages \(L_1, L_2\) over \(\Sigma_1\) such that \(L_1 \circ L_2 \neq L_1\) but \(|L_1 \circ L_2| = |L_1|\), or explain why there are no such examples. A complete solution will include either (1) a precise and clear description of your example languages \(L_1, L_2\) and a precise and clear description of the result of computing \(L_1 \circ L_2\) using relevant definitions to justify this description and to justify the cardinality claims and set (in)equality claims, or (2) a sufficiently general and correct argument why there is no such example, referring back to the relevant definitions.

4. (Graded for correctness) Finite automata (13 points):

   Consider the finite automaton \((Q, \Sigma, \delta, q_0, F)\) whose state diagram is depicted below

   where \(Q = \{q_0, q_1, q_2, q_3\}\), \(\Sigma = \{0,1\}\), and \(F = \{q_0\}\), and \(\delta: Q \times \Sigma \to Q\) is specified by the look-up table

   	\(0\)	\(1\)
   \(q_0\)	\(q_1\)	\(q_0\)
   \(q_1\)	\(q_2\)	\(q_3\)
   \(q_2\)	\(q_0\)	\(q_3\)
   \(q_3\)	\(q_3\)	\(q_3\)

   1. A friend tries to summarize the transition function with the formula \[\delta(q_i,x) = \begin{cases} q_0 &\text{ when $i=0$ and $x=1$} \\ q_3 &\text{ when $0<i\leq 3$ and $x=1$}\\ q_j &\text{ when $j = (i+1) \mod 3$ and $x=0$} \end{cases}\] Are they right? Either help them justify their claim or give a counterexample to disprove it and then fix their formula.

   2. Give a regular expression \(R\) so that \(L(R)\) is the language recognized by this finite automaton. Justify your answer by referring to the definition of the semantics of regular expressions and computations of finite automata. Include an explanation for why each string in \(L(R)\) is accepted by the finite automaton and for why each string not in \(L(R)\) is rejected by the finite automaton.

   3. Keeping the same set of states \(Q = \{q_0, q_1, q_2, q_3\}\), alphabet \(\Sigma = \{0,1\}\), same start state \(q_0\), and same transition function \(\delta\), choose a new set of accepting states \(F_{new}\) so that the new finite automaton that results accepts at least one string that the original one rejected and rejects at least one string that the original one accepted, or explain why there is no such choice of \(F_{new}\). A complete solution will include either (1) a precise and clear description of your choice of \(F_{new}\) and a precise and clear the two example strings using relevant definitions to justify them, or (2) a sufficiently general and correct argument why there is no such example, referring back to the relevant definitions.