HW4CSE105F24: Homework assignment 4

In this assignment,

You will work with context-free languages and their representations. You will also practice analyzing, designing, and working with Turing machines. You will use general constructions and specific machines to explore the classes of recognizable and decidable languages.

Resources: To review the topics for this assignment, see the class material from Weeks 4, 5, and 6. We will post frequently asked questions and our answers to them in a pinned Piazza post.

Reading and extra practice problems: Sipser Chapters 2 and 3. Chapter 2 exercises 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.9, 2.10, 2.11, 2.12, 2.13, 2.16, 2.17. Chapter 3 exercises 3.1, 3.2, 3.5, 3.8.

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,you can (1) use the LaTex tikzpicture environment (see templates in the class notes), or (2)) use the software tools Flap.js or JFLAP described in the class syllabus (and include a screenshot in your PDF), or (3) you can carefully and clearly hand-draw the diagram and take a picture and include it in your PDF. 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 “hw4CSE105F24”.

Assigned questions

1. Push-down automata (PDA) and context-free grammars (CFG) (8 points): On page 14 of the week 3 notes, we have the following list of languages over the alphabet \(\{a,b\}\)

   \(\{a^nb^n \mid 0 \leq n \leq 5 \}\)

   \(\{b^n a^n \mid n \geq 2\}\)

   \(\{a^m b^n \mid 0 \leq m\leq n\}\)

   \(\{a^m b^n \mid m \geq n+3, n \geq 0\}\)	\(\{b^m a^n \mid m \geq 1, n \geq 3\}\)
   \(\{ w \in \{a,b\}^* \mid w = w^\mathcal{R} \}\)	\(\{ ww^\mathcal{R} \mid w\in \{a,b\}^* \}\)

   1. (Graded for completeness) ¹ Pick one of the regular languages and design a regular expression that describes it. Briefly justify your regular expression by connecting the subexpressions of it to the intended language and referencing relevant definitions.

   2. (Graded for completeness) Pick another one of the regular languages and design a deterministic finite automaton (DFA) that recognizes it. Draw the state diagram of your DFA. Briefly justify your design by explaining the role each state plays in the machine, as well as a brief justification about how the strings accepted and rejected by the machine connect to the specified language.

   3. (Graded for completeness) Pick one of the nonregular languages and design a PDA that recognizes it. Draw the state diagram of your PDA. Briefly justify your design by explaining the role each state plays in the machine, as well as a brief justification about how the strings accepted and rejected by the machine connect to the specified language.

   4. (Graded for completeness) Pick one of the nonregular languages and write a CFG that generates it. Briefly justify your design by demonstrating how derivations in the grammar relate to the intended language.

2. General constructions for context-free languages (21 points):

   In class in weeks 4 and 5, we described several general constructions with PDAs and CFGs, leaving their details to homework. In this question, we’ll fill in these details. The first constructions help us prove that the class of regular languages is a subset of the class of context-free languages. The other construction allows us to make simplifying assumptions about PDAs recognizing languages.

   1. (Graded for correctness) ² When we first introduced PDAs we observed that any NFA can be transformed to a PDA by not using the stack of the PDA at all. Suppose a friend gives you the following construction to formalize this transformation:

      Given a NFA \(N = (Q, \Sigma, \delta_N, q_0, F)\) we define a PDA \(M\) with \(L(M) = L(N)\) by letting \(M = ( Q, \Sigma, \Sigma, \delta, q_0, F)\) where \(\delta(~(q,a,b)~) = \delta_N(~(q,a)~)\) for each \(q \in Q\), \(a \in \Sigma_{\varepsilon}\) and \(b \in \Sigma_{\varepsilon}\).

      For each of the six defining parameters for the PDA, explain whether it’s defined correctly or not. If it is not defined correctly, explain why not and give a new definition for this parameter that corrects the mistake.

   2. (Graded for correctness) In the book on page 107, the top paragraph describes a procedure for converting DFAs to CFGs:

      You can convert any DFA into an equivalent CFG as follows. Make a variable \(R_i\) for each state \(q_i\) of the DFA. Add the rule \(R_i \to aR_j\) to the CFG if \(\delta(q_i,a) =q_j\) is a transition in the DFA. Add the rule \(R_i\to \varepsilon\) if \(q_i\) is an accept state of the DFA. Make \(R_0\) the start variable ofthe grammar, where \(q_0\) is the start state of the machine. Verify on your own that the resulting CFG generates the same language that the DFA recognizes.

      Use this construction to get a context-free grammar generating the language \[\{ w \in \{0,1\}^* \mid w \text{ does not end in $101$}\}\] by (1) designing a DFA that recognizes this language and then (2) applying the construction from the book to convert the DFA to an equivalent CFG. A complete and correct submission will include the state diagram of the DFA, a brief justification of why it recognizes the language, and then the complete and precise definition of the CFG that results from applying the construction from the book to this DFA. Ungraded bonus: take a sample string in the language and see how the computation of the DFA on this string translates to a derivation in your grammar.

   3. Let \(M_1 = (Q_1, \Sigma, \Gamma_1, \delta_1, q_1, F_1)\) be a PDA and let \(q_{new}, r_{new}, s_{new}\) be three fresh state labels (i.e. \(Q_1 \cap \{q_{new}, r_{new}, s_{new}\} = \emptyset\)) and let \(\#\) be a fresh stack symbol (i.e. \(\# \notin \Gamma_1\)). We define the PDA \(M_2\) as \[(Q_2, \Sigma, \Gamma_2, \delta_2, q_{new}, \{s_{new}\})\] with \(Q_2 = Q_1 \cup \{q_{new}, r_{new}, s_{new}\}\) and \(\Gamma_2 = \Gamma_1 \cup \{\#\}\) and \(\delta_2 : Q_2 \times \Sigma_\varepsilon \times {\Gamma_2}_\varepsilon \to \mathcal{P}(Q_2 \times {\Gamma_2}_\varepsilon)\) given by \[\delta_2 ( ~(q,a,b)~) = \begin{cases} \{(q_1, \#)\} &\text{if } q = q_{new}, a = \varepsilon, b = \varepsilon\\ \delta_1( ~(q,a,b)~) &\text{if } q\in Q_1 \setminus F_1, a \in \Sigma_{\varepsilon}, b \in {\Gamma_1}_\varepsilon \\ \delta_1( ~(q,a,b)~) &\text{if } q\in F_1, a \in \Sigma, b \in {\Gamma_1}_\varepsilon \\ \delta_1( ~(q,a,b)~) &\text{if } q\in F_1, a =\varepsilon, b \in {\Gamma_1} \\ \delta_1( ~(q,a,b)~) \cup \{(r_{new}, \varepsilon)\} &\text{if } q\in F_1, a =\varepsilon, b =\varepsilon \\ \{(r_{new}, \varepsilon)\} &\text{if } q = r_{new}, a =\varepsilon, b \in \Gamma_{1} \\ \{(s_{new}, \varepsilon)\} &\text{if } q= r_{new}, a = \varepsilon, b = \#\\ \emptyset & \text{otherwise} \end{cases}\] for each \(q \in Q_2\), \(a \in \Sigma_{\varepsilon}\), and \(b \in {\Gamma_2}_\varepsilon\).

      In this question, we’ll apply this construction for a specific PDA and use this example to extrapolate the effect of this construction.

      1. (Graded for correctness) Consider the PDA \(M_1\) with input alphabet \(\{0,1\}\) and stack alphabet \(\{0,1\}\) whose state diagram is

         Draw the state diagram for the PDA \(M_2\) that results from applying the construction to \(M_1\).

      2. (Graded for completeness) Compare \(L(M_1)\) and \(L(M_2)\). Are these sets equal? Does your answer depend on the specific choice of \(M_1\)? Why or why not?

      3. (Graded for completeness) Consider the PDA \(N\) with input alphabet \(\{0,1\}\) and stack alphabet \(\{0,1\}\) whose state diagram is

         Remember that the definition of set-wise concatenation is: for languages \(L_1, L_2\) over the alphabet \(\Sigma\), we have the associated set of strings \[L_1 \circ L_2 = \{ w \in \Sigma^* ~|~ w = uv \text{ for some strings } u \in L_1 \text{ and } v \in L_2 \}\] In class, we discussed how extrapolating the construction that we used to prove that the class of regular languages is closed under set-wise concategation by drawing spontaneous transitions from the accepting states in the first machine to the start state of the second machine doesn’t work. Use the example of \(M_1\) and \(N_1\) to prove this by showing that \[L(M_1) \circ L(N)\] is not the language recognized by the machine results from taking the two machines \(M_1\) and \(N\), setting the start state of \(M_1\) to be the start state of the new machine, setting the set of accepting states of \(N\) to be the set of accepting states of the new machine, and drawing spontaneous arrows from the accepting states of \(M_1\) to the start state of \(N\).

      4. (Graded for completeness) Describe the language recognized by the machine that results from taking the two machines \(M_2\) and \(N\), setting the start state of \(M_2\) to be the start state of the new machine, setting the set of accepting states of \(N\) to be the set of accepting states of the new machine, and drawing spontaneous arrows from the accepting states of \(M_2\) to the start state of \(N\). Use this description to explain why we used the construction of \(M_2\) from \(M_1\) and how this construction could be used in a proof of the closure of the class of context-free languages under set-wise concatenation.

3. Turing machines (12 points):

   Consider the Turing machine \(T\) over the input alphabet \(\Sigma = \{0,1\}\) with the state diagram below (the tape alphabet is \(\Gamma = \{ 0,1,X,\square\}\)). Convention: we do not include the node for the reject state \(qrej\) and any missing transitions in the state diagram have value \((qrej,\square,R)\)

   1. (Graded for correctness) Specify an example string \(w_1\) of length \(4\) over \(\Sigma\) that is accepted by this Turing machine, or explain why there is no such example. A complete solution will include either (1) a precise and clear description of your example string and a precise and clear description of the accepting computation of the Turing machine on this string or (2) a sufficiently general and correct argument why there is no such example, referring back to the relevant definitions.

      To describe a computation of a Turing machine, include the contents of the tape, the state of the machine, and the location of the read/write head at each step in the computation.

      Hint: In class we’ve drawn pictures to represent the configuration of the machine at each step in a computation. You may do so or you may choose to describe these configurations in words.

   2. (Graded for correctness) Specify an example string \(w_2\) of length \(3\) over \(\Sigma\) that is rejected by this Turing machine or explain why there is no such example. A complete solution will include either (1) a precise and clear description of your example string and a precise and clear description of the rejecting computation of the Turing machine on this string or (2) a sufficiently general and correct argument why there is no such example, referring back to the relevant definitions.

   3. (Graded for correctness) Specify an example string \(w_3\) of length \(2\) over \(\Sigma\) on which the computation of this Turing machine is never halts or explain why there is no such example. A complete solution will include either (1) a precise and clear description of your example string and a precise and clear description of the looping (non-halting) computation of the Turing machine on this string or (2) a sufficiently general and correct argument why there is no such example, referring back to the relevant definitions.

      Note: when a Turing machine does not halt on a given input string, we say that it loops on that string.

4. Implementation-level descriptions of deciders and recognizers (9 points):

   For this question, consider the alphabet \(\Sigma = \{a,b,c\}\).

   1. (Graded for correctness) Give an example of an infinite language over \(\Sigma\) (that is not \(\Sigma^*\)) and give two different Turing machines that recognize it: one that is a decider and one that is not. A complete solution will include a precise definition for your example language, along with both a state diagram and an implementation-level description of each Turing machines, along with a brief explanation of why each of them recognizes the language and why one is a decider and there other is not.

   2. (Graded for completeness) True or false: There is a Turing machine that is not a decider that recognizes the empty set. A complete solution will include a witness Turing machine (given by state diagram or implementation-level description or high-level description) and a justification for why it’s not a decider and why it does not accept any strings, or a complete and correct justification for why there is no such Turing machine.

   3. (Graded for completeness) True or false: There is a Turing machine that is not a decider that recognizes the set of all string \(\Sigma^*\). A complete solution will include a witness Turing machine (given by state diagram or implementation-level description or high-level description) and a justification for why it’s not a decider and why it accept each string over \(\{a,b,c\}\), or a complete and correct justification for why there is no such Turing machine.