General comment. The main purpose of most homeworks is to show how well you understand the algorithms.

In many cases, the resulting finite automata, pushdown automata, and Turing machines can be simplified, but please first literally apply the algorithm so that we know that you can use it.

If in addition to this, you also show how to make the corresponding Turing machine or finite automaton or whatever more concise, nothing wrong with that, the TA may even give you some extra points (if he/she has time to grade these additional things). But the most important thing is to show that you can follow the algorithm.

For simple examples that we give you as homeworks:

• you may immediately see how to convert, e.g., a context-free grammar into a Chomsky normal form,
• but if someone gives you a more complex case, you will have to use the algorithm.

If you deviate from the algorithm, how do we know that you learned the algorithms? It was the same with sorting.

• Of course, if someone gives you a list of 4 numbers on the test, you can sort them yourself easily.
• The purpose of the question was that you show that you understand mergesort, quicksort etc., not that you sort 4 numbers.

• If after you show that you understand the algorithm you also provide a simpler answer -- great,
• but not instead of following the algorithm.

1. (Due February 3) In class, we designed automata for recognizing integers and real numbers.

1.1. Use the same ideas to describe an automaton for a body position: sit, stand, or lie down. You start with a normal (sitting) position (N), u (up) brings you to standing (S), down (d) brings you to lying down (L), and vice versa. A sequence of ups and downs is accepted if you end up with in the normal (sitting) position.

A natural idea is to have 3 states:

• the start state N (sitting) which is also final,
• the state S meaning standing, and
• the state L meaning lying down.

• from N, u leads to S and d to L;
• from S, u leads back to S, and d leads to N; and
• from L, u leads to N and d leads back to L.

1.2. Trace, step-by-step, how the finite automaton from Part 1.1 will check whether the following two words (sequences of symbols) lead back to the normal position:

• the word ud (which this automaton should accept) and
• the word uu (which this automaton should reject).

1.3. Write down the tuple <Q, Σ, δ, q₀, F> corresponding to the automaton from Part 1.1:

• Q is the set of all the states,
• Σ is the alphabet, i.e., the set of all the symbols that this automaton can encounter; for simplicity, consider only two symbols: u and d;
• δ: Q x Σ → Q is the function that describes, for each state q and for each symbol s, the state δ(q,s) to which the automaton that was originally in the state q moves when it sees the symbol s (you do not need to describe all possible transitions this way, just describe two of them);
• q₀ is the starting state, and
• F is the set of all final states.

1.4. For each automaton A, let L_A denote the language of all the words accepted by this automaton, i.e., of all the words for which this automaton ends up in a final state. In class, we learned a general algorithm that:

• given two automata A and B that correspond to languages L_A and L_B,
• constructs two new automata for recognizing, correspondingly, the union and intersection of languages L_A and L_B.

• the automaton from Part 1.1 as Automaton A, and
• an automaton for words that end with down as Automaton B.

• the state D meaning that the last move was down, and
• the state N meaning that the last move was not down.

• from N, d leads to D and u leads back to N;
• from D, d leads back to D and u leads to N.

Send solutions to Problem 1 to Melina Salazar-Perez.

Solution to Problem 1

2. (Due February 3)

• Use the general algorithm that we learned in class to design a non-deterministic finite automaton that recognizes the language (u U d)*d of all the words that end with d.
  • u and d are languages consisting of only one 1-symbol word each: u is a language consisting of a single 1-symbol word u; d is a language consisting of a single 1-symbol word d;
  • for any two languages C and D, the notation CD means concatenation;
• transform the resulting non-deterministic finite automaton into a deterministic one.

Send solutions to Problem 2 to Melina Salazar-Perez.

Solution to Problem 2

3. (Due February 5) Apply the general algorithm for transforming the finite automaton into a regular language (i.e., a language described by a regular expression) to Automaton B from Problem 1.4.

Send solutions to Problem 3 to Md. Jahangir Alam.

Solution to Problem 3

4. (Due February 12) Prove that the language L of all the sequences that have exactly the same numbers of ups and downs is not regular. For example, the word uddu is in the language L, while the word duu is not in L.

Send solutions to Problem 4 to Garab Dorji.

Solution to Problem 4

5. (Due February 12) Write and test a method that simulates a general finite automaton. Your program should enable the computer to simulate any given finite automaton and then to simulate, for any given word, step-by-step, how this automaton decides whether this word is accepted by the automaton.

The input to this method should include the full description of the corresponding finite automaton:

• the number N of states; these states are q₀, ..., q_{N − 1}; we assume that q₀ is the start state; for simplicity, we describe the states by the corresponding integers 0, ..., N;
• the number M of symbols; these symbols are s₀, ... s_{M − 1}; for simplicity, we describe the symbols by the corresponding integers 0, ..., M;
• an integer array state[n][m] whose elements describe to what state the finite automaton moves if it was in the state q_n and sees the symbol s_m; and
• a boolean array final[n] whose elements describe, for each state q_n, whether this state is final or not.

For this and other programming assignments, turn in the actual code and the printout of the result of testing this code.

Send solutions to Problem 5 to Garab Dorji.

6. (Due February 12) Show, step by step, how the word uddu is accepted by the following pushdown automaton that accepts all the words that have equal number of d's and u's. This pushdown automaton has 5 states:

• the starting state S,
• the state B (for "balanced") indicating that so far we had equal number of u's and d's,
• the state U indicating that so far, we had more u's than d's;
• the state D indicating that so far, we had more d's than u's; and
• the final state F.

In the stack, in addition to the bottom symbol $, we have:

• either one or several u's -- indicating the difference between the number of u's and d's minus 1,
• or one or several d's -- indicating the difference between the number of d's and u's minus 1.

• From S to B, the transition is ε, ε → $.
• From B to U, the transition is: u, ε → ε.
• From U to B, the transition is: d, $ → $.
• From B to D, the transition is: d, ε → ε.
• From D to B, the transition is: u, $ → $.
• From B to F, the transition is: ε, $ → ε.
• From U to U, we have two transitions: u, ε → u and d, u → ε.
• From D to D, we have two transitions: d, ε → d and u, d → ε.

Send solutions to Problem 6 to Garab Dorji.

Solution to Problem 6

7. (Due February 12) Show, step by step, how the following grammar describing sequences with equal number of u's and d's will generate the expression uddu. The rules are:

• S → ε
• S → uSd
• S → dSu
• S → SS

Send solutions to Problem 7 to Garab Dorji.

Solution to Problem 7

8. (Due February 26) In the corresponding lecture, we described an algorithm that, given a finite automaton, produces a context-free grammar -- a grammar that generate a word if and only if this word is accepted by the given automaton.

• On the example of the automaton B from Homework 1.4, show how this algorithm will generate the corresponding context-free grammar.
• On the example of the word uud accepted by this automaton, show how the tracing of acceptance of this word by the finite automaton can be translated into a generation of this same word by your context-free grammar.

Send solutions to Problem 8 to Melina Salazar-Perez.

Solution to Problem 8

9. (Due February 26) Use a general algorithm to construct a (non-deterministic) pushdown automaton that corresponds to context-free grammar described in Problem 7. Show, step by step, how the expression uddu will be accepted by this automaton.

Send solutions to Problem 9 to Melina Salazar-Perez.

Solution to Problem 9

10. (Due March 5) Transform the grammar from Homework 7 into Chomsky normal form.

Send solutions to Problem 10 to Md. Jahangir Alam.

Solution to Problem 10

11. (Due March 5) Use the general algorithm to transform the pushdown automaton from Problem 6 into a context-free grammar. Before doing this, you need to replace each transition in which you pop and push with two transitions in which you first pop, and then push. Show, step-by-step, how the resulting grammar will generate the word ud.

Send solutions to Problem 11 to Md. Jahangir Alam.

Solution to Problem 11

12. (Due March 19) For the grammar described in Homework 7, show how the expression uddu can be represented as uvxyz in accordance with the pumping lemma for context-free grammars. Show that the corresponding word uvvxyyz will be generated by this grammar.

Send solutions to Problem 12 to Garab Dorji.

Solution to Problem 12

13. (Due March 19) A perfect arrangement would be for a student to have equal number of up and downs, and also to have a sit-and-study experience (s) for each up or down. Show that the language of all the words that have equal number of u's and d's and and twice as many s's is not context-free.

Send solutions to Problem 13 to Garab Dorji.

Solution to Problem 13

14. (Due March 19) Show, step by step, how the stack-based algorithm will transform the expression (2 + 0) * (2 − 5) into a postfix expression, and then how a second stack-based algorithm will compute the value of this postfix expression.

Send solutions to Problem 14 to Melina Salazar-Perez.

Solution to Problem 14

15. (Due March 26) Write a program that, given an arithmetic expression,

• first transforms it to a postfix form, and then
• computes its value (by using the stack-based algorithms that we recalled in class).

Comments:

• as with all programming assignments for this class, submit a file containing the code, and a file containing an example of what this program generates on each step;
• ideally, use Java, but if you want to write it in some other programming language, check with the TA that it is OK; usually, C or C++ are OK.

Send solutions to Problem 15 to Md. Jahangir Alam.

16. (Due April 2) Design a Turing machine that, given a binary number n which is larger than or equal to 4, subtracts 4 from this number. Test it, step-by-step, on the example of n = 5.

Send solutions to Problem 16 to Md. Jahangir Alam.

Solution to Problem 16

17. (Due April 2) Design a Turing machine that, given a unary number n, adds 2 to this number. Test it, step-by-step, on the example of n = 0.

Send solutions to Problem 17 to Md. Jahangir Alam.

Solution to Problem 17

18. (Due April 2) Use the general algorithm to transform a finite automaton B from Homework 1.4 into a Turing machine. Show step-by-step, on an example of a word ud, how this word will be processed by your Turing machine.

Send solutions to Problem 18 to Md. Jahangir Alam.

Solution to Problem 18

19. (Due April 2) As described in the corresponding lecture, every grammar obtained from a finite automaton is LL(1). For the grammar from Homework 8, build the corresponding table. Use this table to check whether the word ud can be derived in this grammar.

Send solutions to Problem 19 to Md. Jahangir Alam.

Solution to Problem 19

20. (Due April 9) As we discussed in the corresponding lecture, a Turing machine can be described as a finite automaton with two stacks:

• the right stack contains, on top, the symbol to which the head points; below is the next symbol to the right, then the next to next symbol to the right, etc.;
• the left stack contains, on top, the symbol directly to the left of the head (if there is a one), under it is the next symbol to the left, etc.

• how each state of the corresponding Turing machine can be represented in terms of two stacks, and
• how each transition from one state to another can be implemented by push and pop operations.

Send solutions to Problem 20 to Garab Dorji.

Solution to Problem 20

21. (Due April 9) Write and test a method that simulates a general Turing machine. Your program should enable the computer to simulate any given Turing machine for accepting-rejecting and then to simulate, for any given word, step-by-step, how this Turing machine decides whether this word is accepted or not.

The input to this method should include:

• the number N of states q₀, ..., q_{N − 1}; we assume that q₀ is the start state, that the last-but-one state q_{N − 2} is the accept state, and the last state q_{N − 1} is the reject state;
• the number M of symbols s₀, ... s_{M − 1}; we assume that s₀ is the blank state _;
• an integer array state[n][m] that describes to what state the head of the Turing machine changes if it was in the state q_n and sees the symbol s_m;
• an integer array symbol[n][m] that describes what symbol should be on the tape after the head in the state q_n sees the symbol s_m (it may be the same symbol as before, or it may be some other symbol written by the Turing machine);
• a character array lr[n][m] that describes, for each state q_n and for each symbol s_m, whether the head moves to the left (L), or to the right (R), or stays in place (blank symbol);
• the integer array of a large size describing the original contents of the tape, i.e., what symbols are written in each cell.

Your program should simulate the work of the Turing machine step-by-step. Return the printout of the method, the printout of the program that you used to test this method, and the printout of the result of this testing. Feel free to use Java, C, C++, Fortran, or any programming language in which the code is understandable.

Send solutions to Problem 21 to Garab Dorji.

22. (Due April 16) Give two examples:

• an example of computation time which makes an algorithm feasible according to the formal definition but not practically feasible, and
• an example of computation time for which the corresponding algorithm is practically feasible, but not feasible according to the formal definition.

Send solutions to Problem 22 to Melina Salazar-Perez.

Solution to Problem 22

23. (Due April 23) What is NP? What is P? What is NP-complete? What is NP-hard? Give brief definitions. Give an example of an NP-complete problem. Is P equal to NP?

Send solutions to Problem 23 to Melina Salazar-Perez.

Solution to Problem 23

24. (Due April 23) Prove that the cubic root of 24 is not a rational number.

Send solutions to Problem 24 to Melina Salazar-Perez.

Solution to Problem 24