16

5. Convert the below BNF into EBNF. (program) → begin (stmt_list) end (stmt_list)→→ (stmt) |(stmt); (stmt_list) (stmt) → (var) = (expression) (var) → A|B|C (expression) → (var) + (var) | (var) - (var) | (var)

8- In database ER diagram, how do we uniquely identify relationships? A. Primary key of participating entities B. Primary key of the relation itself C. By its attributes D. Relationships cannot be uniquely identified

1. (40 pt., 10 pt. each) Construct a Turing machine in JFLAP (version 7.1) that decides each of the following languages. For each language, you must submit one JFLAP file clearly labeled (e.g., 1a.jff). Make sure that you test your Turing machines in JFLAP before submitting. Note: there is no explicit reject state for Turing machines in JFLAP. You should assume that there is a transition to the reject state whenever a state is missing a transition for a particular symbol. a. A = {we {a,b}* | w contains at least one a and at most one b} b. B = {w € {a,b}" | w contains more a's than b's} c. C = {a¹b/c+/|ij≥0} d. D= {0¹1 |n, m≥ 0 and n is divisible by m} For example, 000011 € D (because 4 is divisible by 2) and 00011 # D (because 3 is not divisible by 2).

10- Given a K-map with one 1 circled and the equation is ABCD (ANDed), what is it? (third circle in the first row) A. Minterm B. Maxterm C. Prime implicant D. Essential prime implicant

6- what is changing one requirement, while taking into account competing requirements A. Software Feasibility B. Software Consistency C. Software Maintainability D. Software Stability

Problem 2 (13 pts.) Consider the following claim. Claim. For any sets S and T, (SxT)n (Tx S)-(Sx S) = 0. (a) (11 pts.) Use a proof by contradiction to prove the claim. To get full points you must use a mixture of formal notation and word explanations (e.g. the "column" format). Each step of your proof should have an explanation as to how/why you could make that logical step. When in doubt, more detail is better than less. Grading Notes. While a detailed rubric cannot be provided in advance as it gives away solution details, the following is a general idea of how points are distributed for this problem. We give partial credit where we can. (9) Correctness. If your proof is not correct, this is where you'll get docked. (5) Regardless of how you formulate your proof, somewhere you'll need certain facts without which the proof wouldn't work. E.g. if it weren't true that the sum of two integers is integer, would your proof fail? If so, then that is a fact I need to see stated somewhere. (1) The order of these facts must make sense, so that you're not inferring something before you have all the facts to infer it. E.g. you cannot use the fact that the sum of two integers is integer if you don't already know that you have two integers to begin with. (3) You also must use a proof by contradiction, which clearly states it is a proof by contradiction, states what the contradictory assumption is, finds a contradiction, and clearly states what and where that contradiction is. (2) Communication. We need to see a mix of notation and intuition, preferably in the "column" format with the notation on the left, and the reasons on the right. If you skip too many steps at once, or we cannot follow your proof, or if your solution is overly wordy or confusing, this is where you'll get docked. (b) (2 pts.) Is it possible to prove this claim by contrapositive? If so, what would the statement of the claim be (that you could then apply the contrapositive to)? If not, give a brief explanation why it cannot be done. Grading Notes. This problem is meant for you to think about whether you can modify your proof to be of a different form, and explaining your answer.

4- When the developers make the program in an architecture such that it can be used by other customers in the future A. Customer-specific program B. Program files C. Program generator

Problem 1 (23 pts.) Consider the following claim: Claim. {21n: n € Z} U {14n: n € Z} c{7n:n €Z}. (a) (3 pts.) Write the claim as an (equivalent) if-then statement. Grading Notes. While a detailed rubric cannot be provided in advance as it gives away solution details, the following is a general idea of how points are distributed for this problem. We give partial credit where we can. (3) Correctness. Regardless of how you formulate your proof, you will need to have an if-then statement that is equivalent to the original. (b) (11 pts.) Give a direct proof by cases that the claim is true. As a hint, you might want to prove the if-then statement you constructed in (a). To get full points you must use a mixture of formal notation and word explanations (e.g. the "column" format). Each step of your proof should have an explanation as to how/why you could make that logical step. When in doubt, more detail is better than less. Grading Notes. While a detailed rubric cannot be provided in advance as it gives away solution details, the following is a general idea of how points are distributed for this problem. If you can at least get part-way, we give partial credit where we can. (9) Correctness. If your proof is not correct, this is where you'll get docked. (2) Regardless of how you formulate your proof, you will need clearly labeled exhaustive cases. (6) Regardless of how you formulate your proof, somewhere you'll need certain facts without which the proof wouldn't work. E.g. if it weren't true that the sum of two integers is integer, would your proof fail? If so, then that is a fact I need to see stated somewhere. (1) The order of these facts makes sense, so that you're not inferring something before you have all the facts to infer it. E.g. you cannot use the fact that the sum of two integers is integer if you don't already know that you have two integers to begin with. The order of how you use these does not have to exactly match those in the sample solutions, but there are orders that will not work and you will lose points if, for example, you use "the difference of ints is int" before you use "the product of ints is int". If you combine some steps (such as "the difference and product of ints is int" or "the product of two non-zero ints is a non-zero int") that is fine. Just don't combine all (see below). (2) Communication. We need to see a mix of notation and intuition, preferably in the "column" format with the notation on the left, and the reasons on the right. If you skip too many steps at once, or we cannot follow your proof, or if your solution is overly wordy or confusing, this is where you'll get docked. (c) (3 pts.) State (but do not prove) the contrapositive of your statement from part (a). Grading Notes. While a detailed rubric cannot be provided in advance as it gives away solution details, the following is a general idea of how points are distributed for this problem. We give partial credit where we can. (3) Correctness. You have to have the contrapositive statement of whatever if-then state- ment you wrote in part (a). (d) (3 pts.) State (but do not prove) the converse of your statement from part (a). Grading Notes. While a detailed rubric cannot be provided in advance as it gives away solution details, the following is a general idea of how points are distributed for this problem. We give partial credit where we can. (3) Correctness. Regardless of how you formulate your proof, you will need to have an if-then statement that is the contrapositive of the original. (e) (3 pts.) Give a disproof by counter-example of the converse from part (d). (That is, show that the converse is not true by providing an example that demonstrates it is not true.) Remember that any disproof by counter-example not only provides the counter-example, but also an explanation as to why it is a counter-example. Grading Notes. Note that this problem gives you practice disproving a statement by counter-example, which requires clearly stating a counter-example, and then showing why it is a counter-example.

4. (a) Write a grammar for the following language consisting of strings that have n copies of the letter 'a' followed by one more number of copies of the letter 'b', where n > 0. Examples of such strings are abb, aabbb, etc, but a, ab, ba, and aaabb are not. (b) Draw parse trees for the sentences abb and aabbb in Problem 4.

2. (40 pt., 10 pt. each) Give an implementation-level description of a Turing machine that decides each of the languages in Problem 1. a. A = {w = {a,b}* | w contains at least one a and at most one b} b. B = {w = {a,b}" | w contains more a's than b's} c. C = {a¹b/citii,j ≥ 0} d. D= {01m|n, m≥ 0 and n is divisible by m} For example, 000011 € D (because 4 is divisible by 2) and 00011 # D (because 3 is not divisible by 2).