# Assignment 6: Logic

1. (20 points) Draw a truth table that shows all 16 Boolean functions of 2 variables, A and B. Label the columns with their common names. You know most of these: , , (exclusive or), , , , NAND, NOR, true, false, A, B, A, B. What are the other two? Which of the functions are commutative? Which are associative? Which are complete? (You know that NAND and NOR are complete; are any of the others? Prove/explain your answer in each case.)

2. (10 points) Prove the generalized DeMorgan's laws (equations 12.20(c) and 12.20(d) in the book, p. 677) by induction on k, using the basic laws (equations 12.20(a) and 12.20(b)).

3. (20 points) Draw circuit diagrams for three different 2-bit adders (adders that take two 2-bit inputs and compute a 2-bit output and a carry bit): a ripple-carry adder, a carry-look-ahead (divide-and-conquer adder), and a sum-of-products (disjunctive-normal-form, without trying to eliminate any literals or terms) adder. For each of the adders, give the total number of gates and the depth (the number of gates on the longest path).

4. (Extra Credit: up to 15 points) Determine, for each of the three classes of adders mentioned in the previous questions, a general formula for the total number of gates and the depth, as a function of the number of bits N in each operand. (This is not a big-O question: I'm looking for an exact answer.)

5. (15 points) Minimize the following expression using a Karnaugh map (i.e., remove as many literals and terms as possible): (ABCD)(ABCD) (ABCD) (ABCD) (ABCD). Express the result in product of sums notation. (Hint: You will want to directly generate the product of sums notation from the Karnaugh map). Also, minimize the expression using algebraic laws with the result in sum of products (as opposed to product of sums) notation. Show that the two are equivalent (derive one algebraically from the other).

6. (10 points) Translate the following statements into prenex CNF form (all quantifiers on the outside, insides written as a logical product of sums):
1. A[student(A) (dorm_resident(A) B[class(B) takes(A,B)])
2. AB [(C[takes(A,C) takes(B,C)]) classmates(A,B)]

7. (10 points) Suppose big(N) is interpreted to mean N > 2 and works(A,B,C,N) is interpreted to mean AN + BN = CN. State Fermat's last theorem in predicate calculus, first in whatever form seems most intuitive, then in prenex CNF. (Fermat's last theorem holds that the equation AN + BN = CN has no solutions over the non-zero integers for N > 2. It was posed around 1630, but wasn't solved [by Andrew Wiles] until 1995.)

8. (15 points) Prove the following by contradiction, using resolution. (Restate the premises and the negation of the conclusion in CNF to create a database [set] of terms, then apply resolution to obtain new terms, until you derive the empty term [false].) Premise one: (A B) (C D). Premise two: (A B). Conclusion: C D.

9. (Extra Credit: 10 points) Prove that it is impossible to write a program that takes two arbitrary programs as input and tells you whether they do the same thing (i.e. whether given the same input they either both halt, with the same output, or both run forever.) You may take as a given that the halting problem is undecidable.