1. (Due January 23) Exercise 1.1 from the book.
2. (Due January 25) Prove that the square root of 3 is irrational.
3. (Due January 25) Prove that there exists an irrational number x for which x raised to the square root of 3 power is rational.
4. (Due January 30) Exercise 1.2, Part 3, (a)-(e); Exercise 1.5, Part 2, Part 3(a), and Part 5.
5. (Due February 8) Use the Horn clauses and wave algorithm to solve the following problem: we know that V = V1 + V2, V1 = I * R1, V2 = I * r2, V = I * R, we know the values of I, R1, and R2, we need to find R.
6. (Due February 8) Exercise 2.1, Part 1)-4).
7. (Due February 20) Use resolution to prove the following two statements: (a → b) → (¬b → ¬a) and (a & b → c) → (a & ¬b → ¬c).
8. (Due February 20) Use resolution to prove that if we have ∀x∀y∀z (x > y & y > z → x > z) and ∀x (¬x > x), then ∀x∀y (x > y → ¬y > ¬ x).
9. (Due February 27) Use the Prolog website to derive a simple fact about your own relative.
10. (Due March 21) For the case of interval uncertainty, come up with simple algorithms for checking the validity of modal logic formulas (a < b) and ◊(a < b).
11. (Due March 21) Solve the following tolerance problem: Let us assume that 9 ≤ a ≤ 11. Find all the values b for which (18 ≤ a+b ≤ 21).
12. (Due March 21) Solve the following control problem: Let us assume that −5 ≤ b ≤ 5. Find all the values a for which ◊(−3 ≤ a+b ≤ 4).
13. (Due March 28) select a project.
14. (Due April 2) Write a program that implements a multi-valued logic approach to expert-based control that we covered in class (and which is described in the handouts). This program should take, as input, the difference ΔT between the actual temperature T and the ideal temperature T0, and return the angle φ to which we need to turn the knob of the temperature controlling device to compensate for this difference. Submit a printout of the program, a printout of the results, and be ready to show how your program works.
15. (Due April 9) Finish what we started in class: use the given history to check, step-by-step, that the formula ((p & ¬o) → X(s & (s U o)) & (o → 0) is satisfied at moment t = 2. In this formula:
1 2 3 4 5 6 7 8 9 10 p - - X - - - - X - - s - - - X X X - - - - o - - - - - - X X X X
16. (Due April 9) Use the given history to check, step-by-step, that the formula (t → X◊p) & (p → ¬Xp) is satisfied at moment t = 2. In this formula:
1 2 3 4 5 6 7 8 t - - X - - X - - p - - - - X - X -
17. (Due April 11) Similarly to what we did in class, write a program for computing an, and use program logic to prove this program's correctness.
18. (Due April 16) Come up with two ambiguous English phrases, and propose a reasonable translation of each of these phrases from English to predicate logic.