Written Homework #4

Due April 12

1. [17] State Based Search (again). I was dissapointed with the class's performance on simple state-based search (everything in AI is a search problem!). Let's practice one more time [this, from the 2006 Google Puzzle Challenge]:

"Full House" puzzle: for each puzzle, find the starting square and draw a path moving horizontally and vertically that passes through each open square exactly once. For each straight segment in the path, you must go as far as possible, changing direction only when you are blocked by the grid's edge, a black square, or a square already visited.

 

  • (a) [10 points] Represent this as a state-based search problem. What is a state? What is the initial state? What is the successor-state function? What are the path costs? What is the goal test?
  • (b) [7 points] give an admissible heuristic function.

2. [26] Commonsense knowledge representation:

Consider the following argument:

Wellington heard about Napoleon's death. Therefore, Napoleon could not have heard about Wellington's death.

In this problem, you will axiomatize the commonsense knowledge used in this argument and provide a formal first-order logic derivation of this argument. Note: In this problem, you will use a continuous notion of time rather than the notion of discrete time steps.

  • (a) [5 points] List a set of typed functions/predicates (3 or 4 should suffice) and constants (such as a person's death) needed for a first-order logic representation of the knowledge in this domain. Your domain must include (at least) the following types of entities: people, times, and events. You should use a "strictly after" predicate for time (i.e., t1 > t2).
  • (b) [8 points] Using this vocabulary, write down the basic facts as well as any commonsense axioms used in this argument. You should use the following two time axioms:
  • (1)
    (> is transitive)
  • (2)
    (> is anti-symmetric.)
  • (c)[13 points] Translate your basic facts and axioms into disjunctive clausal form. Using refutation resolution, prove the conclusion from the facts you wrote down in (b). To keep your resolutions from going astray, keep in mind the underlying arguments you are trying to make. We may deduct points for proofs that perform unnecessary resolutions. (15 steps, not counting the original clauses, should be more than sufficient.)

3. [17] Diagnostic Reasoning. The following logic circuit has four wires, W1, W2, W3, and W4. It has an "AND gate", A, and an "inverter", B. The input wires, W1 and W2, can be either "on" or not. If the AND gate A is functioning properly (OK), wire W3 is "on" if and only if wires W1 and W2 are both "on". If the inverter B is functioning properly (OK), wire W4 is "on" if and only if wire W3 is not "on".

  • [5] Use expressions like OK(A) and ON(W1) and so on to describe the functioning of the circuit as defined.
  • [6] Using these formulas, and assuming that all components are functioning properly and that wires W1 and W2 are "on", use resolution to show that wire W4 is not "on".
  • [6] Again, using the formulas describing the functioning of the circuit, and given that W1 and W2 are "on", but that W4 is also "on", use resolution to show that either the AND gate or the inverter is not functioning properly.

5. [6] Using situation calculus, write sentences [i.e. possibility axioms and successor-state axioms] to describe the logical effects of a "SHOOT" action in the Wumpus World. Remember that in situation calculus that sucesssor-state axioms are oriented around descriptions of fluents, not actions. Possibility axioms capture the action preconditions. Assume that the agent has only one arrow with which to shoot.

6. [24] STRIPS-style Planning. Problem 11.4 in the book (Monkey & Bananas).
Points: (a 4) (b 12) (c 4) (d 4)

7. [10] Problem 11.11 from the textbook (Sussman's Anomaly).