Additional Clarification

De Morgan’s rules for quantification

Higher Order Logics

• FOPL allow you to quantify over objects (the entities that exist in the world).

• Higher order logics allow you to quantify over relations (predicates) as well

• This is sometimes very useful (e.g., representing beliefs) but there is no understanding of how to reason effectively with such logics so we stick to FOPL.

Axioms and Theorems

• Mathematicians distinguish between axioms and theorems:
  • Axioms are the basic facts about a domain
  • Theorems are proved by applying inference procedures to axioms (axioms entail theorems)

Building a Knowledge Base

• 5 step methodology
  • 1. Decide what to talk about
  • 2. Decide on Predicates, functions and constants
  • 3. Encode the general knowledge about the domain
  • 4. Encode the specific problem instance
  • 5. Use inference procedures to get the answer

An Example Problem --- the mislabeled boxes puzzle

• Puzzle: A merchant places 3 boxes on a table. Each box contains only 1 kind of fruit: apples, bananas or oranges. You are asked to guess which fruit is in each box. The merchant tells you that each box is mislabeled. Box a is labeled apples, box b oranges, and box c bananas. You are allowed to look in the box labeled oranges and you find that it contains apples. What do you guess the contents of the other 2 boxes are?

Solution

• Objects:
  • fruits: apples, oranges, and bananas
  • boxes: a, b, and c

• Predicates:
  • Contains(), label()

• Functions:
  • ?

Solution continued

• Domain specific knowledge:
  • ?

• Problem specific knowledge:
  • 1. contains(b, apples)
  • 2. label(a, apples)
  • 3. label(b, oranges)
  • 4. label(c, bananas)

Solution continued

Solution continued

Solution continued

Representing Change in the World

• A model-based agent is an agent that keeps track of the world by maintaining an internal model of the relevant aspects of the world.

• The alternative is a reflex agent that maps from percepts to actions

Situation Calculus

• Situation Calculus is a formalism for representing change in the world in FOPL --- representing diachronic rules.

• It identifies situations (the state of the world at a particular time); instead of P(x1,x2,...xn) you have P(x1,x2,...xn,s) where s is a situation variable. Are situations objects?

• Can use situations to represent different states of the world in the same knowledge base.

Situation Calculus continued

• Every property or relation that can change over time is given an extra situation argument; this is crucial to determining the truth of the sentence.

• Example:
  • at(agent, [1,1], s0)
  • at(agent, [1,2], s1)

Effect Axioms

Frame Axioms

Simplified Wumpus World I

OTTER input file

• set(auto).
• formula_list(usable).
• adjacent(start, middle).
• adjacent(middle, finish).
• all s l1 l2 (at(agent, l1, s) & adjacent(l1, l2) -> at(agent, l2, result(goForward, s))).
• at(agent, start, s0).
• -(exists s (at(agent, finish, s))).
• end_of_list.

OTTER Proof

• ------------- PROOF ------------
• 1 [] -at(agent,x,y) | -adjacent(x,z) | at(agent,z,result(goForward,y)).
• 2 [] -at(agent,finish,x).
• 3 [] adjacent(start,middle).
• 4 [] adjacent(middle,finish).
• 5 [] at(agent,start,s0).

OTTER Proof (continued)

• 6 [hyper,5,1,3] at(agent,middle,result(goForward,s0)).
• 7 [hyper,6,1,4] at(agent, finish, result(goForward,result(goForward,s0))).
• 8 [binary,7.1,2.1] $F.

Synchronic Rules

• So far have specified rules about what changes over time also need “same time” rules:
  • causal rules:
    • loc1, loc2, s at(W, loc1, s) ^ adjacent(loc1, loc2) => smelly(loc2)
  • diagnostic rules:
    • loc1, s smelly(loc1) => loc2 at(W, loc2, s) ^ adjacent(loc1, loc2) | loc1 = loc2