Lecture 11 Logic I

Recap §

Improving Backtracking §

• General-purpose ideas to improve speed
• Filtering: Forward Checking, Constraint Propagation, K-Consistency
• Ordering:
  • Which variable assigned next? MRV, Degree
  • Which order values tried? LCV
• Structure:
  • Independent Subproblems

Consistency of a Single Arc §

• $X\to Y$ is consistent if for every x in the tail, there exists some y that can be assigned without violating constraint.

Logic and Reasoning §

Knowledge Base §

• Knowledge Base (KB): Set of sentences in formal language
• “Tell” the agent what it needs to know.
• The agent can then “ask” itself what to do.
• Answers should follow from the KB
• $\forall xBird(x)⟹Fly(x)$
• “For all objects that exists in the world, if the object is a bird, then the object can fly.”
• Penguins can’t fly! Be careful when making KBs.

Entailment §

• Models are formally structured possible worlds with respect to which truth can be evaulated.
• A sentence $a$ is true in model $m$:
  • $m$ is a model of $a$
  • $m$ satisfies $a$
• Entailment: $KB|=a$ iff M(KB) \subseteq M(a)$
• $M(a)$ = set of ALL models of $a$!
• 

Inference §

Procedure / Algorithm to derive conclusions

• $KB|{-}_{i}a$: Sentence $a$ can be derived from $KB$ by procedure $i$
• Soundness: $i$ is sound if whenever $KB|{-}_{i}a$, it is also true that $KB|=a$
• Completeness: $i$ is sound if whenever $KB|=a$, it is also true that $KB|{-}_{i}a$

Propositional Logic §

Syntax §

• Atomic Sentences:
  • Propositional symbols $P_{1,1},W_{2,2}$: A proposition that can be true or false.
• Complex Sentences:
  • If $S$ is a sentence, $\neg S$ is a sentence (negation)
  • $S1,S2$ are sentences, $S1\wedge S2$ is a sentence. (conjunction)
  • $S1\vee S2$ (disjunction)
  • $S1\oplus S2$ (exclusive disjunction, XOR)
  • $S1⟹S2$ (implication)
  • $S1\iff S2$ (biconditional)

Semantics §

• Each model specifies true/false for each proposition symbol.