Propositional Logic

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Propositional Logic

• Reading C. 7.4-7.8, C. 8

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Logic Outline

• Propositional Logic
• Inference in Propositional Logic
• First-order logic

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Agents that reason logically

• A logic is a
• Formal language in which knowledge can be
  expressed
• A means of carrying out reasoning in the
  language
• A Knowledge base agent
• Tell add facts to the KB
• Ask query the KB

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Towards General-Purpose AI

• Problem-specific AI (e.g., Roomba)
• Specific data structure
• Need special implementation
• Can be fast
• General purpose AI (e.g., logic-based)
• Flexible and expressive
• Generic implementation possible
• Can be slow

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Language Examples

• Programming languages
• Formal, not ambiguous
• Lacks expressivity (e.g., partial information)
• Natural Language
• Very expressive, but ambiguous
• Flying planes can be dangerous.
• The teacher gave the boys an apple.
• Inference possible, but hard to automate
• Good representation language
• Both formal and can express partial information
• Can accommodate inference

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Components of a Formal Logic

• Syntax symbols and rules for combining
  them What you can say
• Semantics Specification of the way symbols (and
  sentences) relate to the world What it means
• Inference Procedures Rules for deriving new
  sentences (and therefore, new semantics) from
  existing sentences Reasoning

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Semantics

• A possible world (also called a model) is an
  assignment of truth values to each propositional
  symbol
• The semantics of a logic defines the truth of
  each sentence with respect to each possible world
• A model of a sentence is an interpretation in
  which the sentence evaluates to True
• E.g., TodayIsTuesday -gt ClassAI is true in model
  TodayIsTuesdayTrue, ClassAITrue
• We say TodayIsTuesdayTrue, ClassAITrue is a
  model of the sentence

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Exercise Semantics

• What is the meaning of these two sentences?
• If Shakespeare ate Crunchy-Wunchies for
  breakfast, then Sally will go to Harvard
• If Shakespeare ate Cocoa-Puffs for breakfast,
  then Sally will go to Columbia

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Examples

• What are the models of the following sentences?
• KB1 TodayIsTuesday -gt ClassAI
• KB2 TodayIsTuesday -gt ClassAI, TodayIsTuesday

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Proof by refutation

• A complete inference procedure
• A single inference rule, resolution
• A conjunctive normal form for the logic

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Example Wumpus World

• Agent in 1,1 has no breeze
• KB R2 ? R4 (B1,1lt-gt(P1,2 V P2,1))
  ?B1,1
• Goal show P1,2

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Conversion Example
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Resolution of Example
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Inference Properties

• Inference method A is sound (or truth-preserving)
  if it only derives entailed sentences
• Inference method A is complete if it can derive
  any sentence that is entailed
• A proof is a record of the progress of a sound
  inference algorithm.

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Other Types of Inference

• Model Checking
• Forward chaining with modus ponens
• Backward chaining with modus ponens

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Model Checking

• Enumerate all possible worlds
• Restrict to possible worlds in which the KB is
  true
• Check whether the goal is true in those worlds or
  not

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Wumpus Reasoning

• Percepts nothing in 1,1 breeze in 2,1
• Assume agent has moved to 2,1
• Goal where are the pits?
• Construct the models of KB based on rules of
  world
• Use entailment to determine knowledge about pits

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Constructing the KB
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Properties of Model Checking

• Sound because it directly implements entailment
• Complete because it works for any KB and sentence
  to prove a and always terminates
• Problem there can be way too many worlds to
  check
• O(2n) when KB and a have n variables in total

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Inference as Search

• State current set of sentences
• Operator sound inference rules to derive new
  entailed sentences from a set of sentences
• Can be goal directed if there is a particular
  goal sentence we have in mind
• Can also try to enumerate every entailed sentence

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Example
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Complexity

• N propositions M rules
• Every possible fact can be establisehd with at
  most N linear passes over the database
• Complexity O(NM)
• Forward chaining with Modus Ponens is complete
  for Horn logic

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Example