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Propositional Logic
Reading: C. 7.4-7.8, C. 8
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Announcements Read discussion board frequently
• Questions answered• New posts of client-server• Today: version posted with improved IO on display and
timing Mid-term evaluation on courseworks
• Complete by next Tuesday (1 week) Written homework
• Do not do “predicate logic” on problem 10.5 (will be part of next assignment)
• Should read “but use semantic networks and KL-one type ” .. Do not extend the representation itself.
• Note that section 10.6 covers semantic networks and description logics (another name for KL-one type)
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Logic: Outline
Propositional Logic Inference in Propositional Logic
First-order logic Inference in FOL
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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 themWhat 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 -> ClassAI is true in model {TodayIsTuesday=True, ClassAI=True}
• We say {TodayIsTuesday=True, ClassAI=True} 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 -> ClassAI
KB2: TodayIsTuesday -> 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 Λ R4 =
(B1,1<->(P1,2) V P2,1)) Λ⌐B1,1
Goal: show ⌐P1,2
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Conversion Example