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