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 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

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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 α and always terminates

• Problem: there can be way too many worlds to check

• O(2n) when KB and α 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