Propositional logic
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Propositional Logic. Reading: C. 7.4-7.8, C. 8. Logic: Outline. Propositional Logic Inference in Propositional Logic First-order logic. 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

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Propositional logic

Propositional Logic

Reading: C. 7.4-7.8, C. 8

Logic outline
Logic: Outline

  • Propositional Logic

  • Inference in Propositional Logic

  • First-order logic

Agents that reason logically
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

  • Towards general purpose ai
    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

  • Language examples
    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

  • Components of a formal logic
    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 worldWhat it means

    • Inference Procedures: Rules for deriving new sentences (and therefore, new semantics) from existing sentencesReasoning


    • 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

    Exercise semantics
    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


    • What are the models of the following sentences?

    • KB1: TodayIsTuesday -> ClassAI

    • KB2: TodayIsTuesday -> ClassAI, TodayIsTuesday

    Proof by refutation
    Proof by refutation

    • A complete inference procedure

    • A single inference rule, resolution

    • A conjunctive normal form for the logic

    Example wumpus world
    Example: Wumpus World

    • Agent in [1,1] has no breeze

    • KB = R2Λ R4 = (B1,1<->(P1,2 V P2,1)) Λ⌐B1,1

    • Goal: show ⌐P1,2

    Inference properties
    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.

    Other types of inference
    Other Types of Inference

    • Model Checking

    • Forward chaining with modus ponens

    • Backward chaining with modus ponens

    Model checking
    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

    Wumpus reasoning
    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

    Properties of model checking
    Properties of Model Checking

    • Sound because it directly implements entailment

    • Complete because it works for any KB and sentence to prove αand always terminates

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

      • O(2n) when KB and α have n variables in total

    Inference as search
    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 enumerateevery entailed sentence


    • 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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