Representing Knowledge with Calculus

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Representing Knowledge with Calculus

• Knowledge-Based Agents
• Propositional Logic
• Predicate Logic
• Summary

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Knowledge-Based Agents

• Central component is a knowledge-base
• Knowledge is made of sentences
• Sentences are expressed in a language
• We wish to derive new sentences through
• inference.

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Example
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Declarative vs Procedural

• Two approaches to store knowledge
• Declarative.
• Sentences representing knowledge
• Procedural.
• Encode behavior directly as code.

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Representing Knowledge with Calculus

• Knowledge-Based Agents
• Propositional Logic
• Predicate Logic
• Summary

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Logic

• Important terms
• Syntax .- rules to represent well-formed
• sentences
• Semantics.- define the truth of each sentence.
• Model.- Possible world

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Syntax

• Atomic sentences propositional symbols
• Complex sentences use logical connectives
• Negation,conjunction,disjunction
• Implication and biconditional.

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Syntax
To create well-formed formulas
sentence ? atom complex structure atom
? true false symbol symbol
? P Q R complex s ? sentence
( sentence sentence )
( sentence V
sentence ) (
sentence ? sentence) (
sentence ?? sentence)
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Semantics
Semantics defines the meaning of sentences. The
truth value assignment to a sentence is called an
interpretation. An interpretation maps a symbol
to T,F.
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Semantics

• Review truth assignments for
• Conjunction
• Disjunction
• Implication
• Equivalence

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Historical Background
Newell and Simon developed the Logic Theorist in
1956. It used transformations of expressions to
prove theorems extracted from Principia
Mathematica by Whitehead and Russell.
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Comparison

• Propositional logic
• Less expressive than predicate logic.
• It lacks power to describe situations
  concisely.
• Assumes facts are true or false

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Representing Knowledge with Calculus

• Knowledge-Based Agents
• Propositional Logic
• Predicate Logic
• Summary

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Syntax and Semantics

• What is a model?
• Propositional logic sets of truth values for
  symbols
• First-order logic
• Domain of the model (objects it contains)
• Relations (tuples of objects related)

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Example
Stars, Galaxies, Quasars
A kind of AGN in galaxies
Made of stars
Surrounded by planets
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Symbols and Interpretations
Symbols 1. Constants earth, moon, sun,
milky-way 2. Predicate Symbols
neighbor-planets(x,y) 3. Function Symbols
3rdPlanet(sun)
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Syntax
sentence ? atomic-sentence (
sentence connective sentence )
( quantifier variable, sentence )
sentence atomic-sentence ?
predicate(term) term term term ?
function(term) constant
variable
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Syntax
Connective ? ? V ??
Quantifier ? V
E
Symbols
Constant ? a x1 sun Variable ? A V S
Predicate ? neighborPlanets typeGalaxy
colorStar Function ? firstPlanet
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Semantics
Interpretation Symbol sun refers to star
sun Symbol earth refers to planet earth Star
3x45f refers to specific star
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Terms
Logical expression that refers to an
object. Examples Constant symbols sun, earth,
mars, venus. Function symbols 3rdplanet(sun)
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Sentences
Statements or facts. Examples neighbor-planets
(earth,mars) neighbor-galaxies(milky-way,androme
da)
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Universal Quantifiers
How do we express properties of entire
collections of objects? Universal
quantification V
All stars are burning hydrogen V x star(x) ?
burning-hydrogen(x)
True in all extended interpretations.
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Existential Quantifiers
x Star(x) burning-hydrogen(x)
E
Normally Universal quantifier connects with
? Existential quantifiers connect to
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More on Quantifiers
We can use multiple quantifiers Vx Vy Star(x)
Galaxy(y) InGalaxy(x,y) ?
Contains(y,x)
Vx y Star(x) Star(y) Neighbor(x,y) All
stars have a neighbor star
E
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More on Quantifiers
De Morgans rules apply Vx P x P Vx P
x P Vx P x P x P Vx P
E
E
E
E
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Equality
x,y neighborStars(x,y)
neighborStars(y,x) (x y)
E
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Representing Knowledge with Calculus

• Knowledge-Based Agents
• Propositional Logic
• Predicate Logic
• Summary

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Summary

• Propositional logic talks about facts predicate
• logic talks about relations (more expressive)
• A model is made of objects, relations, and
• functions.
• An interpretation maps symbols to the model
• Complex sentences use connectives and
• quantifiers