Download Using First-Order Logic

First-Order Logic
• Introduction
• Syntax and Semantics
• Using First-Order Logic
• Summary
Introduction
Propositional logic:
o Less expressive than predicate logic.
o Is declarative (as opposed to procedural)
o It lacks power to describe things concisely.
o Assumes facts are true or false
Compared to First-Order Logic…
Builds around objects and relations.
Relations may hold or not.
Even more complex languages:
 Temporal logic. Facts hold at particular times
 High-order logic. Relations and functions are
objects themselves.
Languages
Language
Elements
Belief
Propositions
Facts
True/false/unknown
First-order logic Facts,objects, True/false/unknown
relations
Temporal logic Facts,objects, True/false/unknown
relations, time
Probability
Facts
Degree of belief [0,1]
Fuzzy logic
Facts with
Interval value
degree of truth
First-Order Logic
• Introduction
• Syntax and Semantics
• Using First-Order Logic
• Summary
Syntax and Semantics
What is a model?
Propositional logic: sets of truth values for symbols
First-order logic:
a. Domain of the model (objects it contains)
b. Relations (tuples of objects related)
Example
Stars, Galaxies, Quasars
Made of stars
Surrounded by planets
A kind of
AGN in
galaxies
Symbols and Interpretations
Symbols:
1. Constants: earth, moon, sun, milky-way
2. Predicate Symbols: Neighbor-planets(x,y)
3. Function Symbols: 3rdPlanet(sun)
Syntax
Sentence  AtomicSentence
| ( Sentence Connective Sentence )
| ( Quantifier Variable,… Sentence )
| ~Sentence
AtomicSentence  Predicate(Term) | Term = Term
Term  Function(Term)
| Constant
| Variable
Syntax
Connective   | ^ | V | 
Quantifier  V |
Constant  A | X1 | Sun
Variable  a | x | s | …
Predicate  NeighborPlanets | TypeGalaxy
| ColorStar
Function  1stPlanet, 2ndPlanet
Semantics
Interpretation:
Symbol “sun” refers to star sun
Symbol “earth” refers to planet earth
Star 3x45f refers to “specific star”
Terms
Logical expression that refers to an object.
Examples:
Constant symbols: sun, earth, mars, venus.
Function symbols: 3rdplanet(sun)
Sentences
Statements or facts.
Examples:
Neighbor-Planets(earth,mars) ^
Neighbor-Galaxies(milky-way,andromeda)
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.
Existential Quantifiers
x Star(x) ^ burning-hydrogen(x)
Normally:
Universal quantifier connects with 
Existential quantifiers connect to ^
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
More on Quantifiers
De Morgan’s rules apply:
Vx ~P = ~ x P
~Vx P = x ~P
Vx P = ~ x ~P
x P = ~Vx ~P
Equality
x,y NeighborStars(x,y) ^
NeighborStars(y,x) ^
~(x = y)
First-Order Logic
• Introduction
• Syntax and Semantics
• Using First-Order Logic
• Summary
Using First-Order Logic
Rational Agent:
TELL(KB,Star(sun))
ASK(KB,Star(sun))
ASK(KB, x Star(x))
First-Order Logic
• Introduction
• Syntax and Semantics
• Using First-Order Logic
• Summary
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
 Steps to develop a knowledge base.
Robert Kowalski
Born 1941 in Connecticut.
Professor at Imperial College London since 1975.
He is well known for his many achievements in
logic programming.
Co-developed “event calculus’ and “legal reasoning”
Exercises
Under propositional logic, assume you have 4 propositions
named A, B, C, and D (all Boolean). Observe that V is
disjunction or OR, & is conjunction or AND, and ~ is
negation or NOT. Answer the following:
What is the total number of models defined in this language?
Is the statement A V B valid? why?
Is the statement A V ~A valid? why?
How many models are true under the statement B & D ?