1 predicate (relational) logic 1. introduction the propositional logic is not powerful enough to...
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Predicate (Relational) Logic
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Introduction
The propositional logic is not powerful enough to express certain
types of relationship between propositions such as equivalence.
Can not tell whether it is true or false unless you know the value of X
powerful logic to deal with these problems. PREDICATE LOGIC
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X is greater than 1
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Introduction
Usefulness of Predicate Logic for Natural Language Semantics
While in propositional logic, we can only talk about sentences as a whole, predicate logic allows us to decompose simple sentences into smaller parts: predicates and individuals. John is tall. T(j)
Predicate logic provides a tool to handle expressions of generalization: i.e., quantificational expressions. Every cat is sleeping. Some girl likes David. No one is happy.
Predicate logic allows us to talk about variables (pronouns). The value for the pronoun is some individual in the domain of universe that is contextually determined. It is sleeping. She likes David. He is happy.
Predicate
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A predicate is a verb phrase template that describes a property of objects, or a relationship among objects represented by the variables.
"is blue“ or “B” is a predicate and it describes the property of being blue
The car Tom is driving is blueThe sky is blue
The cover of this book is blue
"B(x)"B(x) reads as "x is blue"
Predicate…
... gives ... to ... is a predicate describes a relationship among three objects
Give( x, y, z ) or G( x, y, z )
“gives a book to" B( x, y )5
John gives the book to MaryJim gives a bread to Tom
Jane gives a lecture to Mary
X gives Y to Z
Predicate…Exercise• Let G(x,y) represent the predicate x > y
G(6,13) means 13 is greater than 6
NO
G(2,0) is true
Yes
G(7,1) means 7 is greater than 1
Yes
“4 is less than 5” can be represented by G(5,4)
Yes
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Predicate…Exercise• Let E(x,y) represent “x sent an e-mail to y”
~E(A,B) means A didn’t sent e-mail to B
Yes
E(A,B) is equivalent to E(B,A)
No
“B sent an e-mail to A” is represented by E(B,A)
Yes
E(x,y) can also be represented by a 3 variable predicate
Yes
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Quantification Forming Propositions from Predicates
universe
universal quantifier
existential quantifier
free variable
bound variable
scope of quantifier
order of quantifiers
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Quantification
A predicate with variables is not a proposition
x > 1
It can be true or false depending on the value of x. A predicate with variables can be made a proposition by applying
assign a value to the variable quantify the variable using a quantifier.
If 3 is assigned to x becomes 3 > 1, and it becomes a true statement, hence a proposition.
A quantification is performed on formulas of predicate logic ( wff ), such as x > 1 or P (x), by using quantifiers on variables.
There are two types of quantifiers: universal quantifier and existential quantifier.
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Quantification
Universe of Discourse (universe)
“the set of objects of interest”
“the domain of the (individual) variables”
set of real numbers, the set of integers, the set of all cars on a parking lot, the set of all students in a classroom
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Quantification
Universal quantifier: turns the statement x > 1 to
"for every object x in the universe, x > 1", which is expressed as
“x, x > 1”
x, P(x) "For all x, P(x) holds", "for each x, P(x) holds" , P(x) is true for every
object x in the universe.
Ex. "All cars have wheels" x, P(x), • P(x) is the predicate denoting: x has wheels• and the universe of discourse is only populated by cars.
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Quantification
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Examples
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Bound & Free variables
bound variable: if either a specific value is assigned to it or it is quantified
Free variable:. If an appearance of a variable is not bound
Scope of the quantifier: The scope of a quantifier is the portion of a formula where it binds its variables, is indicated by square brackets [ ]
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Examples
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t: The scope of the second existential quantifier.
How to read quantified formulas
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Order of Application of Quantifiers
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Well-Formed Formula WFF
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Examples
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Examples
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One way to check whether or not an expression is a wff is to try to state it in English. If you can translate it into a correct
English sentence, then it is a wff.
Reasoning with Predicate Logic
Inference rules of predicate logic Universal instantiation Universal generalization Existential instantiation Existential generalization Negation of quantified statement
Predicate logic is more powerful than propositional logic. It allows one to reason about properties and relationships of individual objects.
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Quantified inference rules
Universal instantiation x P(x) P(A)
Universal generalization P(A) P(B) … x P(x)
Existential instantiation x P(x) P(F) skolem constant F
Existential generalization P(A) x P(x)
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Universal instantiation
If (x) P(x) is true, then P(C) is true, where C is any constant in the domain of x
Example: (x) eats(Ziggy, x) eats(Ziggy, IceCream)
The variable symbol can be replaced by any ground term, i.e., any constant symbol or function symbol applied to ground terms only
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Universal generalization
If P(c) is true, then ( x) P(x) is inferred.
Exampleeats(Ziggy, IceCream) ( x) eats(Ziggy, x)
All instances of the given constant symbol are replaced by the new variable symbol
Note that the variable symbol cannot already exist anywhere in the expression
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Existential instantiation
From (x) P(x) infer P(c)
Example: (x) eats(Ziggy, x) eats(Ziggy, Stuff)
Note that the variable is replaced by a brand-new constant not occurring in this or any other sentence in the KB
Also known as skolemization; constant is a skolem constant
In other words, we don’t want to accidentally draw other inferences about it by introducing the constant
Convenient to use this to reason about the unknown object, rather than constantly manipulating the existential quantifier
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Existential generalization
If P(c) is true, then (x) P(x) is inferred.
Exampleeats(Ziggy, IceCream) (x) eats(Ziggy, x)
All instances of the given constant symbol are replaced by the new variable symbol
Note that the variable symbol cannot already exist anywhere in the expression
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Connections between All and Exists
We can relate sentences involving and using De Morgan’s laws:
(x) P(x) ↔ (x) P(x)
(x) P(x) ↔ (x) P(x)
(x) P(x) ↔ (x) P(x)
(x) P(x) ↔ (x) P(x)
Homework 2
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Thank You!