Chapter 8 FirstOrder Logic

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Chapter 8 First-Order Logic
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Outline

• Why First-Order Logic (FOL)
• Syntax and semantics of FOL
• Using FOL
• Wumpus world in FOL

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Pros and cons of propositional logic

• ? Propositional logic is declarative
• pieces of syntax correspond to facts
• ? Propositional logic allows partial/disjunctive/n
  egated information (unlike most data structure
  and databases)
• ? Propositional logic is compositional
• meaning of is derived from meaning
  of and
• ? Meaning in propositional logic is
  context-independent
• (unlike natural language, where meaning depends
  on context)
• ? Propositional logic has very limited expressive
  power (unlike natural language)
• E.g., cannot say pits causes breezes in
  adjacent squares
• except by writing one sentence for each square

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First-order Logic

• Whereas propositional logic assumes world
  contains facts, first-order logic (like natural
  language) assumes the world contains
• Objects
• nouns and noun phrases
• e.g., people, houses, numbers, theories, Ronald
  McDonald, colors, baseball games, wars, centuries
  
• Relations
• verbs and verb phrases
• e.g., these can be unary relations or properties
  red, round, bogus, prime, multistoried , or
• general n-ary relations brother of, bigger than,
  inside, part of, has color, occurred after, owns,
  comes between,
• Functions
• relations in which there is only one value for
  a given input
• father of, best friend, third inning of, one more
  than, beginning of ..

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Logics in general

• Probability theory
• degree of belief about facts, 0 (total
  disbelief), 1 (total belief)
• e.g., a probabilistic wumpus-world agent might
  believe that the wumpus is in 1, 3 with
  probability 0.75.
• Fuzzy logic
• facts with degree of truth, 0 (false), 1 (true)
• e.g., the sentence Vienna is a large city
  might. might be true in our world only to degree
  0.6.

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Models for FOL

• Models for PL are just set of truth values for
  the propositional symbols.
• Models for FOL contain a set of objects (domain
  elements) and relations among them.

Richard the Lionheart
evil King John

• A model containing 5 objects, 2 binary relations,
  3 unary relations, and 1 unary function (left leg)

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Truth in FOL

• The truth of a sentence is determined by a model
  and an interpretation for the sentences symbols.

• Symbols are basic syntactic elements that stand
  for objects, relations, and functions

• The semantics relate sentences to models using
  interpretation that specifies exactly which
  objects, relations and functions are referred to
  by the constant, predicate, and function symbols.

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Models for FOL Lots!
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Syntax of FOL Basic elements
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Atomic sentences and complex sentences
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Quantifiers Universal quantification
For all
says that P is true for every object x.
e.g.
The implication is true whenever its premise is
false regardless the truth of the conclusion.
Asserting the conclusion of the rule just for
those objects for whom the premise is true and
saying nothing at all about those individuals for
whom the premise is false.
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A common mistake to avoid
is the natural connective with
implication
Common mistake
with
Using conjunction
leads to a overly strong statement!
e.g.
Everybody at APSU is smart.
Everybody is at APSU and everybody is smart.
e.g.
The crown is a king and the crown is a person
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Quantifiers Existential quantification
says that P is true for at least one object x.
There exists an x such that or For some x
e.g.
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A common mistake to avoid
is the natural connective with
conjunction
Common mistake
with
Using implication
leads to a very weak statement!
e.g.
Somebody at APSU is smart.
True if there is somebody who is NOT at APSU.
e.g.
If Richards left leg is a crown then Richards
left leg is on Johns head.
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Nested quantifiers
is the same as
is the same as
is not the same as
is not the same as
For every person, there is someone that person
loves.
There is someone who is loved by everyone.
There is someone who loves everyone (in the
world).
Everyone is loved by someone (at least one
person).
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Connections between and
Quantifier duality each can be represented using
the other
is really a conjunction over the universe of
objects and is a disjunction
P 252 De Morgan rules
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Using FOL The kinship domain

• Kinship family relationship
• Facts
• Rules

e.g.
Mary is the mother of Charles
Charles is the father of William
e.g.
Ones grandmother is the mother of ones parent

• Using First-order Logic

• Objects are people
• Unary predicates Male and Female
• Kinship relations represented by Binary
  predicates Parent, Sibling, Brother, Husband,
  Grandparent, etc.
• Functions Mother and Father

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Using FOL The kinship domain

• Axioms the basic factual information to derive
  conclusions
• Plain facts

e.g.

• Definitions

e.g.
Ones husband is ones male spouse
Parent and child are inverse relations
A sibling is another child of ones parent

• Partial specifications (without complete
  definition)

e.g.
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Using FOL The kinship domain

• Theorems sentences are entailed by the axioms

siblinghood is symmetric
e.g.

• Theorems does not increase the conclusions that
  can be derived, but it reduce the computational
  cost of deriving new sentences

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FOL Knowledge Base

• Assertions sentences added to a KB using TELL

• Queries questions asked using ASK

True
True
True? (Yes?)

• The answer to the last query should be in a
  standard form, substitution or binding list a
  set of variable/term pairs.

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FOL KB for the wumpus world

• Properties of time

• Properties of locations

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FOL KB for the wumpus world

• Diagnostic rules
• infer hidden causes from observed effects

If a square is breezy, some adjacent square must
contain a pit.
If a square is not breezy, no adjacent square
contains a pit.

• Causal rules
• reflect the assumed direction of causality in the
  world

A pit causes all adjacent squares to be breezy.
If all squares adjacent to a given square are
pitless, the square will not be breezy.