First Order Logic

1
First Order Logic

• CS 171/271
• (Chapter 8)
• Some text and images in these slides were drawn
  fromRussel Norvigs published material

2
Propositional Logic Limitations

• Stating similar facts is cumbersome
• Cant make generalizations
• World contains facts, not objects
• Natural language deals with objects (nouns) and
  relations (verbs), hence is more expressive
• Ontological commitment of PL is limited

3
First Order Logic

• World consists of
• Objects
• Relations
• Functions
• Sentences are made up of
• Symbols for constant objects, predicates, and
  functions
• Connectives as in PL
• Quantifiers and variables ?x, ?y

4
FOL Syntax

• Sentence
• Atomic Sentence
• ( Sentence Connective Sentence )
• ?Sentence
• Quantifier Variable Sentence
• Atomic Sentence
• Predicate-Symbol( Term, )
• Term Term

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FOL Syntax

• Term (refers to an object)
• Function-Symbol( Term, )
• Constant-Symbol
• Variable
• Connective ?, ?, ?, ?,
• Quantifier ?, ?
• Variable x, y, z,

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Example
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Symbols

• Constants John, Richard, C, L1, L2
• Predicates
• PersonJohn, Richard
• KingRichard
• CrownC
• Brother(John,Richard),(Richard,John)
• OnHead(C,Richard)
• Functions
• LeftLeg(Richard-gtL1),(John-gtL2)(strictly
  speaking, the function should be total)

8
Models and Interpretations

• A model in FOL consists of the objects (domain
  elements) and relations (including functions)
• See example diagram
• conceptual view of world
• An interpretation associates the symbols to the
  objects, relations, and functions in the model
• Number of interpretations for a given set of
  symbols is combinatorially explosive

9
Semantics

• Truth of a sentence in FOL
• Determined with respect to a model and an
  interpretation
• Analogous notions for entailment, validity, and
  satisfiability
• Model enumeration is impractical in FOL

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Sample Sentences

• Person(John) ? Person(Richard)
• OnHead(C, John)
• LeftLeg(John) L1 ? LeftLeg(Richard) L1
• Richard, and LeftLeg(John) are examples of
  terms(a term is an expression that refers to an
  object)
• Atomic Sentences constructed by equating terms
  () or by a predicate (with terms as arguments)
• Complex Sentences sentences with connectives

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Quantifiers

• Universal Quantification
• ?x P is true in a model m iff P is true with x
  being each possible object in the model
• A conjunction of instantiations
• Existential Quantification
• ?x P is true in a model m iff P is true with x
  being some object in the model
• A disjunction of instantiations

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Sample SentencesUsing Quantifiers

• ?x King(x) ? Person(x)
• ?x Crown(x) ?? OnHead(x,John)
• John has a crown on his head
• ?x?y Brother(x,y) ? Brother(y,x)
• ?x?y Brother(x,Richard) ? Brother(y,Richard) ?
  ?(xy)
• Richard has at least 2 brothers

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Properties of Quantifiers

• Nested Quantifiers
• ?x ?y P equivalent to ?y ?x P
• ?y ?x P equivalent to ?y ?x P
• Does not apply if quantifiers are different
• De Morgans law for quantifiers
• ?x ?P ? ??x P
• ?x P ? ??x ?P
• ?x ?P ? ??x P
• ?x P ? ??x ?P

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More About Quantifiers

• Be careful when
• Using quantifiers (?, ?) in combination with ?, ?
• The domain consists of multiple kinds of objects
• In the quantified sentence(such as ?x P or ?x
  P), P would typically contain terms that are
  variables
• Not just ground terms (terms that have no
  variables)
• ?x P as a query binding list more important than
  truth of the sentence

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Axioms and Theorems

• Axioms are sentences that represent first
  principles
• Plain facts
• Definitions
• Theorems are sentences entailed by axioms

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Some Useful Domains

• Natural Numbers
• Built from 0, successor function S, and Peano
  axioms
• Sets
• ?, ?, ?, ?, and element insertion
• Lists
• Nil, Cons, Append, First, Rest, Find,

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FOL and the Wumpus World

• We can represent the Wumpus World in a more
  compact fashion
• Less sentences needed to represent rules
• We can include time and percept objects in the
  world
• A percept is represented as a list of constant
  symbols
• Predicates with time arguments capture the
  dynamic nature of the agent moving in this world

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Knowledge Engineering

• Identify the task
• Assemble the relevant knowledge
• Decide on a vocabulary
• Encode general knowledge of the domain
• Encode the specific problem instance
• Pose queries to the inference procedure
• Debug the knowledge base