FirstOrder Logic

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First-Order Logic

• Reading C. 8 and C. 9
• Pente specifications handed back at end
• of class

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First-Order Logic Outline

• Expressing Information in first-order logic
• An example
• Inference in FOL
• Resolution theorem proving
• Production systems (forward chaining)
• Logic-based programming (backward chaining)

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Characteristics of FOL

• Declarative
• Expressive
• Partial information
• Negation
• Compositionality

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Ontological Commitment

• Propositional logic
• There are facts that either hold or do not hold
  in the world
• Logic constrains facts
• First-order logic
• The world consists of objects and relations
  between objects
• Logic constrains allowable objects, properties of
  objects, relations between objects

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Ontological commitments of higher order logics

• Temporal logic
• Facts hold at particular times and those times
  are ordered
• Epistemological
• Agents hold beliefs about facts
• Three possible states of knowledge
• The agent believes a fact
• The agent does not believe it
• The agent has no opinion
• Probabilistic
• Facts are true to different degrees (Truth value
  from 0 to 1)

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Problems with propositional logic
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Propositional Logic is lacking in expressiveness

• Cannot represent knowledge of complex
  environments in a concise way
• E.g., Squares adjacent to pits are breezy
• Need objects
• Squares, pits, Kathy
• Need relations
• Adjacent, breezy, smelly, know
• Need functions
• Father-of, mother-of

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Syntax of FOL basic elements

• Constants Vijay, Andrew, Sowmya
• Predicates knows, adjacent, gt
• Functions Sqrt, father-of
• Variables x,y,a,b
• Connectives ?,V,,?,?
• Equality
• Quantifiers ?,?

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Atomic Sentences

• Atomic sentence predicate (term1termm)
  or term1term2
• Term function (term1, , termm) or
  constant or variable
• E.g. know(Kathy,Sowmya), Adjacent (x,y),
  father-of(Kathy) Michael, Andrew, x

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Complex Sentences

• Complex sentences are made from atomic sentences
  using connectivesS, S1?S2, S1VS2, S1?S2,
  S1?S2
• E.g., adjacent(x,y) ? adjacent (y,x),
  knows(Nunzio, Michael),

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Truth in First-order Logic

• Sentences are true with respect to a model and an
  interpretation
• Model contains ? 1 objects (domain elements) and
  relations among them
• Interpretation specifies referents for
• Constant symbols -gt objects
• Predicate symbols -gt relations
• Function symbols -gt functional relations
• An atomic sentence predicate (term1,,termn) is
  true iff the objects referred to by term1,,
  termn are in the relation referred to by
  predicate.

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Universal quantification

• ?ltvariablesgt ltsentencegt
• Everyone at Columbia is smart?x At(x,Columbia)
  ? Smart(x)
• ?x P is true in a model m iff P with x being each
  possible object in the model
• At (Leia, Columbia) ? Smart(Leia)
• At (Ryan, Columbia) ? Smart (Ryan)
• At (Archana, Columbia) ? Smart (Archana)
• At (Stanley, Columbia) ? Smart (Stanley)
• ..

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A common mistake

• Typically, ? is the main connective used with ?
• Common mistake using as the main connective ?
  ?x At(x,Columbia) ? Smart(x)

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Existential Quantification

• ?ltvariablesgt ltsentencegt
• Someone at Columbia is smart?x At(x,Columbia)
  Smart(x)
• ? x P is true in a model m iff P with x being
  each possible object in the model
• Equivalent to the disjunction of instantiations
  of P
• At (Leia, Columbia) ? Smart(Leia)
• V At (Ryan, Columbia) ? ? Smart (Ryan)
• V At (Archana, Columbia) ? ? Smart (Archana)
• V At (Stanley, Columbia) ? ? Smart (Stanley)

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Another Common Mistake

• Typically, ? is the main connective with ?
• Common mistake using ? as the main connective ?
  x At(x,Columbia) ? Smart(x)

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Properties of Quantifiers

• ?x ?y is the same ?y ?x
• ?x ? y is the same as ? y ? x
• ? x ? y is not the same as ? y ? x
• ? x? y Loves(x,y)
• There is a person who loves everyone in the world
• ? y ? x Loves(x,y)
• Everyone is loved by someone.
• Quantifier duality each can be expressed using
  the other ? x Likes (x,Icecream) ? x
  Likes(x,IceCream) ? x Likes(x, Broccoli)
  ?x Likes(x,Broccoli)

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Translation from English to FOL

• A mother is a female parent
• Andrew likes the problem of one of the book
  exercises
• ?

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Example

• Family trees
• What does the model look like?
• Father-of
• Mother-of
• Sibling
• What can we infer?
• Cousin
• Ancestors

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To Make Inferences in FOL

• Method 1
• Unification of variables with literals (in the
  KB)
• Generalized Modus Ponens
• Forward-chaining or Backward-chaining
• Method 2
• Resolution

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Unification

• We want to find a substitution ? such that x and
  y match literals
• Unify (?,?) ? if ?? ??
• Some examples

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Standardizing apart eliminates overlap of
variables, e.g., Knows(z17,Michel)
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Unification for example
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• P1 father-of(Kathy)Michael
• P1 father-of(x)y
• ?x/Kathy,y/Michael
• qancestor(x,y)
• qancestor(Kathy,Michael)

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Example inference using forward chaining
(production systems)
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Properties of forward-chaining

• Sound and complete for first-order definite
  clauses
• Datalog is first-order definite clauses and no
  functions
• May not terminate in general if is not entailed
• This is unavoidable entailment with definite
  clauses is semi-decidable
• Forward chaining is widely used in deductive
  databases

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Example inference using backward chaining
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Properties of backward-chaining

• Depth-first recursive proof search space is
  linear in size of proof
• Incomplete due to infinite loops
• Fix by checking current goal against every goal
  on stack
• Inefficient due to repeated subgoals (both
  success and failure)
• Fix using cache of previous results (extra
  space!)
• Widely used (without improvements!) for logic
  programming (e.g., Prolog)

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Midterm results

• Exams will only be given back to person the owner
  of the exam