Inference in First Order Logic

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Inference in First Order Logic

• CS 171/271
• (Chapter 9)
• Some text and images in these slides were drawn
  fromRussel Norvigs published material

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Inference Algorithms

• Reduction to Propositional Inference
• Lifting and Unification
• Chaining
• Resolution

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Propositionalization

• Strategy convert KB to propositional logic and
  then use PL inference
• Ground atomic sentences become propositional
  symbols
• What about the quantifiers?

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Example

• KB in FOL
• ?x King(x) ? Greedy(x) ? Evil(x)
• King(John)
• Greedy(John)
• Brother(Richard,John)
• The last 3 sentences can be symbols in PL
• Apply Universal Instantiation to the first
  sentence

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Universal Instantiation

• UI says that from a universally quantified
  sentence, we can infer any sentence obtained by
  substituting a ground term for the variable
• Back to Example
• From ?x King(x) ? Greedy(x) ? Evil(x)
• To
• King(John) ? Greedy(John) ? Evil(John)
• King(Richard) ? Greedy(Richard) ? Evil(Richard)

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Issue with UI

• Ground terms all symbols that refer to objects
  as well as function applications (recall that
  function applications return objects)
• For example, suppose Father is a function
• Father(John) and Father(Richard) are also
  objects/ground terms
• But so are Father(Father(John)) and
  Father(Father(Father(John)))
• Infinitely many ground terms/instantiations

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Existential Instantiation

• Whenever there is a sentence, ?x P, introduce a
  new object symbol called the skolem constant and
  then add the unquantified sentence P,
  substituting the variable with that constant
• Example
• From ?x Crown(x) ? OnHead(x, John)
• To Crown(Cnew) ? OnHead(Cnew, John)

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Substitution

• UI and EI apply substitutions
• A substitution is represented by a variable v and
  a ground term g v/g
• Can have sets of these pairs if there are more
  variables involved
• Let ? be a sentence (possibly containing v)
• SUBST( v/g, ? ) stands for the sentence that
  applies the substitution to ?

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UI and EI Defined

• UI ?v a ___ for any ground term g
  SUBST(v/g, a)
• EI ?v a ___ for some constant symbol k
  not SUBST(v/k, a) yet in the knowledge base

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Back to Propositionalization

• Given a KB in FOL, convert KB to PL by
• applying UI and EI to quantified sentences
• converting atomic sentences to symbols
• If there are no functions (Datalog KB), UI
  application does not result in infinitely many
  sentences
• Regular PL Inference can now be carried out
  without problems
• What if there are functions?

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Dealing with Infinitely Many Ground Terms

• Can set a depth-limit for ground terms
• Depth specifies levels of function nesting
  allowed
• Carry out reduction and inference process for
  depth 1, then 2, then 3,
• Stop when entailment can be concluded
• This works if there is such a proof, but goes
  into an endless loop if there is not
• The strategy is complete
• The entailment problem in this sense is
  semidecidable

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Inefficiencies in Propositionalization

• An inordinate number of irrelevant sentences may
  be generated, resulting from UI
• This motivates generating only those sentences
  that are important in entailment

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Example

• Suppose KB contains
• ?x King(x) ? Greedy(x) ? Evil(x)
• ?y Greedy(y)
• King(John)
• Suppose we want to conclude Evil(John)
• Because of the existence of objects other than
  John (such as Richard) and the existence of
  functions, UI will generate many sentences

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Example, continued

• It is sufficient to generate
• King(John) ? Greedy(John) ? Evil(John)
• Greedy(John)
• Which is just
• SUBST( x/John, King(x) ? Greedy(x) ? Evil(x) )
• SUBST( y/John, Greedy(y) )
• Applying the substitution matches the
• Premises King(x) ? Greedy(x)
• With other sentences in the KBGreedy(y),
  King(John)

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Lifted Modus Ponens

• Lifting Raising propositional inference rules to
  first order logic
• Example Generalized Modus Ponens
• If there is a substitution ?, such thatSUBST(?,
  pi) SUBST(?, pi) for all i, then
• p1', p2', , pn, ( p1 ? p2 ? ? pn ? q)
• ________________________________________________
  _______________________________
• SUBST(?,q)
• In our example, ? x/John, y/John

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Unification

• Process that makes logical expressions identical
• Goal match the premises of implications so that
  conclusions can be derived
• UNIFY algorithm takes two sentences and returns a
  unifier (substitution) if it exists

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Unification Algorithm
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Unification Algorithm
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About UNIFY

• UNIFY returns a Most General Unifier (MGU)
• There are efficiency issues withOCCUR-CHECK
  function
• May need to standardize apart rename variables
  to avoid name clashes
• Unification is a key component of all first-order
  algorithms

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Whats Next?

• Forward and backward chaining algorithms for FOL
  that use unification
• Resolution-based theorem proving systems