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Inference in First-Order Logic. Proofs Unification Generalized modus ponens Forward and backward chaining Completeness Resolution Logic programming. Inference in First-Order Logic. Proofs – extend propositional logic inference to deal with quantifiers Unification

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inference in first order logic
Inference in First-Order Logic
  • Proofs
  • Unification
  • Generalized modus ponens
  • Forward and backward chaining
  • Completeness
  • Resolution
  • Logic programming

CS 561, Session 16-18

inference in first order logic1
Inference in First-Order Logic
  • Proofs – extend propositional logic inference to deal with quantifiers
  • Unification
  • Generalized modus ponens
  • Forward and backward chaining – inference rules and reasoning

program

  • Completeness – Gödel’s theorem: for FOL, any sentence entailed by

another set of sentences can be proved from that set

  • Resolution – inference procedure that is complete for any set of

sentences

  • Logic programming

CS 561, Session 16-18

remember propositional logic
Remember:propositionallogic

CS 561, Session 16-18

proofs
Proofs

CS 561, Session 16-18

proofs1
Proofs

The three new inference rules for FOL (compared to propositional logic) are:

  • Universal Elimination (UE):

for any sentence , variable x and ground term ,

x 

{x/}

  • Existential Elimination (EE):

for any sentence , variable x and constant symbol k not in KB,

x 

{x/k}

  • Existential Introduction (EI):

for any sentence , variable x not in  and ground term g in ,

x {g/x}

CS 561, Session 16-18

proofs2
Proofs

The three new inference rules for FOL (compared to propositional logic) are:

  • Universal Elimination (UE):

for any sentence , variable x and ground term ,

x  e.g., from x Likes(x, Candy) and {x/Joe}

{x/} we can infer Likes(Joe, Candy)

  • Existential Elimination (EE):

for any sentence , variable x and constant symbol k not in KB,

x  e.g., from x Kill(x, Victim) we can infer

{x/k} Kill(Murderer, Victim), if Murderer new symbol

  • Existential Introduction (EI):

for any sentence , variable x not in  and ground term g in ,

 e.g., from Likes(Joe, Candy) we can infer

x {g/x} x Likes(x, Candy)

CS 561, Session 16-18

example proof
Example Proof

CS 561, Session 16-18

example proof1
Example Proof

CS 561, Session 16-18

example proof2
Example Proof

CS 561, Session 16-18

example proof3
Example Proof

CS 561, Session 16-18

unification
Unification

CS 561, Session 16-18

unification1
Unification

CS 561, Session 16-18

generalized modus ponens gmp
Generalized Modus Ponens (GMP)

CS 561, Session 16-18

soundness of gmp
Soundness of GMP

CS 561, Session 16-18

properties of gmp
Properties of GMP
  • Why is GMP and efficient inference rule?

- It takes bigger steps, combining several small inferences into one

- It takes sensible steps: uses eliminations that are guaranteed

to help (rather than random UEs)

- It uses a precompilation step which converts the KB to canonical

form (Horn sentences)

Remember: sentence in Horn from is a conjunction of Horn clauses

(clauses with at most one positive literal), e.g.,

(A  B)  (B  C  D), that is (B  A)  ((C  D)  B)

CS 561, Session 16-18

horn form
Horn form
  • We convert sentences to Horn form as they are entered into the KB
  • Using Existential Elimination and And Elimination
  • e.g., x Owns(Nono, x)  Missile(x) becomes

Owns(Nono, M)

Missile(M)

(with M a new symbol that was not already in the KB)

CS 561, Session 16-18

forward chaining
Forward chaining

CS 561, Session 16-18

forward chaining example
Forward chaining example

CS 561, Session 16-18

backward chaining
Backward chaining

CS 561, Session 16-18

backward chaining example
Backward chaining example

CS 561, Session 16-18

completeness in fol
Completeness in FOL

CS 561, Session 16-18

historical note
Historical note

CS 561, Session 16-18

resolution
Resolution

CS 561, Session 16-18

resolution inference rule
Resolution inference rule

CS 561, Session 16-18

remember normal forms
Remember: normal forms

“product of sums of

simple variables or

negated simple variables”

“sum of products of

simple variables or

negated simple variables”

CS 561, Session 16-18

conjunctive normal form
Conjunctive normal form

CS 561, Session 16-18

skolemization
Skolemization

CS 561, Session 16-18

resolution proof
Resolution proof

CS 561, Session 16-18

resolution proof1
Resolution proof

CS 561, Session 16-18

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