Natural Language Processing  First-Order Predicate Logic - Predicates  Short Introduction to Description Logics  PowerLoom

Semantic Representations Semantic Representation based on some form of (formal) Representation Language. –Semantics Networks –Conceptual Dependency Graphs –Case Frames –Ontologies –DL and similar KR languages –First-Order Predicate Logic

Example Define Hierarchy of Concepts mother personfemale has-child mother (defconcept mother (person AND female AND (  has-child.person))

Example Define Hierarchy of Concepts mother 1.  x: mother(x)  person(x)IS-A 2.  x: mother(x)  female(x)IS-A 3.  x: mother(x)   y: has-child(x,y)Role  person(y)

Example Define Hierarchy of Concepts mother personfemale has-child mother (defconcept mother (person AND female AND (  has-child.person))

First-Order Predicate Logic (FOPL) Predicates core of formulae applied to variables (constants, terms) correspond to classes, concepts, sets e.g. student(x) describes the class / set of all students unary, binary, n-ary predicates possible in FOPL e.g. siblings(x,y) or brothers(x,y) represents pairs of siblings or pairs of brothers.

First-Order Predicate Logic (FOPL) Terms inside predicate in formulae functions, variables (x,y,...), constants (c,d,...) constants stand for concrete values or objects variables are referring to (sets of) values functions represent functions: e.g. plus(x,y) represents a function with two arguments; could mean + in arithmetic; could also mean something else, e.g.  depends on Interpretation!

First-Order Predicate Logic (FOPL) Formulae wff atomic formulaP(x), P(c), P( ), P(,..., ) complex formula  ;  ;  ;    quantified formula  x,y: P(x,y) ;  x: P(x) Interpretation into domain assigns meaning to formulae!

First-Order Predicate Logic (FOPL) Interpretation I into domain D (or  ), D={d 1, d 2,..., d n } term I (c)  D ; I (x)  D ; I (f(t 1,...,t n ) ) = I (f) ( I (t 1 ),..., I (t n )) I (f) is a function in Domain D atomic formula I (P(t 1,...,t n )) true if ( I (t 1 ),..., I (t n ))  I (P) I (P) is a relation in Domain D complex formula I (  ) true if I (  ) not true; I (  ) true if I (  ) true and I (  ) true; equivalent for  and   

First-Order Predicate Logic (FOPL) Interpretation I into domain D (or  ), D={d 1, d 2,..., d n } quantified formula(just simple atomic formula here)  x:  (x) true if I (  ) is true if you substitute x in  with d, for one d  I (x).  x:  (x) true if I (  ) is true if you substitute x in  with d, for all d  I (x). Note: for nested quantifiers, apply I from left to right:  x  y:  (x,y)true if I (  (d,d')) is true when you 1.substitute x in  with d for all d  I (x), and then 2.find for each d one d'  I (y) so that I (  (d,d')) is true.

References see Tutorial - Horrocks and Sattler