First-Order Logic

Goals for a KR language

• Expressive and concise

• Unambiguous and independent of context

• Efficient and effective inference procedure

First-Order Logic

• The most studied and well understood KR language

• It can express anything that can be programmed

Ontological commitments

• Represents the world in terms of:
  • objects: people, houses, games, places ...
  • properties (of objects): yellow, big, hard, far-away, ...
  • relations (among objects): part-of, brother-of has-color, ...
  • functions (from one object to another): father-of, left-leg-of, ...

• Examples: family, Wumpus

Syntax of FOPL

Syntax continued

• Terms (represent objects)
  • If x is a variable or a constant, x is a term
  • If x1, ..., xn are all terms and fun is a n-ary function, then fun(x1, ..., xn) is a term

Syntax continued

Syntax continued

• Difference between functions and predicates:
  • a function is used to build terms out of other terms; E.g.: father-of(bill)
  • a predicate is used to define sentences (which are true or false), E.g.: father-of(bill, david)
  • = (the equality symbol) is a predicate that is almost always part of FOPL; states that 2 terms refer to the same object

Syntax continued

Syntax continued

• Some terminology
  • atomic sentence --- a sentence with no connectives or quantifiers
  • literal --- an atomic sentence or its negation
  • ground term --- a literal without variables

Semantics of FOPL

• Interpretation: mapping between elements of the language and the entities in the world/model

• Begin with an interpretation function that maps predicates, functions and constants to their “meanings” in some world/model

• Basically 2 elements:
  • objects referred to by terms
  • claims about objects (their properties, relations); these are expressed by sentences (wff)

Semantics continued

Semantics continued

Semantics continued

• sentences/claims about objects
  • atomic sentences are relations with arguments where the arguments are specified by terms
  • examples:
    • color-of(chair, blue)
    • mother-of(bill, mary)
  • usual interpretation: T or F (true or false in the world)

Functions versus predicates revisited

• father(john) --- {<john, bill>, <sue,david>, ...}

• father(john,bill) --- {<john, bill>, <sue,david>, ...}

• function expression correspond to terms; they denote objects in the world/model

• predicate expressions correspond to sentences; they are true or false in the world/model

Semantics continued

• Complex sentences: atomic sentences and connectives
  • compositional semantics
  • the same as proposition logic

Semantics continued

Semantics continued