Tarskian Interpretations

In FOF

• The Egalitarian student problem

• New role interpretation

  fof(equality_lost,interpretation, 

• Specify the domain

      ( ! [X] : ( X = "a" | X = "f" | X = "john" | X = "gotA") 

  Note the use of "quoted terms" that are known to be distinct.
• Interpret functions

      & ( a = "a"
        & f = "f"
        & john = "john"
        & grade_of("a") = "a"
        & grade_of("f") = "a"
        & grade_of("john") = "f"
        & grade_of("gotA") = "a" )  

  Note the interpretation of meaningless terms.
• Interpret predicates

      & ( ~ human("a")
        & ~ human("f")
        & human("john")
        & human("gotA")
        & ~ created_equal("a","a")
        & ~ created_equal("a","f")
        & ~ created_equal("a","john")
        & ~ created_equal("a","gotA")
        & ~ created_equal("f","a")
        & ~ created_equal("f","f")
        & ~ created_equal("f","john")
        & ~ created_equal("f","gotA")
        & ~ created_equal("john","a")
        & ~ created_equal("john","f")
        & created_equal("john","john")
        & created_equal("john","gotA")
        & ~ created_equal("gotA","a")
        & ~ created_equal("gotA","f")
        & created_equal("gotA","john")
        & created_equal("gotA","gotA") ) ) ). 

  Note the interpretation of meaningless atoms.

In TFF (and beyond)

• The Garfield problem

• Keep problem types

  tff(human_type,type,      human: $tType ).
  tff(cat_type,type,        cat: $tType ).
  tff(jon_decl,type,        jon: human ).
  tff(garfield_decl,type,   garfield: cat ).
  tff(arlene_decl,type,     arlene: cat ).
  tff(nermal_decl,type,     nermal: cat ).
  tff(loves_decl,type,      loves: cat > cat ).
  tff(owns_decl,type,       owns: ( human * cat ) > $o ). 

• Declare new domain types to declare domain elements

  tff(d_human_type,type,    d_human: $tType ).
  tff(d_cat_type,type,      d_cat: $tType ).
  tff(d_jon_decl,type,      d_jon: d_human ).
  tff(d_garfield_decl,type, d_garfield: d_cat ).
  tff(d_arlene_decl,type,   d_arlene: d_cat ).
  tff(d_nermal_decl,type,   d_nermal: d_cat ). 

  ... and add type promotion functions ...

  tff(d2human_decl,type,    d2human: d_human > human ).
  tff(d2cat_decl,type,      d2cat: d_cat > cat ). 

• New role interpretation

  tff(garfield,interpretation, 

• Specify domains

      ( ! [H: human] : ? [DH: d_human] : H = d2human(DH)
      & ! [DH: d_human] : ( DH = d_jon )
      & ! [DH1: d_human,DH2: d_human] : ( d2human(DH1) = d2human(DH2) => DH1 = DH2 )
      & ! [C: cat] : ? [DC: d_cat] : C = d2cat(DC)
      & ! [DC: d_cat]: ( DC = d_garfield | DC = d_arlene | DC = d_nermal )
      & $distinct(d_garfield,d_arlene,d_nermal) 
      & ! [DC1: d_cat,DC2: d_cat] : ( d2cat(DC1) = d2cat(DC2) => DC1 = DC2 ) 

• Interpret functions (using type promoted domain elements)

      & jon = d2human(d_jon)
      & garfield = d2cat(d_garfield)
      & arlene = d2cat(d_arlene)
      & nermal = d2cat(d_nermal)
      & loves(d2cat(d_garfield)) = d2cat(d_garfield)
      & loves(d2cat(d_arlene)) = d2cat(d_garfield)
      & loves(d2cat(d_nermal)) = d2cat(d_garfield) 

• Interpret predicates

      & owns(d2human(d_jon),d2cat(d_garfield))
      & ~ owns(d2human(d_jon),d2cat(d_arlene))
      & ~ owns(d2human(d_jon),d2cat(d_nermal)) ) ). 

• You can reuse problem types and get confused

Complete Examples

Egalitarian Students

Jon Owns Garfield's Lovers