# 1. parenthesize every sentence into unary function applications, either leftward or rightward
# 2. the type of a sentence in boolean
# 3. thing42:chair, thing23:chair
#    chair is predicate that is true of things that are chairs and false of things that are not
#    chairs
#    the type of a noun is thing->boolean
# 4. A N --> A(N)
#    D N --> D(N)
#    N PP ---> PP(N)
#    NP VP --> VP(NP)
#    P NP --> P(NP)

primitive type boolean
primitive type thing
typedef N = thing->boolean = thing->S
typedef A = N->N = (thing->S)->(thing->S) = (thing->boolean)->(thing->boolean)
typedef D = N->NP
typedef P = NP->PP = NP->(N<-N) = NP->((thing->S)<-(thing->S))
          = NP->((thing->boolean)<-(thing->boolean))
typedef Vintrans = VP = S<-NP = boolean<-NP
typedef Vtrans = NP->VP = NP->(S<-NP) = NP->(boolean<-NP)
typedef NP =
typedef VP = S<-NP = boolean<-NP
typedef S = boolean
typedef PP = N<-N = (thing->S)<-(thing->S) = (thing->boolean)<-(thing->boolean)

chair:thing->boolean
table:thing->boolean

big
red
round

big table
big chair
red table
red chair
round table
round chair

the table
the chair
the big table
the big chair
the red table
the red chair
the round table
the round chair

on the table
on the chair

table on the table
table on the chair
chair on the table
chair on the chair

the table
the chair
the table on the table
the table on the chair
the chair on the table
the chair on the chair

table
chair
table on the table
table on the chair
chair on the table
chair on the chair
N(PP)

on the table
on the chair

the table
the chair

the man ate the apple
the man ate

ate the apple