meeting -*- Outline -*-

* semantics (3.2)

  This is in essence, a denotational semantics.

  a set-theoretic model
    a type S denotes a set in some universe U of nonempty sets
    a term t:S denotes an element in the meaning of S

** Semantics of types

   Use M.S for the meaning of S

   Q: Can the meaning of a type S, M.S, be empty?  Why?
      no, we never want to deal with types that can't have members

   Q: What is the universe, U, closed under?
      non-empty subset, cartesian product, power set,
      and function space construction

   Q: What distinguised sets are assumed in the universe, U?
      an infinite set of individuals, I,
      a two-element set, 2 == {tt,ff}

   Q: What's a model of a type structure?
      a map from operators, f \in Op_n, to fuctions M.f: U^n -> U
      such that
           M.Bool == 2
           M.(S -> T) == {f | f : M.S -> M.T}           

** Semantics of terms
   Q: What is a model of a signature, SG, over a type structure AT?
      A model of AT and a meaning function, M, such that
       for all c:S \in SG, M.c:S \in M.S
       and for all types S, M.(==):S->S->Bool is equality on M.S

   Q: What is a model of a type environment?
      an environment, which relates variables to values



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          MEANING OF TERMS

def: Let TE be a type environment.
    A TE-environment, E, is a function
      E : dom(TE) -> U
    such that for all (v:S) \in TE,
         E.v \in M.S

def: Let be a signature.  Let TE |- t: S.
     Let E be a TE-environment.
     Then M_E.t:S is defined by

  M_E.c:S        == M.c if c:S \in SG, and
                         c \not\in dom(TE)
  M_E.v:S        == E.v if v:S \in TE
  M_E.(t1.t2):S  == (M_E.t1).(M_E.t2)
  M_E.(\v:S.t):S == (\a.M_{E,(v:a)}.t)
  M_E.(t:S):S    == M_E.t : S

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