The Semantics of SIMPLE

P in Prog
L in SL
S in Stat
E in Exp
V in Val
I in Ide

// Programs
Prog => L
// Lists of statements
SL => S
    |  S ; L
// Statements
Stat => I = E
      |  if E then L fi
      |  if E then L0 else L1 fi
      |  while E do L od
      |  read(I)
      |  write(E)
// Expressions
Exp => E1 + E2
     |  E1 * E2
     |  E1 - E2
     |  E1 / E2
     |  (E)
     |  V
     |  I

n in N                   Natural Numbers
env in U = Id -> N       Environments
inp in I = N*            Input (a sequence of numbers)
out in O = N*            Output (a sequence of numbers)
cont in C = U -> I -> O  Continuations

meaningProg : Prog -> I -> O
meaningSL : SL -> C-> U -> I -> -> O
meaningStat : Stat -> C -> U -> I -> O
meaningExp : Exp -> U -> N
meaningVal : Val -> N

meaningProg[[SL]] = meaningSL[[SL]] cleanup basicEnv

meaningSL[[S;L]] =
  \cont . \env . \inp .
  meaningS[[S]] (meaningSL[[L]]] cont) env inp
meaningSL[[S]] = meaningS[[S]]

meaningStat[[read(I)]] =
  \cont . \env . \inp .
    cont (setenv env I (car inp)) (cdr inp)
meaningStat[[write(E)]] =
  \cont . \env . \inp .
    cons (meaningExp E env) (cont env inp)
meaningStat[[I = E]] =
  \cont . \env . \inp .
    cont (setenv env I (meaningExp E env)) inp
meaningStat [[if E then L1 else L2 fi]] =
  \cont . \env . \inp .
  (meaningExp E env) = 0 -> meaningSL L2 cont env inp
                          , meaningSL L1 cont env inp
meaningStat [[if E then L fi]] =
  \cont . \env . \inp .
  (meaningExp E env) = 0 -> cont env inp
                          ; meaningSL L cont env inp
meaningStat[[while E do L od]] = F(E,L) s.t.
  F(E,L) : C -> U -> I -> O
  F(E,L) cont env inp =
    ((meaningExp E env) == 0) -> cont env inp,
    meaningSL L (\e . \i . F(E,L) cont e i ) env inp

meaningExp [[E1 + E2]] =
  \env .
  (meaningExp E1 env) +
  (meaningExp E2 env)
meaningExp [[E1 - E2]] =
  \env .
  (meaningExp E1 env) -
  (meaningExp E2 env)
meaningExp [[E1 * E2]] =
  \env .
  (meaningExp E1 env) *
  (meaningExp E2 env)
meaningExp [[E1 / E2]] =
  \env .
  (meaningExp E1 env) /
  (meaningExp E2 env)
meaningExp [[I]] =
  \env . (env I)
meaningExp [[V]] =
  \env . meaningVal V
meaningExp [[(E)]] =
  meaningExp E

// Build a basic environment that assigns 0 to every variable.
basicEnv : U
basicEnv = \x . 0
// A continuation used to ``clean up'' at the end of the program.
cleanup : C
cleanup = \env . \inp . nil
// Set the value associated with an identifier in an environment.
setenv : U -> Id -> N -> U
setenv env id num =
  \x . (x == id) -> num , (env id)