Denotational Semantics

1
Denotational Semantics

• The meaning of a program is given by specifying
  the function (from input to output) that
  corresponds to the program.
• The denotational semantic definition, thus maps
  syntactical things into functions PProgram?Inp
  ut?Output
• We need to specify each of these
• Syntactic domains (describing program syntax
  elements that are assigned common valuation
  functions)
• Semantic domains (that are used to express
  meanings of programs and their components).
• Semantic functions (that map syntax into
  semantics)

2
Elements for The Expression Language

• Syntactic DomainsE Expression N NumberD
  Digit
• Semantic Domainsv Integer ...,-2,-1,0,1,2,...
  Operations Integer ? Integer ?
  Integer -Integer ? Integer ? Integer Integer
  ? Integer ? Integer
• Semantic FunctionsE Expression ?
  IntegerEE1 E2 EE1 EE2EE1
  E2 EE1 EE2EE1 - E2
  EE1 - EE2E( E ) EEEN
  NNN Number ? IntegerNND
  10NN NDNND DDD Digit ?
  IntegerD0 0, D1 1, ... D9
  9

3
Adding an Environment to the Expression Language

• Need to create a lifted domain from the Integers.
  A lifted domain has all the elements of some
  partially ordered domain together with a unique
  bottom element (?), that is less than any other
  element in the original domain. We call the
  lifted integers Integer?.
• New Semantic DomainsEnv Environment
  Identifier ? Integer?
• Modified Semantic FunctionsE Expression ?
  Environment ? Integer?EE1 E2(Env)
  EE1(Env) EE2 (Env) EE1
  E2(Env) EE1(Env) EE2(Env) EE1 -
  E2(Env) EE1(Env) - EE2(Env) E( E
  )(Env) EE(Env)EI(Env) Env(I)
  EN(Env) NNNote that E is now a
  higher-order function (when given an Expression,
  it yields a function mapping an Environment to an
  Integer). Such functions can be constructed from
  normal functions by Currying. This is useful in
  this context, because it lets us explain what the
  meaning of an expression is, no matter what the
  environment in which we evaluate that expression.

4
Adding Statements

• New Syntactic DomainsL Statement-listS
  StatementI IdentifierL Letter
• New Semantic FunctionsP Program ?
  EnvironmentPL LL(Env0))L
  Statement-list?Environment?EnvironmentLL1
  L2 LL2 ?LL1LS SSS
  Statement ? Environment ? EnvironmentSI
  E(Env) EnvI EE(Env)

5
Mathematics of Functional Programming

• We can define a functionby extension (by
  providing the graph of the function, namely, the
  set of ordered pairs that represent the
  functions mapping) orby comprehension (by
  providing a rule that tells us how to calculate
  the value of the function at any point).
• Consider the normal definition of the factorial
  function f(n) if n 0 then 1 else ng(n-1)
• Think of a recursive definition of a function as
  really naming two functions, the function used in
  the body, and the function being defined f(n)
  if n 0 then 1 else nf(n-1)
• Now f has two arguments, n, and g. Think of
  defining a higher order function H H(g)(n) if
  n 0 then 1 else ng(n-1)
• The factorial function f is a fixed-point of H,
  that is, a value of H such that H(g) g.
  Why? H(f)(n) if n 0 then 1 else nf(n-1)
  f(n)

6
Least-Fixed-Points

• If we think of a function being represented by
  extension, then we can talk about a partial
  ordering of all functions, with the bottom
  element being the empty set ?.
• Think of the function that just maps 0 to 1,
  (0,1), and the function that in addition maps 1
  to 1, (0,1),(1,1), and the function that
  further maps 2 to 2, (0,1),(1,1),(2,2), and the
  function that further maps 3 to 6,
  (0,1),(1,1),(2,2),(3,6).We are constructing a
  sequence of functions, f0, f1, f2, f3, that begin
  a sequence of functions that represent the
  factorial function.
• If we defined a function f that was just like f,
  but in addition mapped negative values to -1, f
  would still satisfy the definition of the
  factorial function, however, f would be a larger
  function in the natural lattice order of the
  function definitions because it contains every
  pair in f, but it also contains many other
  pairs.Thats why we say the factorial function
  is the least-fixed-point of the definition of
  H(f).

7
Adding Control Flow

• Defining if statements is relatively
  straightforwardSif E then L1 else
  L2(Env) if EE(Env) gt 0 then LL1(Env)
  else LL1(Env)
• While and other repetitive statements, however
  present somewhat of a problem Swhile E do
  L od(Env) if EE(Env) ? 0 then Env
  else SL while E do L od(Env)
• The recursive nature of the while is typically
  handled by defining the meaning as the
  least-fixed-point of the specified semantic
  function.