Denotational Semantics

1
Denotational Semantics

• Syntax-directed approach, generalization of
  attribute grammars
• Define context-free abstract syntax
• Specify syntactic categories (terminals,
  non-terminals)
• Specify value domains for semantic (meaning)
  functions
• Define meaning functions to map syntactic
  constructs into their intended meaning
• Syntax-directed approach provides compositional
  rules for meanings.

2
Example decimal numerals

• Syntactic categories
• D ? Digits the decimal digits 0,1
• N ? Num decimal numerals (strings)
• Productions
• D 0 1 2.
• N D N D
• Value Domain
• Nat 0,1, the natural numbers
• Semantics defines mapping of strings into numbers

3
The meaning of decimal numerals

• Semantic functions
• D Digits ? Nat
• M Num ? Nat
• Semantic equations
• D 0 0, D 1 1,
• MD D D
  production for M
• M N D 10 M N M D production for
  M

4
Propositional logic

• Boolean expressions with variables and logical
  connectives.
• Categories
• P ? Prop propositions (variable names)
• F ? Form Formulae
• Productions
• F P F F or F F and F F gt F
  (implication)
• Value Domains
• Bool True, False boolean values
• r ? Assign P ? Bool assignments (values of
  names)

5
The meaning of propositional formulae

• The value of a formula depends on the current
  assignments. Curried form is
• M Form ? Assign ? Bool
• Semantic equations
• M P r r (P) -- meaning of
  variable is its value
• M F r M F r
• M F1 or F2 r M F1 r or M F2 r
• M F1 and F2 r M F1 r and M F2 r
• M F1 gt F2 r M F1 r or M F2 r
• Must distinguish or in syntax from or in
  semantics

6
A language of expressions

• Syntactic categories
• Id ? Ident Identifiers
• Num ? N numeric literals
• E ? Exp Expressions
• Productions
• E num Id (E1 E2) (E1 E2)
• E let Id E1 in E2 end
  local binding
• Value domains
• Semantics describes integer values of
  expressions, in terms of current bindings of
  identifiers
• Int -2, -1, 0, 1, 2 integers
• r ? Env Ident ? Int environments

7
The meaning of simple expressions

• Semantics describes rule of computation. Value is
  a function of environment
• E Exp ? Env ? Int
• Semantic functions
• E Id r r (Id)
• E num r M num given
  previously
• E E1 E2 r E E1 r E E2 r
• E E1 - E2 r E E1 r - E E2 r

8
Bindings and environments

• A binding is modeled by an update function, best
  described in a programming language
• fun Update (Env, Id, Val) (x)
• if Id x then Val else Env
  (x)
• E let Id E1 in E2 end r E E2 Update (r,
  Id, E E1)

9
A domain with an error value

• To model unbound names or type errors in a
  dynamically typed language, extend domain with
  undefined value
• Int -2, -2, 0, 1, 2,
  integers
• r ? Env Ident ? (Int ? )
  environments
• Type checking is explicit
• E Id r let n r (Id) in if Is_Int (n)
  then n else ? end
• E E1 E2 r
• let n1 E E1 r
• n2 E E2 r
• in
• if Is_int (n1) andalso Is_Int (n2) then
  n1 n2 else ?
• end

10
Denotational Semantics of Imperative languages

• Need domains to describe the state of the
  computation, including sequences of instructions
  (the program counter) and memory (the mapping of
  names to locations).
• C ? Com commands, describes control structures
• C C1 C2
• C if E then C1 else C2
• C while E do C
• Value domain
• s ? States configuration of
  computation

11
The meaning of commands

• The value of a command is a state
• C if E then C1 else C2 s
• if IsTrue (E E s) then C C1 s else
  C C2 s
• C C1 C2 s let s C C1 s in C C2
  s end
• C while E do C s
• let
• fun p (s) if IsTrue (E E s )
  then p (C C s ) else s
• in
• p (s)
• end

12
Describing state

• New syntactic category
• L ? left-hand side for bindings and
  references
• Value domains
• a ? Loc
  locations (addresses)
• s ? States Loc ? (Int ? ) memory
• r ? Env Ident ? (Int Loc ? )
  environments
• Semantic functions
• C Com ? Env ? States ? States
• E Exp ? Env ? States ? Int
• L Left ? Env ? States ? Loc

13
Manipulating state

• C C1 C2 if E then C1 else C2
• L E
  assignment
• local V E in C end
  local variable
• L Id
  (lhs is simple name)
• C L E r s let
• a L L r s
  meaning of l.h.s is a location
• n E E r s
  meaning of rhs is a value
• in
• update (s, a, n) meaning of
  assignment is new state
• end
• side effects of E are
  ignored

14
Local variables

• Assume pool of available locations
• C local V E in C end r s
• let
• a ? Loc such that s ( a ) unused
• n E E r s
• in
• C C update (r, V, a) update (s, a,
  n)
• end
• Enlarge environment with new variable,
  enlarge state with its initial value, evaluate
  command in new environment / state

15
Dereferencing

• Obtaining the value of a variable is a two-step
  affair name -gt loc, loc -gt value
• E !L if E1 then E2 else E3
• E !L r s let a L L r s in s (a) end

16
Programs

• New syntactic category P
• P program (I) C end
• program is a command with a read /
  write parameter
• Semantic function P Prog -gt int -gt int
• P program (I) C end n
• let fun r (J) undefined for all J
• fun s (a) unused for all a
• a ? Loc
• sf C C update (r, I, a)
  update (s, a, n)
• in
• sf (a)
• end

17
Function calls

• Semantic domains
• A function affects the state
• f ? Fun States -gt int gt int States
• The name of the function denotes a function, not
  a location
• Denote int Loc Fun
• r ? Env Ident -gt (Denote ?)
• A function does not depend on the environment
  static binding of identifiers.

18
Syntax and semantics of functions

• M function f (a) E1 in E2 end r s
• E1 is body of function, E2
  contains calls
• let
• fun g s n M E1 update (r,
  a, n) s
• in
• M E2 update (r, f, g)
• end
• M call f (E) r s
• let
• (n, s) M E r s func
  r (f)
• in
• func s n
• end

19
Transfer of control

• In the presence of gotos and exceptions, state
  must include notion of program counter.
• C C1 C2 if E then C1 else C2 L E
• local I E in C end skip goto L
  L C
• M program (I) C end
• New semantic domains and functions
• q ? CC States -gt int (next instruction
  to execute)
• C Com -gt Env -gt CC -gt States -gt Int

20
Commands have continuations

• C C1 C2 r q s
• let fun q (s) C C2 r q s
• in
• C C1 r q s
• A label captures a continuation
• C L C r q s
• let fun q (s) C C update (r, L, q)
  q s
• in
• q (s)
• end

21
The goto resets the continuation

• C goto L r q s (r L) s
• The meaning of the goto is the result of applying
  the continuation of the label to the state.
• C Here goto Here r q s q (s)
• And q (s) q (s) for any s gt infinite loop