Louden

1
Loudens Simple Language for Describing Formal
Semantics

• program ? stmt-liststmt-list ? stmt
  stmt-list stmtstmt ? assign-stmt if-stmt
  while-stmtassign-stmt ?identifier
  exprif-stmt ? if expr then stmt-list else
  stmt-list fiwhile-stmt ? while expr do
  stmt-list odidentifier ? a b c ...
  z expr ? expr term expr - term
  termterm ? term factor factorfactor ? (
  expr ) number identifiernumber ? number
  digit digitdigit ? 0 1 2 3
  4 5 6 7 8 9
• One oddity about Loudens little language In
  if-stmt (resp. while-stmt) the expr is evaluated
  in the current environment. If it is greater
  than 0, the then (resp. do) is executed otherwise
  the else is executed (resp. while is terminated).

2
A variety of notations

• For our language, the semantic domain
  Environment, the set of all environments, is the
  collection of all (partial) functions mapping
  identifiers into Integer values. That
  is Environment Identifier ? Integer.
• We might express an element of this set by a name
  such as Env or s.
• The empty environment would be an empty map, one
  that doesnt map any identifier into any Integer
  value.
• A function can be modified by changing its map at
  exactly one element of its domain I, to have
  value n. This can be expressed variously
  as Envn/I sn/I (relatively
  standard) n/iEnv n/Is (somewhat
  standard) I nEnv I ns (somewhat
  nonstandard) EnvIn (very nonstandard)

3
Logical Inference Rules

• Inference rules are all of the following form
• This inference rule embodies the transitive
  property

4
Structural Operational Semantics

• Meaning is assigned to a program by reducing it
  (and its input) to the value it yields.
• A reduction rule has either a syntactic component
  (or a mixture of sentactic components and
  semantic values) on the left side and a semantic
  value on the right side. Consider the rules for
  digits 0 gt 0 1 gt 1 2 gt 2 ... 9
  gt 9and the rules for numbers V 0 gt
  10V V 1 gt 10V 1 V 2 gt 10V
  2 ... V 9 gt 10V 9 V1 V2 gt V1 V2
  V1 - V2 gt V1 - V2 V1 V2 gt V1 V2
  ( V ) gt V ... etc.

2 1
gt 2 1
gt 2 1
gt 2 1
gt 3
5
Adding Assignment Changes Things Slightly
lta 2 3 Env0gt
gt a 5
gt lta 5 Env0gt
gt Env0a 5
gt a 5
lta 2 3 b a 4 a b - 5 Env0gt
gt ltb a 4 a b - 5 a 5gt
gt ltb 5 4 a b - 5 a 5gt
gt ltb 20 a b - 5 a 5gt
gt lta b - 5 a 5, b 20gt
gt lta 20 - 5 a 5, b 20gt
gt lta 20 - 5 a 5, b 20gt
gt lta 15 a 5, b 20gt
gt a 15, b 20
6
Control Flow is Not Much Harder