abstract interpretation

abstract interpretation

   <theory> A partial execution of a program which gains
   information about its {semantics} (e.g. control structure,
   flow of information) without performing all the calculations.
   Abstract interpretation is typically used by compilers to
   analyse programs in order to decide whether certain
   optimisations or transformations are applicable.

   The objects manipulated by the program (typically values and
   functions) are represented by points in some {domain}.  Each
   abstract domain point represents some set of real
   ("{concrete}") values.

   For example, we may take the abstract points "+", "0" and "-"
   to represent positive, zero and negative numbers and then
   define an abstract version of the multiplication operator, *#,
   which operates on abstract values:

   	*# | + 0 -
   	---|------
   	+  | + 0 -
   	0  | 0 0 0
   	-  | - 0 +

   An interpretation is "safe" if the result of the abstract
   operation is a safe approximation to the abstraction of the
   concrete result.  The meaning of "a safe approximation"
   depends on how we are using the results of the analysis.

   If, in our example, we assume that smaller values are safer
   then the "safety condition" for our interpretation (#) is

   	a# *# b# <= (a * b)#

   where a# is the abstract version of a etc.

   In general an interpretation is characterised by the {domains}
   used to represent the basic types and the abstract values it
   assigns to constants (where the constants of a language
   include primitive functions such as *).  The interpretation of
   constructed types (such as user defined functions, {sum types}
   and {product types}) and expressions can be derived
   systematically from these basic domains and values.

   A common use of {abstract interpretation} is {strictness
   analysis}.

   See also {standard interpretation}.

   (1994-11-08)