Consider her way -- Re: Dimensionality Checking

["Alexandre E. Kopilovitch" <aek@vib.usr.pu.ru>]

Recent discussion about the dimensions/unit analysis seems to me... well, not
directed by the spirit of the Ada language. As far as I understand, all
participants of the discussion presume that if the dimensional/unit analysis
should be done for an Ada program then it must be performed entirely with the
facilities of Ada language itself. I think that that assumption is plain wrong.

  Let us consider what the dimensional/unit analysis is, in general terms.
For every dimension/unit U we have the mapping Deg_U, which assigns the
rational numbers to the subtypes (that is, the domain of the Deg_U is the set
of all subtypes in our Ada program, and the range of the Deg_U is the set of
rational numbers). We extend this mapping to the variables and user-defined
functions, using the subtype of a variable and the subtype that a function
returns. The mapping Deg_U is "logarithmic", that is, for any variables (or 
user-defined functions) X, Y, Z:
  Z is compatible with X * Y implies Deg_U(Z) = Deg_U(X) + Deg_U(Y), and
  Z is compatible with X / Y implies Deg_U(Z) = Deg_U(X) - Deg_U(Y),
Then we require for all assignments (including an argument passing), additions
and subtractions in our program, that the values of the mapping Deg_U for the
right operand (side) must be equal to the corresponding value for the left
operand. This requirement permits us to define Deg_U over the rational
expressions. Taking into account that the "logarithmic" rule effectively
determines the values of the mapping for the square root, we conclude that 
the mapping Deg_U is defined over the algebraic expressions. The final step
is to assign zero value to all standard (predefined) transcendent functions
(such as Sin etc.). So, the mapping Deg_U is defined for all expressions,
and the condition to be verified is already formulated: for each assignment,
addition and subtraction the value of the mapping Deg_U on the right operand
must be equal to the corresponding value on the left operand.
  This is a definition of the basic "linear" dimensional analysis. One may
construct other, more sophisticated forms of the dimensional analysis for the
specific purposes.

  Now let's recall the fact that the Ada is not a problem-oriented language,
but rather a "superassembler". It intentionally and carefully avoids all
paradigms that aren't closely related to the real computer architectures or
to the general software engineering, even if those paradigms are heavily used
in some significant application area. (Note that I'm speaking here about the
paradigms; the data representation issues are another matter, that is a natural
job for an assembler.)
  Obviously, the dimensionality paradigm is one of the kind that Ada avoids:
it belongs neither to a computer architecture nor to the general software
engineering, but to the particular application area, no matter how significant
it is in the real world. Therefore Ada most probably will not take any move
to support it directly.

  The proper way to do the dimensional/unit analysis for the Ada programs is
to use the ASIS and some suitable language processor, I guess that the SML
might be the best for this purpose (because it is well-suited for the
manipulations with the algebraic type systems). So the configuration of the
whole tool chain may look like that:

       dimension/unit values for the subtypes
                       |
                       |
                      \|/
                       |
  |------|         |---------|	      |-------------|
  | ASIS | ------> | SML     | ------ | SML         |
  | tool |         | program |        | interpreter |
  |------|         |---------|        |-------------|
     |                 |
    /|\                |
     |                \|/
     |                 |
  |----------|    diagnostic output
  | Ada      |      
  | compiler |
  |----------|
     |
    /|\
     |
     |
 program to be verified
   

  Finally, I'll try to explain why the subtypes, and not the types, are the
natural carriers for the dimensionality info. Briefly, with the physical
magnitudes, all the dimensionalities are imaginary, and only the repetition
count within some underlying measurement process is real. That count is obviously
dimensionless. In fact, a dimension of physical magnitude is the abstraction
for the class of the instruments with which we can measure the magnitude.
  If you do not like such vague metaphysical arguments then consider the
following question: if X and X*X belong to the different types, how will you
treat (interpret) the usual approximations by the Taylor series?
  And as for the units, I hope everyone will agree that there is no fundamental
difference (in physics) between miles and kilometers.


Alexander Kopilovitch                      aek@vib.usr.pu.ru
Saint-Petersburg
Russia