What is Expressiveness in a Computer Language

[Torben Ægidius Mogensen]

"Joe Marshall" <eval.apply at gmail.com> writes:


> > On the Expressive Power of Programming Languages, by Matthias
> > Felleisen, 1990.
> > http://www.ccs.neu.edu/home/cobbe/pl-seminar-jr/notes/2003-sep-26/expressive-slides.pdf
> 
> The gist of the paper is this: Some computer languages seem to be
> `more expressive' than others.  But anything that can be computed in
> one Turing complete language can be computed in any other Turing
> complete language.  Clearly the notion of expressiveness isn't
> concerned with ultimately computing the answer.
>
> Felleisen's paper puts forth a formal definition of expressiveness
> in terms of semantic equivilances of small, local constructs.  In
> his definition, wholescale program transformation is disallowed so
> you cannot appeal to Turing completeness to claim program
> equivalence.

I think expressiveness is more subtle than this.  Basically, it boils
down to: "How quickly can I write a program to solve my problem?".

There are several aspects relevant to this issue, some of which are:

 - Compactness: How much do I have to type to do what I want?

 - Naturality: How much effort does it take to convert the concepts of
   my problem into the concepts of the language?

 - Feedback: Will the language provide sensible feedback when I write
   nonsensical things?

 - Reuse: How much effort does it take to reuse/change code to solve a
   similar problem?

Compactness is hard to measure.  It isn't really about the number of
characters needed in a program, as I don't think one-character symbols
instead of longer keywords make a language more expressive.  It is
better to count lexical units, but if there are too many different
predefined keywords and operators, this isn't reasonable either.
Also, the presence of opaque one-liners doesn't make a language
expressible.  Additionally, as mentioned above, Turing-completeness
(TC) allows you to implement any TC language in any other, so above a
certain size, the choice of language doesn't affect size.  But
something like (number of symbols in program)/log(number of different
symbols) is not too bad.  If programs are allowed to use standard
libraries, the identifiers in the libraries should be counted in the
number of different symbols.

Naturality is very difficult to get a grip on, and it strongly depends
on the type of problem you want to solve.  So it only makes sense to
talk about expressiveness relative to a set of problem domains.  If
this set is small, domain-specific languages win hands down, so if you
want to compare expressiveness of general-purpose languages, you need
a large set of very different problems.  And even with a single
problem, it is hard to get an objective measure of how difficult it is
to map the problem's concepts to those of the language.  But you can
normally observe whether you need to overspecify the concept (i.e.,
you are required to make arbitrary decisions when mapping from concept
to data), if the mapping is onto (i.e., can you construct data that
isn't sensible in the problem domain) and how much redundancy your
representation has.

Feedback is a mixture of several things.  Partly, it is related to
naturality, as a close match between problem concepts and language
concepts makes it less likely that you will express nonsense (relative
to the problem domain) that makes sense in the language.  For example,
if you have to code everything as natural numbers, untyped pure lambda
calculus or S-expressions, there is a good chance that you can get
nonsense past the compiler.  Additionally, it is about how difficult
it is to tie an observation about a computed result to a point in the
program.

Measuring reuse depends partly on what is meant by problems being
similar and also on whether you at the time you write the original
code can predict what types of problems you might later want to solve,
i.e., if you can prepare the code for reuse.  Some languages provide
strong mechanisms for reuse (templates, object hierarchies, etc.), but
many of those require that you can predict how the code is going to be
reused.  So, maybe, you should measure how difficult it is to reuse a
piece of code that is _not_ written with reuse in mind.

This reminds me a bit of last years ICFP contest, where part of the
problem was adapting to a change in specification after the code was
written.

> Expressiveness isn't necessarily a good thing.  For instance, in C,
> you can express the addresses of variables by using pointers.  You
> cannot express the same thing in Java, and most people consider this
> to be a good idea.

I think this is pretty much covered by the above points on naturality
and feedback: Knowing the address of a value or object is an
overspecification unless the address maps back into something in the
problem domain.

On a similar note, is a statically typed langauge more or less
expressive than a dynamically typed language?  Some would say less, as
you can write programs in a dynamically typed language that you can't
compile in a statically typed language (without a lot of encoding),
whereas the converse isn't true.  However, I think this is misleading,
as it ignores the feedback issue: It takes longer for the average
programmer to get the program working in the dynamically typed
language.

        Torben