Strachey, referential transparency, Haskell

2012-08-05 - Magnus Therning

This is my input into the recent discussion on referential transparency (RT). I’m nowhere near as well versed in the subject as others, but how am I ever to learn anything unless I put my thoughts out there for them to be laughed at and ridiculed? ;)

It all started with a [post on stackoverflow.com](http://stackoverflow.com/questions/210835/what-is-referential-transparency, which received several very long and detailed responses, in particular from Uday Reddy (here and here. His answers were also linked to from Reddit. His second response contains a link to an excellent paper by Strachey, Fundamental concepts in programming languages. I’d go as far as saying that, despite it being lecture notes rather than a fully worked paper, it ought to be required reading for all software developers.

The rest of what I write here hinges on me actually understanding what Strachey writes in his paper. Of course I’m looking forward to comments/corrections/etc that help me correct my understanding.

What Strachey says about RT

In section 3.2.1 he introduces RT like this:

One of the most useful properties of expressions is that called by Quine referential transparency. In essence this means that if we wish to find the value of an expression which contains a sub-expression, the only thing we need to know about the sub-expression is its value. Any other features of the sub-expression, such as its internal structure, the number and nature of its components, the order in which they are evaluated or the colour of the ink in which they are written, are irrelevant to the value of the main expression.

There is however a crucial bit just before that:

Like the rest of mathematics, we shall be concerned only with R-values.

That is, he starts out with a very limited subset of what most people would consider a usable imperative programming language.

He then dives into some more details in section 3.2.2 by adding the concept of environment, which is handled through the use of a where-clause, or alternatively using let-statements (this ought to be making any Haskell developer feel right at home). After a few interesting sections on stuff like applicative structure, evaluation, and conditional expressions he finally tackles the issue of variables in section 3.3.1. There are two pieces to the trick, the first is to take advantage of his earlier insight that lead to a split of values into L-values and R-values:

If we consider L-values as well as R-values, however, we can preserve referential transparency as far as L-values are concerned. This is because L-values, being generalised addresses, are not altered by assignment commands. Thus the command x := x+1 leaves the address of the cell representing x (L-value of x) unchanged although it does alter the contents of this cell (R-value of x). So if we agree that the values concerned are all L-values, we can continue to use where-clauses and lambda-expressions for describing parts of a program which include assignments.

The cost of this is that the entire theory constructed earlier for operations taking R-values now has to be revised to incorporate L-values. The outline for this is in the rest of section 3.3 and it basically comes down to include an abstract store in the environment. However, before doing that he mentions that:

I think these problems are inevitable and although much of the work remains to be done, I feel hopeful that when completed it will not seem so formidable as it does at present, and that it will bring clarification to many areas of programming language study which are very obscure today. In particular the problems of side effects will, I hope, become more amenable.

He does reach his goal, but it’s a bit unfortunate that he stops short of considering the wider of problem of side effects. My assumption is that this would have to be dealt with in a similar way to assignment, but that would mean that rather than just adding an store to the environment the world, or a subset of it, would need to be added.

An open question (to me) is if anyone has built on Strachey’s work in this area and thought of the details of RT and general side effects?

RT in Haskell

The original question described RT as

it means you can replace equals with equals

which I actually think is a rather good, and very short, description of it. It’s not the full story, there are further important details, but it’s a good first intuition. Also, it’s a description usable in Haskell. Well, to be slightly more nuanced, it good for Haskell without IO (Haskell-IO). However, this is where the strict type system of Haskell really starts to shine because (here I’m probably a bit imprecise) we only have R-values in Haskell-IO. If we want to use assignment we add the use of a state monad, and we do that explicitly.

A former colleague of mine said that in Haskell we need to build up our own computational models ourselves. For instance, if we need assigment we use State, if we need to record progress we use Writer, etc. In other languages the language designer has already made all those choices for us, we don’t get to make them ourselves. For RT it means that Haskell is more explicit in what the environment of a function is.

Moving on to general side effects those are also more explicit in Haskell since they have to happen inside the IO monad. That alone is a great boon for RT in Haskell since it becomes explicit where RT as worked out by Strachey applies directly, and where there are (hopefully amenable) problems of side effects left. Even further, in Haskell it’s possible to make subsets of IO (by wrapping IO, see e.g. my own posts on wrapping IO, part 1 and wrappint IO, part 2. I’m sure that if including the world in the environment is the way to achieve RT with general side effects, then it’s highly beneficial to be able to create subsets of the world.

RT in Haskell vs. RT in (common) imperative languages

Uday writes in his first answer that:

But, today, functional programmers claim that imperative programming languages are not referentially transparent. Strachey would be turning in his grave.

This may well be true, but I think that when a Haskell programmer says it, he’s only twitching slightly. The reason? Strachey writes:

Any departure of R-value referential transparency in a R-value context should either be eliminated by decomposing the expression into several commands and simpler expressions, or, if this turns out to be difficult, the subject of a comment.

Which is something that Haskell programmers do naturally by use of IO. That is, in Haskell you either have an R-value, and you clearly see that you do, or you put in a comment, which is encoded in the type of the function.

This rather lengthy post basically arrives at the following, which is what I suspect the user pacala is saying about RT on Reddit:

Imperative languages may well be RT, but when trying to understand a code base the environment of each function is so large that understanding is an intractable problem. I don’t have this problem in Haskell.

(x > y -> SIN, COS) (z)                        // ... -> ... , ... is conditional expression, apply to z

Array (x, y) := z                              // array update (note that arrays are not native)

TRIG := (x > y -> SIN, COS)                    // excellent way to perplex any reader
                                               // but can achieve something like anonymous funs