As Amr Sabry aptly observed more than a decade ago discussions of purity and referential transparency usually lead to confusion and disagreement. His JFP paper provided the needed rigor and argued that Haskell even _with regular file (stream) IO_ is pure. As was shown yesterday, with Lazy IO, Haskell is not pure. Before we discuss definitions, let us note the motivations. Suppose I have a Haskell program. As every single (compiled) Haskell program it has the main function, of the type IO a for some a. It must use IO, at least to print the final result, the observation of the program. Suppose the program contains the expression "e1 + e2" adding two integer expressions. Can I replace this expression with "e2 + e1" and be sure the observations of my program are unaffected? If one says that ``you should assume that every IO operation returns a random result'' then we have no ability to reason about any real Haskell program. We can't be sure of soundness of any compiler optimization. So, Haskell is far worse than C!? With C, Xavier Leroy has managed to create a provably correct optimizing compiler, and mechanically _prove_ the soundness of all optimizations. A C program, just like a Haskell program, must use IO at least to print the final result. We can never hope to build a provably correct compiler for Haskell? We cannot prove observational equivalences of any real Haskell program? Isn't that sad? The formal question is thus: are the following two expressions f1 and f2, of a *pure type* Int->Int->Int f1, f2:: Int -> Int -> Int f1 = \e1 e2 -> case (e1,e2) of (1,_) -> e1 - e2 (_,_) -> e1 - e2 f2 = \e1 e2 -> case (e1,e2) of (_,1) -> e1 - e2 (_,_) -> e1 - e2 equal? Before one invokes an equational theory or says that both these expressions are just integer subtraction, let me clarify the question: are f1 and f2 at least weakly observationally equivalent? That is, for any program context C[] such that C[f1] is well-typed, the program C[f2] must too be well-typed, and if one can observe the result of C[f1] and of C[f2], the two observations must be identical. The observation of a program may (and often does) involve side-effects and IO (more on it below). The message posted yesterday exhibited a context C[] that distinguishes f1 from f2. Thus, in presence of Lazy IO, any equational theory that equates f1 and f2 cannot be sound. I don't think one can design such a context C[] using the regular, eager file IO. Amr Sabry in a paper ``What is a Purely Functional Language?'' J. Functional Programming, 8(1), 1-22, Jan. 1998. http://www.cs.indiana.edu/~sabry/papers/purelyFunctional.ps proposed the definition of purity, see p 2 and Definition 4.7. The definition essentially states that call-by-name, call-by-value and call-by-need evaluation functions must be weakly equivalent. These evaluation functions map programs to observables. The part of evaluation dealing with observing the answers may have side effects! Sec 5.1 rigorously shows that lambda-calculus with global state and destructive mutation may be called pure functional, if effects are regarded as values and the program is written in what we now call a monadic style (whose idea was proposed by Reynolds back in 1981 when studying Idealized Algol). The end of Sec 5.1 remarks that IO can be treated that way -- and in fact, Appendix D of Haskell report 1.2 indeed had described such a semantics of IO: http://haskell.org/definition/haskell-report-1.2.ps.gz (the appendix starts on p. 139). Thanks to Paul Hudak for pointing this out three years ago. Thus Haskell even with IO may be called pure functional. With Lazy IO, it can no longer be called pure functional as different orders of evaluation of arguments of a function lead to different program observations.