Re: [Haskell-cafe] Is (==) commutative?

Dear Alexander, On 07/26/2012 01:09 PM, Alexander Solla wrote:

On 7/25/12, Christian Sternagel wrote:

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On 07/26/2012 11:53 AM, Alexander Solla wrote:

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The classically valid inference:

(x == y) = _|_ => (y == x) = _|_ Btw: whether this inference is valid or not depends on the semantics of (==) and that's exactly what I was asking about. In Haskell we just have type classes, thus (==) is more or less arbitrary (apart from its type).

Indeed. This is true for the interpretation of any function. But you apparently want to treat (==) as equality. This may or may not be possible, depending on the interpretation you choose. Does _|_ == _|_, or _|_ =/= _|_, or do these questions not even make semantic sense in the object language? That's what you need to answer, and the solution to your problem will become clear. Note that picking any of these commits you to a "philosophical" position, insofar as the commitment will induce a metalanguage which excludes expressions from other metalanguages. for my specific case (HOLCF) there is already a fixed metalanguage which has logical equality (=) and differentiates between type 'bool' (True, False) for logical truth and type 'tr' (TT, FF, _|_) for truth-valued computable functions (including nontermination/error). Logical equality satisfies '_|_ = _|_'. Now in principle (==) is just an arbitrary function (for each instance) but I gather that there is some intended use for the type class Eq (and I strongly suspect that it is to model equality ;), I merely want to find out to what extend it does so).

Currently the axioms of the formal eq class include (_|_ == y) = _|_ (x == _|_) = _|_ (and this decision was just based on how ghci evaluates corresponding expressions). The equations you use above would be (roughly) written '(_|_ == _|_) = TT' and '(_|_ /= _|_) = TT' in HOLCF and neither of them is satisfied, since both expressions are logically equivalent to _|_ (in HOLCF). (But still both expressions make sense.) During the discussion I revisited the other two axioms I was using for eq... and now I am wondering whether those make sense. The other two axioms where (x == y) = TT ==> x = y (x == y) = FF ==> not (x = y) The second one should be okay, but the first one is not true in general, since it assumes that we would only ever use Eq instances implementing syntactic equality (which should be true for deriving?). E.g., for the data type data Frac = Frac Int Int we would have 'not (Frac 1 2 = Frac 2 4)' in HOLCF (since we have injectivity of all constructors). But nobody prevents a programmer from writing an Eq instance of Frac that compares 'normal forms' of fractions. cheers chris