Question

What,if anything, prevents the execution of a Haskell term from being a proof in equational logic?

Below are some things thay may impede treating Haskell as a logical system with terms being true equations. Any feedback on these reasons would be welcome. Are they are valid reasons? What is their possible impact? Are there more reason?

--------------------Reason (1)------------------

There are some equations that are not expressible in Haskell e.g. mirror (mirror x) = x . It is not possible at the value level, because Haskell does not support dependent types and thus cannot express the type of the proposition "forall a . forall x:Tree a, mirror (mirror x) = x", and therefore a proof term also cannot be constructed. Another example from Torrini et al discusses specifying order. A partial order that is also linear is called a total order. The class linorder of total orders is specified using the usual total order axioms. They conclude that such axiomatizations are not possible in Haskell.

--------------------Reason (2)------------------

According to Thompson the equations in Miranda (and I assume Haskell) are not pure mathematical equations due to *where* and other reasons. According to Danielsson the fact that they are not always structurally equations does not prevent functional programmers from

using them as if they were valid equations. Danielsson show that this informal *fast and loose* use of equational axioms and theorems is *morally correct*. In particular, it is impossible to transform a terminating program into a looping one. These results justify the informal reasoning that functional programmers use.

Kieburtz states that neither the syntax nor type system discriminate between values and latent computations.

To prove some property of a Haskell program the verification logic must support such distinctions. To address this shortcomming of Haskell Kieburtz has included P-Logic in the Programatica Project to provide a verification logic for Haskell

--------------------Reason (3)------------------

There is no formal specification for the meaning of a Haskell program (i.e. its meaning is not defined in a logic). At the level of precise logical specification and logical interoperability this could be a problem (e.g. semantic web likes logic). This should not be a problem for programming tasks, after all most languages like Java do not have a formal semantic definition in logic (like Maude or CafeOBJ ).