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Showing posts from 2009

Type Checking and Cyclic Proof

Standard functional programming languages like Haskell and SML use polymorphic type systems. These type systems however, are not sound. What this means is that the type-systems themselves can not be used to prove properties of the software in the sense of "total correctness". To see that this is true, we can get a "proof" (read program) of inhabitation of any arbitrary type A, by simply using the program:

data Bot {- Look Ma! No constructors! -}
bot :: Bot
bot = bot

Clearly when we say that bot is an inhabitant of Bot, we dont mean that it actually produces a value of type Bot, since there aren't any as Bot has no constructors! We can easily use this type of proof to prove something like A ∧ ¬ A which leads to a pretty degenerate logic. However, the type system is still *useful* in the sense that if the program ever *does* terminate, it's sure to do so with the appropriate type. This means we can get the full class of Turing complete programs, a very…