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I'm looking at some lambda calculus at the moment and came across this question:

0:R

1:R

plus: R->R->R

(lambda f:T . lambda g:U . (f 0) (g 0)) (plus 1) (plus (plus 1 1))

Is it well typed given appropriate types for T and U?

I'm new to lambda calc but I gather that we substitute (plus 1) for f and (plus (plus 1 1)) for g.

This will reduce to
(plus 1 0) (plus (plus 1 1) 0) The right hand side (plus 1 1) will reduce to an R and then we will have (plus R 0) which will reduce to an R. The left side will reduce to an R so the whole expression reduces to the form R R.

Is this correct? Then f and g are of type R -> R (from currying) to make it valid. Is it an issue that the final expression will be of the form R R. Does this mean R gets applied to R then, so R would have to be an 'automorphic' function (mapping from a type to the same type)... is this something related to not having recursion in simply typed lambda calc?

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While your way of calculating the types isn't quite how it works, it gives the right answer here. As you say, you end up with f 0 having type R and g 0 having type R. Since we apply f 0 to g 0, R must be of the form R -> X for some X. In other words, you have the equi-recursive type R = R -> X. Such types are almost always disallowed. Thus, this would be a type error. If R is a fixed primitive type, e.g. a type of primitive natural numbers, then it is simply not the case that R = R -> X and so you would also get a type error.

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  • $\begingroup$ Thanks, this is what I was thinking of when I said contradicting with the lack of recursion. So, for example $\lambda_{:T} v . v v$ is always invalid in simply typed lambda calculus, ie. we cannot find an appropriate type for T? $\endgroup$ – Rob M Jan 9 '18 at 18:48
  • $\begingroup$ There's a difference between having value-level recursion, e.g. a fixed point primitive, which is quite common in simply typed lambda calculi meant for programming, and having type-level equi-recursion, which is very uncommon. Iso-recursive types, where you have to explicitly "wrap" and "unwrap" between a type and its expansion is fairly common. Equi-recursive and iso-recursive types don't require value-level recursion, though they often provide an indirect way of creating it. For the simply typed lambda calculus without equi-recursive types, $\lambda v\!:\!T.vv$ is always a type error. $\endgroup$ – Derek Elkins Jan 9 '18 at 18:57
  • $\begingroup$ There are other type systems that can give a type to $\lambda v\!:\!T.vv$. Intersection type systems can give it a type like $(T \& (T \to X)) \to X$. In polymorphic type systems with higher-rank types, you can set $T=\forall a.a \to a$, but the type $\forall a.a\to a$ is very strict, e.g. in many systems there is only one value for that type, the identity function. $\endgroup$ – Derek Elkins Jan 9 '18 at 19:00

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