阿格达的“严格积极”

Question

I'm trying to encode some denotational semantics into Agda based on a program I wrote in Haskell.

data Value = FunVal (Value -> Value)
           | PriVal Int
           | ConVal Id [Value]
           | Error  String

In Agda, the direct translation would be;

data Value : Set where
    FunVal : (Value -> Value) -> Value
    PriVal : ℕ -> Value
    ConVal : String -> List Value -> Value
    Error  : String -> Value

but I get an error relating to the FunVal because;

Value is not strictly positive, because it occurs to the left of an arrow in the type of the constructor FunVal in the definition of Value.

What does this mean? Can I encode this in Agda? Am I going about it the wrong way?

Thanks.

Answer 1

HOAS doesn't work in Agda, because of this:

apply : Value -> Value -> Value
apply (FunVal f) x = f x
apply _ x = Error "Applying non-function"

w : Value
w = FunVal (\x -> apply x x)

Now, notice that evaluating apply ww gives you apply ww back again. The term apply ww has no normal form, which is a no-no in agda. Using this idea and the type:

data P : Set where
    MkP : (P -> Set) -> P

We can derive a contradiction.

One of the ways out of these paradoxes is only to allow strictly positive recursive types, which is what Agda and Coq choose. That means that if you declare:

data X : Set where
    MkX : F X -> X

That F must be a strictly positive functor, which means that X may never occur to the left of any arrow. So these types are strictly positive in X :

X * X
Nat -> X
X * (Nat -> X)

But these are not:

X -> Bool
(X -> Nat) -> Nat  -- this one is "positive", but not strictly
(X * Nat) -> X

So in short, no you can't represent your data type in Agda. You can use de Bruijn encoding to get a term type you can work with, but usually the evaluation function needs some sort of "timeout" (generally called "fuel"), eg a maximum number of steps to evaluate, because Agda requires all functions to be total. Here is an example due to @gallais that uses a coinductive partiality type to accomplish this.