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Stream: theory: type theory

Topic: APG type theory

Julius Hamilton (Aug 24 2024 at 18:29):

apg.jpg

The set $\mathcal{V}$ of typed values are terms in the following grammar:

$v : t ::= () : 1 | inl_{t_2} (v : t_1) : t_1 + t_2 | inr_{t_1} (v : t_2) : t_1 + t_2 | (v_1 : t_1, v_2 : t_2) : t_1 \times t_2 | \text{ Prim }v_t : t (t \in P, v \in PV(p)) | \text{ Elmt }e : \lambda(e) (e \in \mathcal(E)$

This says:

There is a set $\mathcal{V}$ whose elements are values
Each value of $\mathcal{V}$ has a type $t$ .
- This is written $v : t$ .
- This collection of types is denoted $\mathcal{T}$ .
$v$ can be of any of the following types:
- The unit type $1$
- The sum type $t_1 + t_2$ , where $t1, t2$ are types in $\mathcal{T}$ .
- The product type $t_1 \times t_2$ .
- The primitive type $t$
- The label type $\lambda(e)$ , for an element $e$

Looking at the whole picture:

You have a schema $S$ , which is a pair $(\mathcal{L}, \sigma)$ . $\mathcal{L}$ is a set of labels, and $\sigma$ assigns each label to a type. Thus, a schema is a bunch of typed labels. Your labels could be “Car”, “Elizabeth”, “meets”. The only types are unit, sum, product, primitive, and label; so I guess $\sigma$ might assign individuals to $1$ , essential concepts to “primitive”, any sort of function, relation or grouping to a product type, any sort of “optional grouping” to a sum type, and I’m not sure what the label type is doing in there, since the labels should be in $\mathcal{L}$ , whereas all these other typed values should be in $\mathcal{V}$ .

2.jpg

You have elements.
You map elements to their labels.
You map values to their types.
You map elements to their values.

And I guess the takeaway here is that our labels have type annotations, and our values have type annotations, and every element has a label and a value.

Is this correct?

Ryan Wisnesky (Aug 24 2024 at 18:47):

sounds about right. there are some built-in examples in the CQL IDE ('APG' being the name of the example)

Julius Hamilton (Aug 24 2024 at 19:20):

Thanks.

These conjectures are unproven?

3.jpg

Ryan Wisnesky (Aug 24 2024 at 19:45):

I'm not sure. @Joshua Meyers would be the person to ask. An earlier version of the paper had a bunch of relatively easy theorems proved in Coq, https://arxiv.org/pdf/1909.04881v1 , and then in attempt to connect the work to Poly, which would have made for cooler theorems, progress stalled out, for reasons I don't remember but might be Coq related and not math related.