Tuesday, May 19, 2015

Constructors and Co-constructors in HoTT

The HoTT book's treatment of constructors, eliminators, etc. deviates somewhat from the treatment commonly found in texts on constructive logic.  Of eliminators, it says that they say how to use elements of a type.  A computation rule "expresses how an eliminator acts on a constructor."  In all cases, the book usually presents these things as functions.

The contrast is with the way intro and elim rules are usually presented in logic, as inference rules.  For example, &-intro licenses an inference from A, B to A&B.  This correponds cleanly to the HoTT pair constructor.  But &-intro is usually accompanied by two &-elim rules, call them &-elim1 and &-elim2, which are also inference rules.

&-elim1:  A&B |- A
&-elim2:  A&B |- B

This could also be express more HoTTly:

elim1:  (a,b):AxB |- a:A
elim2:  (a,b):AxB |- b:B

The HoTT book does not propose eliminators like this.  Instead it defines projection functions pr1 and pr2, which are like elim rules but are not basic to the type definition.  It offers an "elimination rule for products" that answers the question "How can we use pairs, i.e. how can we define functions out of a product type?"  The technique is to define a function on pairs f:AxB->C in terms of an iterated function g:A->B->C.

So the contrast in this case is fairly clear: the logical tradition treats an eliminator as an inference that "undoes" a constructor; HoTT treats an eliminator as a function that "uses" a construction.  In the case of &/x (i.e. product types), that means HoTT must introduce derived functions (projectors) in order to "undo" the constructor.

Note that this makes HoTT's terminology a bit abusive; after all a function on products does not necessarily eliminate anything: just look at the Identity function.

Also note that in both cases the relation between constructor and eliminator is asymmetric; constructors are "more primitive" than eliminators.  This follows Gentzen's original treatment of intro and elim rules: intro rules are viewed as defining (in a sense) the logical operators, and elim rules are derived (in a sense) from intro rules.

The HoTT book's strategy seems follow from the decision to treat these things as functions.  The projection functions cannot be primitive (to the product type) since we do not yet (in the middle of defining product types, so to speak) know how to apply a function to a product type.  So you have to start by defining how any such function f:AxB->C works, and then use that result to define the specific projection functions.

The proposal here is to reconceptualize constructors and eliminators.  This involves several basic moves: first, foreground the notion of inference rather then functions; second, do not treat constructors and eliminators as functions, and third, treat eliminators as co-constructors rather than undoers or users of constructors.

Foregrounding inference:  read "a:A" as "from token a infer type A" (there's a lot to be said about this approach but I'll leave that for another post) and treat the rules defining types like AxB as inference rules.  Not traditional inference rules that go from true premises to true conclusion, but inferency inference rules, that go from good inferences to good inferences.  For example, read pair constructor (intro rule) as something like "if the inference from a to A is good and the inference from b to B is good, then the inference from (a,b) to AxB is good".

Constructors and co-constructors: the idea is pretty simple and easily illustrated for the case of product types:

()-ctor:   $$a:A,\ b:B \vdash (a,b):A\times B$$
pr1-co-ctor:   $$x:A\times B \vdash pr1(x):A$$
pr2-co-ctor:   $$x:A\times B \vdash pr2(x):B$$

In the case of ctors, the rule is not to be construed in functional terms: it is not a function that produces a "result" of type AxB, given inputs of types A and B.  It does not "produce" anything at all; it just licenses the inference from (a,b) to AxB, given inferences from a to A and from b to B.  Strictly speaking we should not even think of this as constructing anything either, except perhaps the syntactic form (a,b).

In the case of co-constructors, the idea is that they, like constructors, license inferences.  As in the case of so-called constructors, we're not really constructing anything; $$pr1(x)$$ has type $$A$$ if $$x$$ has type $$A\times B$$.  That is all.  But the construction metaphor is convenient nonetheless: co-constructors may be viewed as "devices" that construct something, just like constructors.

Notice that we could write these using a function type notation, but here (as in HoTT) there is no function definition for constructors and co-constructors.

A critical contrast with the HoTT (and the traditional logical) approach is that co-constructors need not operate on constructors.  Constructors and co-constructors are perfectly symmetrical; nothing in the concept of types and tokens warrants (a priori) the primacy of constructors over co-constructors.  In other words it is not the case that one must first construct an element in order to have something that an eliminator can operate on.

Also note that with this approach a HoTT-style definition of how functions on product types may be defined still works, but it is not to be construed as an elimination rule on products.

It follows from the definitions that $$x:A\times B \vdash (pr1(x), pr2(x)) : A\times B$$.  This does not mean that every $$x:A\times B$$ is a pair, but it does mean that a pair can be inferred from every such $$x$$.  Which dovetails with the HoTT book's strategy: to define a function on a product type it suffices to define it for all pairs.

What about the uniqueness principle uppt?  In the HoTT book, it depends on the induction principle for pairs.  But as just mentioned that follows directly from the constructor and co-constructors.