A hands-on introduction to cubicaltt

Some months ago I gave a series of hands-on lectures on cubicaltt at Inria Sophia Antipolis that can be found at:

https://github.com/mortberg/cubicaltt/tree/master/lectures

The lectures cover the main features of the system and don’t assume any prior knowledge of Homotopy Type Theory or Univalent Foundations. Only basic familiarity with type theory and proof assistants based on type theory is assumed. The lectures are in the form of cubicaltt files and can be loaded in the cubicaltt proof assistant.

cubicaltt is based on a novel type theory called Cubical Type Theory that provides new ways to reason about equality. Most notably it makes various extensionality principles, like function extensionality and Voevodsky’s univalence axiom, into theorems instead of axioms. This is done such that these principles have computational content and in particular that we can transport structures between equivalent types and that these transports compute. This is different from when one postulates the univalence axiom in a proof assistant like Coq or Agda. If one just adds an axiom there is no way for Coq or Agda to know how it should compute and one looses the good computational properties of type theory. In particular canonicity no longer holds and one can produce terms that are stuck (e.g. booleans that are neither true nor false but don’t reduce further). In other words this is like having a programming language in which one doesn’t know how to run the programs. So cubicaltt provides an operational semantics for Homotopy Type Theory and Univalent Foundations by giving a computational justification for the univalence axiom and (some) higher inductive types.

Cubical Type Theory has a model in cubical sets with lots of structure (symmetries, connections, diagonals) and is hence consistent. Furthermore, Simon Huber has proved that Cubical Type Theory satisfies canonicity for natural numbers which gives a syntactic proof of consistency. Many of the features of the type theory are very inspired by the model, but for more syntactically minded people I believe that it is definitely possible to use cubicaltt without knowing anything about the model. The lecture notes are hence written with almost no references to the model.

The cubicaltt system is based on Mini-TT:

"A simple type-theoretic language: Mini-TT" (2009)
Thierry Coquand, Yoshiki Kinoshita, Bengt Nordström and Makoto Takeya
In "From Semantics to Computer Science; Essays in Honour of Gilles Kahn"

Mini-TT is a variant Martin-Löf type theory with datatypes and cubicaltt extends Mini-TT with:

  1. Path types
  2. Compositions
  3. Glue types
  4. Id types
  5. Some higher inductive types

The lectures cover the first 3 of these and hence correspond to sections 2-7 of:

"Cubical Type Theory: a constructive interpretation of the univalence axiom"
Cyril Cohen, Thierry Coquand, Simon Huber and Anders Mörtberg
To appear in post-proceedings of TYPES 2016
https://arxiv.org/abs/1611.02108

I should say that cubicaltt is mainly meant to be a prototype implementation of Cubical Type Theory in which we can do experiments, however it was never our goal to implement a competitor to any of the more established proof assistants. Because of this there are no implicit arguments, type classes, proper universe management, termination checker, etc… Proofs in cubicaltt hence tend to get quite verbose, but it is definitely possible to do some fun things. See for example:

  • binnat.ctt – Binary natural numbers and isomorphism to unary numbers. Example of data and program refinement by doing a proof for unary numbers by computation with binary numbers.
  • setquot.ctt – Formalization of impredicative set quotients á la Voevodsky.
  • hz.ctt\mathbb{Z} defined as an (impredicative set) quotient of nat * nat.
  • category.ctt – Categories. Structure identity principle. Pullbacks. (Due to Rafaël Bocquet)
  • csystem.ctt – Definition of C-systems and universe categories. Construction of a C-system from a universe category. (Due to Rafaël Bocquet)

For a complete list of all the examples see:

https://github.com/mortberg/cubicaltt/tree/master/examples

For those who cannot live without implicit arguments and other features of modern proof assistants there is now an experimental cubical mode shipped with the master branch of Agda. For installation instructions and examples see:

https://agda.readthedocs.io/en/latest/language/cubical.html
https://github.com/Saizan/cubical-demo

In this post I will give some examples of the main features of cubicaltt, but for a more comprehensive introduction see the lecture notes. As cubicaltt is an experimental prototype things can (and probably will) change in the future (e.g. see the paragraph on HITs below).

The basic type theory

The basic type theory on which cubicaltt is based has Π and ∑ types (with eta and surjective pairing), a universe U, datatypes, recursive definitions and mutually recursive definitions (in particular inductive-recursive definitions). Note that general datatypes and (mutually recursive) definitions are not part of the version of Cubical Type Theory in the paper.

Below is an example of how natural numbers and addition are defined:

data nat = zero
         | suc (n : nat)

add (m : nat) : nat -> nat = split
  zero -> m
  suc n -> suc (add m n)

If one loads this in the cubicaltt read-eval-print-loop one can compute things:

> add (suc zero) (suc zero)
EVAL: suc (suc zero)

Path types

The homotopical interpretation of equality tells us that we can think of an equality proof between a and b in a type A as a path between a and b in a space A. cubicaltt takes this literally and adds a primitive Path type that should be thought of as a function out of an abstract interval \mathbb{I} with fixed endpoints.

We call the elements of the interval \mathbb{I} names/directions/dimensions and typically use i, j, k to denote them. The elements of the interval \mathbb{I} are generated by the following grammar (where dim is a dimension like i, j, k…):

r,s := 0
     | 1
     | dim
     | - r
     | r /\ s
     | r \/ s

The endpoints are 0 and 1, – corresponds to symmetry (r in \mathbb{I} is mapped to 1-r), while /\ and \/ are so called “connections”. The connections can be thought of mapping r and s in \mathbb{I} to min(r,s) and max(r,s) respectively. As Path types behave like functions out of the interval there is both path abstraction and application (just like for function types). Reflexivity is written:

refl (A : U) (a : A) : Path A a a = <i> a

and corresponds to a constant path:

        <i> a
   a -----------> a

with the intuition is that <i> a is a function \(i : \mathbb{I}) -> a. However for deep reasons the interval isn’t a type (as it isn’t fibrant) so we cannot write functions out of it directly and hence we have this special notation for path abstraction.

If we have a path from a to b then we can compute its left end-point by applying it to 0:

face0 (A : U) (a b : A) (p : Path A a b) : A = p @ 0

This is of course convertible to a. We can also reverse a path by using symmetry:

sym (A : U) (a b : A) (p : Path A a b) : Path A b a =
  <i> p @ -i

Assuming that some arguments could be made implicit this satisfies the equality

sym (sym p) == p

judgmentally. This is one of many examples of equalities that hold judgmentally in cubicaltt but not in standard type theory where sym would be defined by induction on p. This is useful for formalizing mathematics, for example we get the judgmental equality C^op^op == C for a category C that cannot be obtained in standard type theory with the usual definition of category without using any tricks (see opposite.ctt for a formal proof of this).

We can also directly define cong (or ap or mapOnPath):

cong (A B : U) (f : A -> B) (a b : A) (p : Path A a b) :
     Path B (f a) (f b) = <i> f (p @ i)

Once again this satisfies some equations judgmentally that we don’t get in standard type theory where this would have been defined by induction on p:

cong id p == p
cong g (cong f p) == cong (g o f) p

Finally the connections can be used to construct higher dimensional cubes from lower dimensional ones (e.g. squares from lines). If p : Path A a b then <i j> p @ i /\ j is the interior of the square:

                  p
        a -----------------> b
        ^                    ^
        |                    |
        |                    |
  <j> a |                    | p
        |                    |
        |                    |
        |                    |
        a -----------------> a
                <i> a

Here i corresponds to the left-to-right dimension and j corresponds to the down-to-up dimension. To compute the left and right sides just plug in i=0 and i=1 in the term inside the square:

<j> p @ 0 /\ j = <j> p @ 0 = <j> a   (p is a path from a to b)
<j> p @ 1 /\ j = <j> p @ j = p       (using eta for Path types)

These give a short proof of contractibility of singletons (i.e. that the type (x : A) * Path A a x is contractible for all a : A), for details see the lecture notes or the paper. Because connections allow us to build higher dimensional cubes from lower dimensional ones they are extremely useful for reasoning about higher dimensional equality proofs.

Another cool thing with Path types is that they allow us to give a direct proof of function extensionality by just swapping the path and lambda abstractions:

funExt (A B : U) (f g : A -> B)
       (p : (x : A) -> Path B (f x) (g x)) :
       Path (A -> B) f g = <i> \(a : A) -> (p a) @ i

To see that this makes sense we can compute the end-points:

(<i> \(a : A) -> (p a) @ i) @ 0 = \(a : A) -> (p a) @ 0
                                = \(a : A) -> f a
                                = f

and similarly for the right end-point. Note that the last equality follows from eta for Π types.

We have now seen that Path types allows us to define the constants of HoTT (like cong or funExt), but when doing proofs with Path types one rarely uses these constants explicitly. Instead one can directly prove things with the Path type primitives, for example the proof of function extensionality for dependent functions is exactly the same as the one for non-dependent functions above.

We cannot yet prove the principle of path induction (or J) with what we have seen so far. In order to do this we need to be able to turn any path between types A and B into a function from A to B, in other words we need to be able to define transport (or cast or coe):

transport : Path U A B -> A -> B

Composition, filling and transport

The computation rules for the transport operation in cubicaltt is introduced by recursion on the type one is transporting in. This is quite different from traditional type theory where the identity type is introduced as an inductive family with one constructor (refl). A difficulty with this approach is that in order to be able to define transport in a Path type we need to keep track of the end-points of the Path type we are transporting in. To solve this we introduce a more general operation called composition.

Composition can be used to define the composition of paths (hence the name). Given paths p : Path A a b and q : Path A b c the composite is obtained by computing the missing top line of this open square:

        a                   c
        ^                   ^
        |                   |
        |                   |
  <j> a |                   | q
        |                   |
        |                   |
        |                   |
        a ----------------> b
                p @ i

In the drawing I’m assuming that we have a direction i : \mathbb{I} in context that goes left-to-right and that the j goes down-to-up (but it’s not in context, rather it’s implicitly bound by the comp operation). As we are constructing a Path from a to c we can use the i and put p @ i as bottom. The code for this is as follows:

compPath (A : U) (a b c : A)
  (p : Path A a b) (q : Path A b c) : Path A a c =
   <i> comp (<_> A) (p @ i)
                    [ (i = 0) -> <j> a
                    , (i = 1) -> q ]

One way to summarize what compositions gives us is the so called “box principle” that says that “any open box has a lid”. Here “box” means (n+1)-dimensional cube and the lid is an n-dimensional cube. The comp operation takes as second argument the bottom of the box and then a list of sides. Note that the collection of sides doesn’t have to be exhaustive (as opposed to the original cubical set model) and one way to think about the sides is as a collection of constraints that the resulting lid has to satisfy. The first argument of comp is a path between types, in the above example this path is constant but it doesn’t have to be. This is what allows us to define transport:

transport (A B : U) (p : Path U A B) (a : A) : B =
  comp p a []

Combining this with the contractibility of singletons we can easily prove the elimination principle for Path types. However the computation rule does not hold judgmentally. This is often not too much of a problem in practice as the Path types satisfy various judgmental equalities that normal Id types don’t. Also, having the possibility to reason about higher equalities directly using path types and compositions is often very convenient and leads to very nice and new ways to construct proofs about higher equalities in a geometric way by directly reasoning about higher dimensional cubes.

The composition operations are related to the filling operations (as in Kan simplicial sets) in the sense that the filling operations takes an open box and computes a filler with the composition as one of its faces. One of the great things about cubical sets with connections is that we can reduce the filling of an open box to its composition. This is a difference compared to the original cubical set model and it provides a significant simplification as we only have to explain how to do compositions in open boxes and not also how to fill them.

Glue types and univalence

The final main ingredient of cubicaltt are the Glue types. These are what allows us to have a direct algorithm for composition in the universe and to prove the univalence axiom. These types add the possibility to glue types along equivalences (i.e. maps with contractible fibers) onto another type. In particular this allows us to directly define one of the key ingredients of the univalence axiom:

ua (A B : U) (e : equiv A B) : Path U A B =
  <i> Glue B [ (i = 0) -> (A,e)
             , (i = 1) -> (B,idEquiv B) ]

This corresponds to the missing line at the top of:

        A           B
        |           |
      e |           | idEquiv B
        |           |
        V           V
        B --------> B
              B

The sides of this square are equivalences while the bottom and top are lines in direction i (so this produces a path from A to B as desired).

We have formalized three proofs of the univalence axiom in cubicaltt:

  1. A very direct proof due to Simon Huber and me using higher dimensional glueing.
  2. The more conceptual proof from section 7.2 of the paper in which we show that the unglue function is an equivalence (formalized by Fabian Ruch).
  3. A proof from ua and its computation rule (uabeta). Both of these constants are easy to define and are sufficient for the full univalence axiom as noted in a post by Dan Licata on the HoTT google group.

All of these proofs can be found in the file univalence.ctt and are explained in the paper (proofs 1 and 3 are in Appendix B).

Note that one often doesn’t need full univalence to do interesting things. So just like for Path types it’s often easier to just use the Glue primitives directly instead of invoking the full univalence axiom. For instance if we have proved that negation is an involution for bool we can directly get a non-trivial path from bool to bool using ua (which is just a Glue):

notEq : Path U bool bool = ua boob bool notEquiv

And we can use this non-trivial equality to transport true and compute the result:

> transport notEq true
EVAL: false

This is all that the lectures cover, in the rest of this post I will discuss the two extensions of cubicaltt from the paper and their status in cubicaltt.

Identity types and higher inductive types

As pointed out above the computation rule for Path types doesn’t hold judgmentally. Luckily there is a neat trick due to Andrew Swan that allows us to define a new type that is equivalent to Path A a b for which the computation rule holds judgmentally. For details see section 9.1 of the paper. We call this type Id A a b as it corresponds to Martin-Löf’s identity type. We have implemented this in cubicaltt and proved the univalence axiom expressed exclusively using Id types, for details see idtypes.ctt.

For practical formalizations it is probably often more convenient to use the Path types directly as they have the nice primitives discussed above, but the fact that we can define Id types is very important from a theoretical point of view as it shows that cubicaltt with Id is really an extension of Martin-Löf type theory. Furthermore as we can prove univalence expressed using Id types we get that any proof in univalent type theory (MLTT extended with the univalence axiom) can be translated into cubicaltt.

The second extension to cubicaltt are HITs. We have a general syntax for adding these and some of them work fine on the master branch, see for example:

  • circle.ctt – The circle as a HIT. Computation of winding numbers.
  • helix.ctt – The loop space of the circle is equal to Z.
  • susp.ctt – Suspension and n-spheres.
  • torsor.ctt – Torsors. Proof that S1 is equal to BZ, the classifying
    space of Z. (Due to Rafaël Bocquet)
  • torus.ctt – Proof that Torus = S1 * S1 in only 100 loc (due to Dan
    Licata).

However there are various known issues with how the composition operations compute for recursive HITs (e.g. truncations) and HITs where the end-points contain function applications (e.g. pushouts). We have a very experimental branch that tries to resolve these issues called “hcomptrans”. This branch contains some new (currently undocumented) primitives that we are experimenting with and so far it seems like these are solving the various issues for the above two classes of more complicated HITs that don’t work on the master branch. So hopefully there will soon be a new cubical type theory with support for a large class of HITs.

That’s all I wanted to say about cubicaltt in this post. If someone plays around with the system and proves something cool don’t hesitate to file a pull request or file issues if you find some bugs.

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Type theoretic replacement & the n-truncation

This post is to announce a new article that I recently uploaded to the arxiv:

https://arxiv.org/abs/1701.07538

The main result of that article is a type theoretic replacement construction in a univalent universe that is closed under pushouts. Recall that in set theory, the replacement axiom asserts that if F is a class function, assigning to any set X a new set F(X), then the image of any set A, i.e. the set \{F(X)\mid X\in A\} is again a set. In homotopy type theory we consider instead a map f : A\to X from a small type A:U into a locally small type X, and our main result is the construction of a small type \mathrm{im}(f) with the universal property of the image of f.

We say that a type is small if it is in U, and for the purpose of this blog post smallness and locally smallness will always be with respect to U. Before we define local smallness, let us recall the following rephrasing of the `encode-decode method’, which we might also call the Licata-Shulman theorem:

Theorem. Let A be a type with a:A, and let P: A\to U be a type with p:P(a). Then the following are equivalent.

  1. The total space \Sigma_{(b:A)} P(b) is contractible.
  2. The canonical map \Pi_{(b:A)} (a=b)\to P(b) defined by path induction, mapping \mathrm{refl}_a to p, is a fiberwise equivalence.

Note that this theorem follows from the fact that a fiberwise map is a fiberwise equivalence if and only if it induces an equivalence on total spaces. Since for path spaces the total space will be contractible, we observe that any fiberwise equivalence establishes contractibility of the total space, i.e. we might add the following equivalent statement to the theorem.

  • There (merely) exists a family of equivalences e:\Pi_{(b:B)} (a=b)\simeq P(b). In other words, P is in the connected component of the type family b\mapsto (a=b).

There are at least two equivalent ways of saying that a (possibly large) type X is locally small:

  1. For each x,y:X there is a type (x='y):U and an equivalence e_{x,y}:(x=y)\simeq (x='y).
  2. For each x,y:X there is a type (x='y):U; for each x:X there is a term r_X:x='x, and the canonical dependent function \Pi_{(y:X)} (x=y)\to (x='y) defined by path induction by sending \mathrm{refl}_x to r_x is an equivalence.

Note that the data in the first structure is clearly a (large) mere proposition, because there can be at most one such a type family (x='y), while the equivalences in the second structure are canonical with respect to the choice of reflexivity r_x. To see that these are indeed equivalent, note that the family of equivalences in the first structure is a fiberwise equivalence, hence it induces an equivalence on total spaces. Therefore it follows that the total space \Sigma_{(y:X)} (x='y) is contractible. Thus we see by Licata’s theorem that the canoncial fiberwise map is a fiberwise equivalence. Furthermore, it is not hard to see that the family of equivalences e_{x,y} is equal to the canonical family of equivalences. There is slightly more to show, but let us keep up the pace and go on.

Examples of locally small types include any small type, any mere proposition regardless of their size, the universe is locally small by the univalence axiom, and if A is small and X is locally small then the type A\to X is locally small. Observe also that the univalence axiom follows if we assume the `uncanonical univalence axiom’, namely that there merely exists a family of equivalences e_{A,B} : (A=B)\simeq (A\simeq B). Thus we see that the slogan ‘identity of the universe is equivalent to equivalence’ actually implies univalence.

Main Theorem. Let U be a univalent universe that is closed under pushouts. Suppose that A:U, that X is a locally small type, and let f:A\to X. Then we can construct

  • a small type \mathrm{im}(f):U,
  • a factorization
    image
  • such that i_f is an embedding that satisfies the universal property of the image inclusion, namely that for any embedding g, of which the domain is possibly large, if f factors through g, then so does i_f.

Recall that f factors through an embedding g in at most one way. Writing \mathrm{Hom}_X(f,g) for the mere proposition that f factors through g, we see that i_f satisfies the universal property of the image inclusion precisely when the canonical map

\mathrm{Hom}_X(i_f,g)\to\mathrm{Hom}_X(f,g)

is an equivalence.

Most of the paper is concerned with the construction with which we prove this theorem: the join construction. By repeatedly joining a map with itself, one eventually arrives at an embedding. The join of two maps f:A\to X and g:B\to X is defined by first pulling back, and then taking the pushout, as indicated in the following diagram

join_maps

In the case X \equiv \mathbf{1}, the type A \ast_X B is equivalent to the usual join of types A \ast B. Just like the join of types, the join of maps with a common codomain is associative, commutative, and it has a unit: the unique map from the empty type into X. The join of two embeddings is again an embedding. We show that the last statement can be strengthened: the maps f:A\to X that are idempotent in a canonical way (i.e. the canonical morphism f \to f \ast f in the slice category over X is an equivalence) are precisely the embeddings.

Below, I will indicate how we can use the above theorem to construct the n-truncations for any n\geq -2 on any univalent universe that is closed under pushouts. Other applications include the construction of set-quotients and of Rezk-completion, since these are both constructed as the image of the Yoneda-embedding, and it also follows that the univalent completion of any dependent type P:A\to U can be constructed as a type in U, namely \mathrm{im}(P), without needing to resort to more exotic higher inductive types. In particular, any connected component of the universe is equivalent to a small type.

Theorem. Let U be a univalent universe that is closed under pushouts. Then we can define for any n\geq -2

  • an n-truncation operation \|{-}\|_n:U\to U,
  • a map |{-}|:\Pi_{(A:U)} A\to \|A\|_n
  • such that for any A:U, the type \|A\|_n is n-truncated and satisfies the (dependent) universal property of n-truncation, namely that for every type family P:\|A\|_n\to\mathrm{Type} of possibly large types such that each P(x) is n-truncated, the canonical map
    (\Pi_{(x:\|A\|_n)} P(x))\to (\Pi_{(a:A)} P(|a|_n))
    given by precomposition by |{-}|_n is an equivalence.

Construction. The proof is by induction on n\geq -2. The case n\equiv -2 is trivial (take A\mapsto \mathbf{1}). For the induction hypothesis we assume an n-truncation operation with structure described in the statement of the theorem.

First, we define \mathcal{Y}_n:A\to (A\to U) by \mathcal{Y}_n(a,b):\equiv \|a=b\|_n. As we have seen, the universe is locally small, and therefore the type A\to U is locally small. Therefore we can define

\|A\|_{n+1} :\equiv \mathrm{im}(\mathcal{Y}_n)
|{-}|_{n+1} :\equiv q_{\mathcal{Y}_n}.

For the proof that \|A\|_{n+1} is indeed (n+1)-truncated, and satisfies the universal property of the n-truncation we refer to the article.

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Parametricity, automorphisms of the universe, and excluded middle

Specific violations of parametricity, or existence of non-identity automorphisms of the universe, can be used to prove classical axioms. The former was previously featured on this blog, and the latter is part of a discussion on the HoTT mailing list. In a cooperation between Martín Escardó, Peter Lumsdaine, Mike Shulman, and myself, we have strengthened these results and recorded them in a paper that is now on arXiv.

In this blog post, we work with the full repertoire of HoTT axioms, including univalence, propositional truncations, and pushouts. For the paper, we have carefully analysed which assumptions are used in which theorem, if any.

Parametricity

Parametricity is a property of terms of a language. If your language only has parametric terms, then polymorphic functions have to be invariant under the type parameter. So in MLTT, the only term inhabiting the type \prod_{X:\mathcal{U}}X \to X of polymorphic endomaps is the polymorphic identity \lambda (X:\mathcal{U}). \mathsf{id}_X.

In univalent foundations, we cannot prove internally that every term is parametric. This is because excluded middle is not parametric (exercise 6.9 of the HoTT book tells us that, assuming LEM, we can define a polymorphic endomap that flips the booleans), but there exist classical models of univalent foundations. So if we could prove this internally, excluded middle would be false, and thus the classical models would be invalid.

In the abovementioned blog post, we observed that exercise 6.9 of the HoTT book has a converse: if f:\prod_{X:\mathcal{U}}X\to X is the flip map on the type of booleans, then excluded middle holds. In the paper on arXiv, we have a stronger result:

Theorem. There exist f:\prod_{X:\mathcal{U}}X\to X and a type X and a point x:X with f_X(x)\neq x if and only if excluded middle holds.

Notice that there are no requirements on the type X or the point x. We have also applied the technique used for this theorem in other scenarios, for example:

Theorem. There exist f:\prod_{X:\mathcal{U}}X\to \mathbf{2} and types X, Y and points x:X, y:Y with f_X(x)\neq f_Y(y) if and only if weak excluded middle holds.

The results in the paper illustrate that different violations of parametricity have different proof-theoretic strength: some violations are impossible, while others imply varying amounts of excluded middle.

Automorphisms of the universe

In contrast to parametricity, which proves that terms of some language necessarily have some properties, it is currently unknown if non-identity automorphisms of the universe are definable in univalent foundations. But some believe that this may not be the case.

In the presence of excluded middle, we can define non-identity automorphisms of the universe. Given a type X, we use excluded middle to decide if X is a proposition. If it is, we map X to \neg X, and otherwise we map X to itself. Assuming excluded middle, we have \neg\neg X=X for any proposition, so this is an automorphism.

The above automorphism swaps the empty type \mathbf{0} with the unit type \mathbf{1} and leaves all other types unchanged. More generally, assuming excluded middle we can swap any two types with equivalent automorphism ∞-groups, since in that case the corresponding connected components of the universe are equivalent. Still more generally, we can permute arbitrarily any family of types all having the same automorphism ∞-group.

The simplest case of this is when all the types are rigid, i.e. have trivial automorphism ∞-group. The types \mathbf{0} and \mathbf{1} are both rigid, and at least with excluded middle no other sets are; but there can be rigid higher types. For instance, if G is a group that is a set (i.e. a 1-group), then its Eilenberg-Mac Lane space B G is a 1-type, and its automorphism ∞-group is a 1-type whose \pi_0 is the outer automorphisms of G and whose \pi_1 is the center of G. Thus, if G has trivial outer automorphism group and trivial center, then BG is rigid. Such groups are not uncommon, including for instance the symmetric group S_n for any n\neq 2,6. Thus, assuming excluded middle we can permute these BS_n arbitrarily, producing uncountably many automorphisms of the universe.

In the converse direction, we recorded the following.

Theorem. If there is an automorphism of the universe that maps some inhabited type to the empty type, then excluded middle holds.

Corollary. If there is an automorphism g:\mathcal{U}\to\mathcal{U} of the universe with g(\mathbf{0})\neq\mathbf{0}, then the double negation

\neg\neg\prod_{P:\mathcal{U}}\mathsf{isProp}(P)\to P+\neg P

of the law of excluded middle holds.

This corollary relates to an unclaimed prize: if from an arbitrary equivalence f:\mathcal{U}\to\mathcal{U} such that f(X) \neq X for a particular X:\mathcal{U} you get a non-provable consequence of excluded middle, then you get X-many beers. So this corollary wins you 0 beers. Although perhaps sober, we think this is an achievement worth recording.

Using this corollary, in turn, we can win \mathsf{LEM}_\mathcal{U}-many beers, where \mathsf{LEM}_\mathcal{U} is excluded middle for propositions in the universe \mathcal{U}. If \mathcal{U} :\mathcal{V} we have \mathsf{LEM}_\mathcal{U}:\mathcal{V}. Suppose g is an automorphism of \mathcal{V} with g(\mathsf{LEM}_\mathcal{U})\neq\mathsf{LEM}_\mathcal{U}, then \neg\neg\mathsf{LEM}_\mathcal{U}. For suppose that \neg\mathsf{LEM}_\mathcal{U}, and hence \mathsf{LEM}_\mathcal{U}=\mathbf{0}. So by the corollary, we obtain \neg\neg\mathsf{LEM}_\mathcal{V}. But \mathsf{LEM}_\mathcal{V} implies \mathsf{LEM}_\mathcal{U} by cumulativity, so \neg\neg\mathsf{LEM}_\mathcal{U} also holds, contradicting our assumption that \neg\mathsf{LEM}_\mathcal{U}.

To date no one has been able to win 1 beer.

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HoTT MRC

From June 4 — 10, 2017, there will be a workshop on homotopy type theory as one of the AMS’s Mathematical Research Communities (MRCs).

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HoTTSQL: Proving Query Rewrites with Univalent SQL Semantics

SQL is the lingua franca for retrieving structured data. Existing semantics for SQL, however, either do not model crucial features of the language (e.g., relational algebra lacks bag semantics, correlated subqueries, and aggregation), or make it hard to formally reason about SQL query rewrites (e.g., the SQL standard’s English is too informal). This post focuses on the ways that HoTT concepts (e.g., Homotopy Types, the Univalence Axiom, and Truncation) enabled us to develop HoTTSQL — a new SQL semantics that makes it easy to formally reason about SQL query rewrites. Our paper also details the rich set of SQL features supported by HoTTSQL. 

You can download this blog post’s source (implemented in Coq using the HoTT library). Learn more about HoTTSQL by visiting our website.

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Combinatorial Species and Finite Sets in HoTT

(Post by Brent Yorgey)

My dissertation was on the topic of combinatorial species, and specifically on the idea of using species as a foundation for thinking about generalized notions of algebraic data types. (Species are sort of dual to containers; I think both have intereseting and complementary things to offer in this space.) I didn’t really end up getting very far into practicalities, instead getting sucked into a bunch of more foundational issues.

To use species as a basis for computational things, I wanted to first “port” the definition from traditional, set-theory-based, classical mathematics into a constructive type theory. HoTT came along at just the right time, and seems to provide exactly the right framework for thinking about a constructive encoding of combinatorial species.

For those who are familiar with HoTT, this post will contain nothing all that new. But I hope it can serve as a nice example of an “application” of HoTT. (At least, it’s more applied than research in HoTT itself.)

Combinatorial Species

Traditionally, a species is defined as a functor F : \mathbb{B} \to \mathbf{FinSet}, where \mathbb{B} is the groupoid of finite sets and bijections, and \mathbf{FinSet} is the category of finite sets and (total) functions. Intuitively, we can think of a species as mapping finite sets of “labels” to finite sets of “structures” built from those labels. For example, the species of linear orderings (i.e. lists) maps the finite set of labels \{1,2, \dots, n\} to the size-n! set of all possible linear orderings of those labels. Functoriality ensures that the specific identity of the labels does not matter—we can always coherently relabel things.

Constructive Finiteness

So what happens when we try to define species inside a constructive type theory? The crucial piece is \mathbb{B}: the thing that makes species interesting is that they have built into them a notion of bijective relabelling, and this is encoded by the groupoid \mathbb{B}. The first problem we run into is how to encode the notion of a finite set, since the notion of finiteness is nontrivial in a constructive setting.

One might well ask why we even care about finiteness in the first place. Why not just use the groupoid of all sets and bijections? To be honest, I have asked myself this question many times, and I still don’t feel as though I have an entirely satisfactory answer. But what it seems to come down to is the fact that species can be seen as a categorification of generating functions. Generating functions over the semiring R can be represented by functions \mathbb{N} \to R, that is, each natural number maps to some coefficient in R; each natural number, categorified, corresponds to (an equivalence class of) finite sets. Finite label sets are also important insofar as our goal is to actually use species as a basis for computation. In a computational setting, one often wants to be able to do things like enumerate all labels (e.g. in order to iterate through them, to do something like a map or fold). It will therefore be important that our encoding of finiteness actually has some computational content that we can use to enumerate labels.

Our first attempt might be to say that a finite set will be encoded as a type A together with a bijection between A and a canonical finite set of a particular natural number size. That is, assuming standard inductively defined types \mathbb{N} and \mathsf{Fin},

\displaystyle \Sigma (A:U). \Sigma (n : \mathbb{N}). A \cong \mathsf{Fin}(n).

However, this is unsatisfactory, since defining a suitable notion of bijections/isomorphisms between such finite sets is tricky. Since \mathbb{B} is supposed to be a groupoid, we are naturally led to try using equalities (i.e. paths) as morphisms—but this does not work with the above definition of finite sets. In \mathbb{B}, there are supposed to be n! different morphisms between any two sets of size n. However, given any two same-size inhabitants of the above type, there is only one path between them—intuitively, this is because paths between \Sigma-types correspond to tuples of paths relating the components pointwise, and such paths must therefore preserve the particular relation to \mathsf{Fin}(n). The only bijection which is allowed is the one which sends each element related to i to the other element related to i, for each i \in \mathsf{Fin}(n).

So elements of the above type are not just finite sets, they are finite sets with a total order, and paths between them must be order-preserving; this is too restrictive. (However, this type is not without interest, and can be used to build a counterpart to L-species. In fact, I think this is exactly the right setting in which to understand the relationship between species and L-species, and more generally the difference between isomorphism and equipotence of species; there is more on this in my dissertation.)

Truncation to the Rescue

We can fix things using propositional truncation. In particular, we define

\displaystyle U_F := \Sigma (A:U). \|\Sigma (n : \mathbb{N}). A \cong \mathsf{Fin}(n)\|.

That is, a “finite set” is a type A together with some hidden evidence that A is equivalent to \mathsf{Fin}(n) for some n. (I will sometimes abuse notation and write A : U_F instead of (A, p) : U_F.) A few observations:

  • First, we can pull the size n out of the propositional truncation, that is, U_F \cong \Sigma (A:U). \Sigma (n: \mathbb{N}). \|A \cong \mathsf{Fin}(n)\|. Intuitively, this is because if a set is finite, there is only one possible size it can have, so the evidence that it has that size is actually a mere proposition.
  • More generally, I mentioned previously that we sometimes want to use the computational evidence for the finiteness of a set of labels, e.g. enumerating the labels in order to do things like maps and folds. It may seem at first glance that we cannot do this, since the computational evidence is now hidden inside a propositional truncation. But actually, things are exactly the way they should be: the point is that we can use the bijection hidden in the propositional truncation as long as the result does not depend on the particular bijection we find there. For example, we cannot write a function which returns the value of type A corresponding to 0 : \mathsf{Fin}(n), since this reveals something about the underlying bijection; but we can write a function which finds the smallest value of A (with respect to some linear ordering), by iterating through all the values of A and taking the minimum.
  • It is not hard to show that if A : U_F, then A is a set (i.e. a 0-type) with decidable equality, since A is equivalent to the 0-type \mathsf{Fin}(n). Likewise, U_F itself is a 1-type.
  • Finally, note that paths between inhabitants of U_F now do exactly what we want: a path (A,p) = (B,q) is really just a path A = B between 0-types, that is, a bijection, since p = q trivially.

Constructive Species

We can now define species in HoTT as functions of type U_F \to U. The main reason I think this is the Right Definition ™ of species in HoTT is that functoriality comes for free! When defining species in set theory, one must say “a species is a functor, i.e. a pair of mappings satisfying such-and-such properties”. When constructing a particular species one must explicitly demonstrate the functoriality properties; since the mappings are just functions on sets, it is quite possible to write down mappings which are not functorial. But in HoTT, all functions are functorial with respect to paths, and we are using paths to represent the morphisms in U_F, so any function of type U_F \to U automatically has the right functoriality properties—it is literally impossible to write down an invalid species. Actually, in my dissertation I define species as functors between certain categories built from U_F and U, but the point is that any function U_F \to U can be automatically lifted to such a functor.

Here’s another nice thing about the theory of species in HoTT. In HoTT, coends whose index category are groupoids are just plain \Sigma-types. That is, if \mathbb{C} is a groupoid, \mathbb{D} a category, and T : \mathbb{C}^{\mathrm{op}} \times \mathbb{C} \to \mathbb{D}, then \int^C T(C,C) \cong \Sigma (C : \mathbb{C}). T(C,C). In set theory, this coend would be a quotient of the corresponding \Sigma-type, but in HoTT the isomorphisms of \mathbb{C} are required to correspond to paths, which automatically induce paths over the \Sigma-type which correspond to the necessary quotient. Put another way, we can define coends in HoTT as a certain HIT, but in the case that \mathbb{C} is a groupoid we already get all the paths given by the higher path constructor anyway, so it is redundant. So, what does this have to do with species, I hear you ask? Well, several species constructions involve coends (most notably partitional product); since species are functors from a groupoid, the definitions of these constructions in HoTT are particularly simple. We again get the right thing essentially “for free”.

There’s lots more in my dissertation, of course, but these are a few of the key ideas specifically relating species and HoTT. I am far from being an expert on either, but am happy to entertain comments, questions, etc. I can also point you to the right section of my dissertation if you’re interested in more detail about anything I mentioned above.

Posted in Applications, Higher Inductive Types, Programming | Tagged , , | 5 Comments

Parametricity and excluded middle

Exercise 6.9 of the HoTT book tells us that, and assuming LEM, we can exhibit a function f:\Pi_{X:\mathcal{U}}(X\to X) such that f_\mathbf{2} is a non-identity function \mathbf{2}\to\mathbf{2}. I have proved the converse of this. Like in exercise 6.9, we assume univalence.

Parametricity

In a typical functional programming career, at some point one encounters the notions of parametricity and free theorems.

Parametricity can be used to answer questions such as: is every function

f : forall x. x -> x

equal to the identity function? Parametricity tells us that this is true for System F.

However, this is a metatheoretical statement. Parametricity gives properties about the terms of a language, rather than proving internally that certain elements satisfy some properties.

So what can we prove internally about a polymorphic function f:\Pi_{X:\mathcal{U}}X\to X?

In particular, we can see that internal proofs (claiming that f must be the identity function for every type Xcannot exist: exercise 6.9 of the HoTT book tells us that, assuming LEM, we can exhibit a function f:\Pi_{X:\mathcal{U}}(X\to X) such that f_\mathbf{2} is \mathsf{flip}:\mathbf{2}\to\mathbf{2}. (Notice that the proof of this is not quite as trivial as it may seem: LEM only gives us P+\neg P if P is a (mere) proposition (a.k.a. subsingleton). Hence, simple case analysis on X\simeq\mathbf{2} does not work, because this is not necessarily a proposition.)

And given the fact that LEM is consistent with univalent foundations, this means that a proof that f is the identity function cannot exist.

I have proved that LEM is exactly what is needed to get a polymorphic function that is not the identity on the booleans.

Theorem. If there is a function f:\Pi_{X:\mathcal U}X\to X with f_\mathbf2\neq\mathsf{id}_\mathbf2, then LEM holds.

Proof idea

If f_\mathbf2\neq\mathsf{id}_\mathbf2, then by simply trying both elements 0_\mathbf2,1_\mathbf2:\mathbf2, we can find an explicit boolean b:\mathbf2 such that f_\mathbf2(b)\neq b. Without loss of generality, we can assume f_\mathbf2(0_\mathbf2)\neq 0_\mathbf2.

For the remainder of this analysis, let P be an arbitrary proposition. Then we want to achieve P+\neg P, to prove LEM.

We will consider a type with three points, where we identify two points depending on whether P holds. In other words, we consider the quotient of a three-element type, where the relation between two of those points is the proposition P.

I will call this space \mathbf{3}_P, and it can be defined as \Sigma P+\mathbf{1}, where \Sigma P is the suspension of P. This particular way of defining the quotient, which is equivalent to a quotient of a three-point set, will make case analysis simpler to set up. (Note that suspensions are not generally quotients: we use the fact that P is a proposition here.)

Notice that if P holds, then \mathbf{3}_P\simeq\mathbf{2}, and also (\mathbf{3}_P\simeq\mathbf{3}_P)\simeq\mathbf{2}.

We will consider f at the type (\mathbf{3}_P\simeq\mathbf{3}_P) (not \mathbf{3}_P itself!). Now the proof continues by defining

g:=f_{\mathbf{3}_P\simeq\mathbf{3}_P}(\mathsf{ide}_{\mathbf{3}_P}):\mathbf{3}_P\simeq\mathbf{3}_P

(where \mathsf{ide_{\mathbf3_P}} is the equivalence given by the identity function on \mathbf3_P) and doing case analysis on g(\mathsf{inr}(*)), and if necessary also on g(\mathsf{inl}(x)) for some elements x:\Sigma P. I do not believe it is very instructive to spell out all cases explicitly here. I wrote a more detailed note containing an explicit proof.

Notice that doing case analysis here is simply an instance of the induction principle for +. In particular, we do not require decidable equality of \mathbf3_P (which would already give us P+\neg P, which is exactly what we are trying to prove).

For the sake of illustration, here is one case:

  • g(\mathsf{inr}(*))= \mathsf{inr}(*): Assume P holds. Then since (\mathbf{3}_P\simeq\mathbf{3}_P)\simeq\mathbf{2}, then by transporting along an appropriate equivalence (namely the one that identifies 0_\mathbf2 with \mathsf{ide}_{\mathbf3_P}), we get f_{\mathbf{3}_P\simeq\mathbf{3}_P}(\mathsf{ide}_{\mathbf{3}_P})\neq\mathsf{ide}_{\mathbf{3}_P}. But since g is an equivalence for which \mathsf{inr}(*) is a fixed point, g must be the identity everywhere, that is, g=\mathsf{ide}_{\mathbf{3}_P}, which is a contradiction.

I formalized this proof in Agda using the HoTT-Agda library

Acknowledgements

Thanks to Martín Escardó, my supervisor, for his support. Thanks to Uday Reddy for giving the talk on parametricity that inspired me to think about this.

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