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What is T0? What is t0?
If t0:T0 then p(t0) : B(T0), so it seems that it can't be sent to qp(t0)
or pq(t0) which belong to B(B(T0)).
How does this yield an instance of the previous claim? What is B? What is p?
Dear all,Let’s say a type theory TT is initial if its term model C_TT is initial among TT-models, where TT-models are models of the categorical semantics of type theory (e.g. CwFs/C-systems etc.) with enough extra structure to model the rules of TT.
But there are two distinct TT-model homomorphisms from C_TT to C_TT*, one which sends p(t0) to pq(t0) and one which sends p(t0) to qp(t0) (where p(t0) is regarded as an element of Tm_{C_TT} (empty, B(B(T0))), i.e. of the set of terms of B(B(T0)) in the empty context as they are interpreted in the term model C_TT).
On Oct 13, 2017, at 11:50 AM, Michael Shulman <shu...@sandiego.edu> wrote:On Thu, Oct 12, 2017 at 5:09 PM, Steve Awodey <steve...@gmail.com> wrote:in order to have an (essentially) algebraic notion of type theory, which
will then automatically have initial algebras, etc., one should have the
typing of terms be an operation, so that every term has a unique type. In
particular, your (R1) violates this. Cumulativity is a practical convenience
that can be added to the system by some syntactic conventions, but the real
system should have unique typing of terms.
I'm not convinced of that. When we define the syntactic model, a
morphism from A to B (say) is defined to be a term x:A |- t:B, where
the types A and B are given. So it's not clear that it matters
whether the same syntactic object t can also be typed as belonging to
some other type. I thought that the fundamental structure that we
induct over to prove initiality is the *derivation* of a typing
judgment, which includes the type that the term belongs to: two
derivations of x:A |- t:B and x:A |- t:C will necessarily be different
if B and C are different. In an ideal world, a judgment x:A |- t:B
would have at most one derivation, so that we could induct on
derivations and still consider the syntactic model to be built out of
terms rather than derivations. If not, then we need a separate step
of showing that different derivations of the same judgment yield the
same interpretation; but still, it's not clear to me that the
simultaneous derivability of x:A |- t:C is fatal.
So I don’t think we can say “These theories aren’t initial.” — but more like “We’re not sure what the correct initiality statement is for these theories, and some versions one might try are false.”
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Begin forwarded message:
From: Dimitris Tsementzis <dtse...@princeton.edu>Subject: Re: [HoTT] A small observation on cumulativity and the failure of initialityDate: October 13, 2017 at 4:53:35 PM CDTTo: Alexander Altman <alexand...@me.com>
E.g., if you had a type theory with judgemental subtyping, not just judgemental equality, and one of the subtyping rules given was that each universe is a subtype of the next, would that still violate the conditions needed for initiality?
As far as I can understand the terminology yes I believe the observation would still apply even with sub-typing rules of this kind.Dimitris
On Oct 12, 2017, at 20:44, Alexander Altman <alexand...@me.com> wrote:
How does outright explicit subtyping play into this? E.g., if you had a type theory with judgemental subtyping, not just judgemental equality, and one of the subtyping rules given was that each universe is a subtype of the next, would that still violate the conditions needed for initiality?
On Thursday, October 12, 2017 at 7:09:06 PM UTC-5, Steve Awodey wrote:
in order to have an (essentially) algebraic notion of type theory, which will then automatically have initial algebras, etc., one should have the typing of terms be an operation, so that every term has a unique type. In particular, your (R1) violates this. Cumulativity is a practical convenience that can be added to the system by some syntactic conventions, but the real system should have unique typing of terms.
Steve
On Oct 12, 2017, at 2:43 PM, Dimitris Tsementzis <dtse...@princeton.edu> wrote:
Dear all,Let’s say a type theory TT is initial if its term model C_TT is initial among TT-models, where TT-models are models of the categorical semantics of type theory (e.g. CwFs/C-systems etc.) with enough extra structure to model the rules of TT.
Then we have the following, building on an example of Voevodsky’s.OBSERVATION. Any type theory which contains the following rules (admissible or otherwise)Γ |- T Type
———————— (C)
Γ |- B(T) Type
Γ |- t : T
———————— (R1)
Γ |- t : B(T)
Γ |- t : T
———————— (R2)
Γ |- p(t) : B(T)
together with axioms that there is a type T0 in any context and a term t0 : T0 in any context, is not initial.
PROOF SKETCH. Let TT be such a type theory. Consider the type theory TT* which replaces (R1) with the ruleΓ |- t : T
———————— (R1*)
Γ |- q(t) : B(T)i.e. the rule which adds an “annotation” to a term t from T that becomes a term of B(T). Then the category of TT-models is isomorphic (in fact, equal) to the category of TT*-models and in particular the term models C_TT and C_TT* are both TT-models. But there are two distinct TT-model homomorphisms from C_TT to C_TT*, one which sends p(t0) to pq(t0) and one which sends p(t0) to qp(t0) (where p(t0) is regarded as an element of Tm_{C_TT} (empty, B(B(T0))), i.e. of the set of terms of B(B(T0)) in the empty context as they are interpreted in the term model C_TT).COROLLARY. Any (non-trivial) type theory with a “cumulativity" rule for universes, i.e. a rule of the form
Γ |- A : U0
———————— (U-cumul)
Γ |- A : U1is not initial. In particular, the type theory in the HoTT book is not initial (because of (U-cumul)), and two-level type theory 2LTT as presented here is not initial (because of the rule (FIB-PRE)).The moral of this small observation, if correct, is not of course that type theories with the guilty rules cannot be made initial by appropriate modifications to either the categorical semantics or the syntax, but rather that a bit of care might be required for this task. One modification would be to define their categorical semantics to be such that certain identities hold that are not generally included in the definitions of CwF/C-system/…-gadgets (e.g. that the inclusion operation on universes is idempotent). Another modification would be to add annotations (by replacing (R1) with (R1*) as above) and extra definitional equalities ensuring that annotations commute with type constructors.
But without some such explicit modification, I think that the claim that e.g. Book HoTT or 2LTT is initial cannot be considered obvious, or even entirely correct.Best,Dimitris
PS: Has something like the above regarding cumulativity rules has been observed before — if so can someone provide a relevant reference?
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http://staff.math.su.se/palmgren/Named_variables_in_cwfs_v02.pdf
________________________________________
Från: homotopyt...@googlegroups.com <homotopyt...@googlegroups.com> för Thomas Streicher <stre...@mathematik.tu-darmstadt.de>
Skickat: den 14 oktober 2017 11:52
Till: Andrej Bauer
Kopia: Homotopy Type Theory; Univalent Mathematics
Ämne: Re: [HoTT] A small observation on cumulativity and the failure of initiality
Andrej,
Thomas
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A quite detailed interpretation of type theory in set theory is presented
in the paper of Peter Aczel "On Relating Type Theories and Set Theories".
On can define directly the interpretation of lambda terms (provided that abstraction is
typed), and using set theoretic coding, one can use a global application operation
(so that the interpretation is total). One can then checked by induction on derivations
that all judgements are valid for this interpretation.
Thierry
In an ideal world, a judgment x:A |- t:B
would have at most one derivation, so that we could induct on
derivations and still consider the syntactic model to be built out of
terms rather than derivations.
Sent from my iPhone
On 16 Oct 2017, at 06:31, Andrew Polonsky <andrew....@gmail.com> wrote:
In the set-thoretic model -- which is the simplest, most "standard" model one can think of -- the universes are indeed cumulative, and "coherence" is just the observation that conversion (definitional equality) of raw terms is preserved by the interpretation function.
Set theory is a form of a mathematical assembly language where you cannot hide anything. Equalities in the set theoretic translation of Type Theory are accidents of implementation choices. Making them the guideline for the design of Type Theory seems to to put the cart in front of the horse.
In categorical models, strict equality between interpreted objects is perhaps a more subtle concept.
Still, it would seem to be a natural requirement of *ANY* class or flavor of semantics, that expressions which are definitionally equal in the object language, are evaluated to entities which are again (judgmentally) equal on the meta-level.
This really amounts to nothing more than asking that the semantics in question actually validate the conversion rule -- one of the structural rules of the theory.
Best,
Andrew
On Sunday, October 15, 2017 at 11:26:22 AM UTC+2, Thomas Streicher wrote:> A quite detailed interpretation of type theory in set theory is presented
>
> in the paper of Peter Aczel "On Relating Type Theories and Set Theories".
>
> On can define directly the interpretation of lambda terms (provided that abstraction is
>
> typed), and using set theoretic coding, one can use a global application operation
>
> (so that the interpretation is total). One can then checked by induction on derivations
>
> that all judgements are valid for this interpretation.
Thanks, Thierry, for pointing this out. But Peter's method does not
extend to arbitrary split models of dependent type theory. What Peter
uses here intrinsically is that everything is a set since otherwise he
couldn't interpret type theoretic quantification.
My interpretation got partial on pseudoexpressions since pseudoterms
can't be understood as pseudo-type-expressions.
I don't see how Peter's method extends to interpretation of
realizability or (pre)sheaf models of dependent type theories not to
speak of arbitrary contextual cats or other split categorical models
of dependent type theories.
Thomas
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Sent from my iPhone
On 16 Oct 2017, at 00:49, Andrej Bauer <andrej...@andrej.com> wrote:
The theorem that beta-reduction preserves matching brackets is mixing
levels of abstraction in bad way. As we are happy to reason about terms as
trees knowing that we can parse strings into trees, we should think about
intrinsic terms (what Thomas calls "derivations") knowing that by type
checking and scope checking we can convert trees into derivations.
Assigning meaning to purely syntactic entities is a confusing levels of
abstractions in the same way as the matching brackets theorem.
I do not understand these remarks. Did I say somewhere that we should
worry about matching brackets in beta-reductions, or that we should do
semantics at the wrong level of abstraction?
Sorry if this was not very clear. I was trying to make an analogy. Some novice programmers seem to think that strings are the only data structure. From this point of view we may define beta reduction as an operation on strings and then prove that it preserves matching brackets.
I hope we all agree that this makes little sense. Instead beta reduction should be defined on trees or even better alpha-equivalence classes of trees. Analogously we should think of derivations and not preterms when studying the semantics of Type Theory.
Equalities in the set theoretic translation of Type Theory are accidents of implementation choices. Making them the guideline for the design of Type Theory seems to to put the cart in front of the horse.
What you call "intrinsically typed syntax" is what I favor. I want
initiality theorems that are proven by a straightforward structural
induction over something that is inductively generated, and
derivations are the basic inductively generated object of a deductive
system.
It seems most natural to me to introduce terms as a
convenient 1-dimensional notation for derivations, and departing from
that is what causes all the problems.
Just to clarify: by set theory we mean ZFC, not the set-level fragment of HoTT.
I am not sure what is in general the “native meaning” of type constructors. Ok, it is pretty clear for function types but not in general.
Choosing a clever encoding you could make products strictly monoidal, that is Ax(BxC) = (AxB)xC. Is this now the true equality or not?
Looking at inductive types you can have representations where F(mu F) = mu F or you choose that this is just an isomorphism. Either of them can be justified by set theoretic encodings which is no help in deciding which ones should hold.
Thorsten
From:
<homotopyt...@googlegroups.com> on behalf of Andrew Polonsky <andrew....@gmail.com>
Date: Monday, 16 October 2017 at 11:42
To: Homotopy Type Theory <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] A small observation on cumulativity and the failure of initiality
Equalities in the set theoretic translation of Type Theory are accidents of implementation choices. Making them the guideline for the design of Type Theory seems to to put the cart in front of the horse.
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On Mon, Oct 16, 2017 at 5:30 AM, Neel Krishnaswami
<neelakantan....@gmail.com> wrote:
> 1. The addition of universes is an open problem. Basically the logical
> strength of the theory goes up and the proof of Harper and Pfenning
> needs to be redone. (They exploited the fact that LF doesn't have
> large eliminations to do a recursion on the size of the type.)
>
> I would be rather surprised if this couldn't be made to work, though.
Me too.