[TYPES] What's a program? (Seriously)
Tarmo Uustalu
tarmo at cs.ioc.ee
Thu May 20 09:10:53 EDT 2021
Hi!
On Thu, 20 May 2021 10:31:56 +0100
Neel Krishnaswami <neelakantan.krishnaswami at gmail.com> wrote:
> Tadeusz already explained what's going on, but let me unpack his
> remarks a bit.
>
> The basic idea is that the distinctive feature of intuitionistic logic
> is the existence property. However, classical and intuitionistic
> proofs coincide on the ⊤, ∧, ⊃ fragment.
Not quite. Think of Peirce's formula, which is a purely implicative
classical tautology not valid intuitionistically.
Any translation from classical logic to intuitionistic
logic for reducing derivability in the former to derivability
in the latter must take this into account in some way.
Tarmo U
> On 20/05/2021 02:52, Jason Gross wrote:
> > [ The Types Forum,
> > http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
> >> If you ask for for a proof of (A \/ not A), you get a "fake"
> >> proof of
> > (not A); if you ever manage to build a proof of A and try to use it
> > to get a contradiction using this (not A), it will "cheat" by
> > traveling back in time to your "ask", and serve you your own proof
> > of A.
> >
> > I don't understand how this semantics works; it seems to me that it
> > invalidates the normal reduction rules.
> > Consider the following:
> > Axiom LEM : forall A, A + (A -> False).
> >
> > Definition term1
> > := match LEM nat as LEM_nat return _ -> match LEM_nat with inl _
> > => _ | _ => _ end with
> > | inl v => fun _ => v
> > | inr bad => fun f => f bad
> > end (fun bad => let _ := bad 0 in bad 1).
> > Definition term2
> > := match LEM nat as LEM_nat return _ -> match LEM_nat with inl _
> > => _ | _ => _ end with
> > | inl v => fun _ => v
> > | inr bad => fun f => f bad
> > end (fun bad => bad 1).
> > Lemma pf : term1 = term2. Proof. reflexivity. Qed.
> >
> > However, if I understand your interpretation correctly, then term1
> > should reduce to 0 but term2 should reduce to 1.
> >
> > Another issue is that typechecking requires normalization under
> > binders, but normalization under binders seems to invalidate the
> > semantics you suggest, because the proof of A might be not be
> > well-scoped in the context in which you asked for it. (Trivially,
> > it seems like eta-expanding the proof of fake proof of (not A)
> > results in invoking the continuation if you try to fully normalize
> > a term.)
> >
> > What am I missing/misunderstanding?
> >
> > Best,
> > Jason
> >
> >
> > On Wed, May 19, 2021 at 11:27 AM Gabriel Scherer
> > <gabriel.scherer at gmail.com> wrote:
> >
> >> I am not convinced by the example of Jason and Thomas, which
> >> suggests that I am missing something.
> >>
> >> We can interpret the excluded middle in classical abstract
> >> machines (for example Curien-Herbelin-family mu-mutilda, or
> >> Parigot's earlier classical lambda-calculus), or in presence of
> >> control operators (classical abstract machines being nicer syntax
> >> for non-delimited continuation operators). If you ask for for a
> >> proof of (A \/ not A), you get a "fake" proof of (not A); if you
> >> ever manage to build a proof of A and try to use it to get a
> >> contradiction using this (not A), it will "cheat" by traveling
> >> back in time to your "ask", and serve you your own proof of A.
> >>
> >> This gives a computational interpretation of (non-truncated)
> >> excluded middle that seems perfectly in line with Talia's notion
> >> of "program". Of course, what we don't get that we might expect is
> >> a canonicity property: we now have "fake" proofs of (A \/ B) that
> >> cannot be distinguished from "real" proofs by normalization alone,
> >> you have to interact with them to see where they take you. (Or, if
> >> you see those classical programs through a double-negation
> >> translation, they aren't really at type (A \/ B), but rather at
> >> its double-negation translation, which has weirder normal forms.)
> >>
> >>
> >>
> >>
> >> On Wed, May 19, 2021 at 5:07 PM Jason Gross
> >> <jasongross9 at gmail.com> wrote:
> >>> [ The Types Forum,
> >>> http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
> >>>
> >>> Non-truncated Excluded Middle (that is, the version that returns
> >>> an informative disjunction) cannot have a computational
> >>> interpretation in Turing machines, for it would allow you to
> >>> decide the halting problem. More generally, some computational
> >>> complexity theory is done with reference
> >>> to oracles for known-undecidable problems. Additionally, I'd be
> >>> suspicious
> >>> of a computational interpretation of the consistency of ZFC or PA
> >>> ---- would having a computational interpretation of these mean
> >>> having a type theory that believes that there are ground terms of
> >>> type False in the presence of a contradiction in ZFC?
> >>>
> >>> On Wed, May 19, 2021, 07:38 Talia Ringer
> >>> <tringer at cs.washington.edu> wrote:
> >>>
> >>>> [ The Types Forum,
> >>> http://lists.seas.upenn.edu/mailman/listinfo/types-list
> >>>> ]
> >>>>
> >>>> Somewhat of a complementary question, and proof to the world
> >>>> that I'm
> >>> up at
> >>>> 330 AM still thinking about this:
> >>>>
> >>>> Are there interesting or commonly used logical axioms that we
> >>>> know for
> >>> sure
> >>>> cannot have computational interpretations?
> >>>>
> >>>> On Wed, May 19, 2021, 3:24 AM Neel Krishnaswami <
> >>>> neelakantan.krishnaswami at gmail.com> wrote:
> >>>>
> >>>>> [ The Types Forum,
> >>>> http://lists.seas.upenn.edu/mailman/listinfo/types-list
> >>>>> ]
> >>>>>
> >>>>> Dear Sandro,
> >>>>>
> >>>>> Yes, you're right -- I didn't answer the question, since I was
> >>>>> too taken by the subject line. :)
> >>>>>
> >>>>> Anyway, I do think that HoTT with a non-reducible univalence
> >>>>> axiom is still a programming language, because we can give a
> >>>>> computational interpretation to that language: for example, you
> >>>>> could follow the strategy of Angiuli, Harper and Wilson's POPL
> >>>>> 2017 paper, *Computational Higher-Dimensional Type Theory*.
> >>>>>
> >>>>> Another, simpler example comes from Martin Escardo's example
> >>>>> upthread of basic Martin-Löf type theory with the function
> >>>>> extensionality axiom. You can give a very simple realizability
> >>>>> interpretation to the equality type and extensionality axiom,
> >>>>> which lets every compiled program compute.
> >>>>>
> >>>>> What you lose in both of these cases is not the ability to give
> >>>>> a computational model to the language, but rather the ability to
> >>>>> identify normal forms and to use an oriented version of the
> >>>>> equational theory of the language as the evaluation mechanism.
> >>>>>
> >>>>> This is not an overly shocking phenomenon: it occurs even in
> >>>>> much simpler languages than dependent type theories. For
> >>>>> example, once you add the reference type `ref a` to ML, it is
> >>>>> no longer the case that the language has normal forms, because
> >>>>> the ref type does not have introduction and elimination rules
> >>>>> with beta- and eta- rules.
> >>>>>
> >>>>> Another way of thinking about this is that often, we *aren't
> >>>>> sure* what the equational theory of our language is or should
> >>>>> be. This is because we often derive a language by thinking
> >>>>> about a particular semantic model, and don't have a clear idea
> >>>>> of which equations are properly part of the theory of the
> >>>>> language, and which ones are accidental features of the
> >>>>> concrete model.
> >>>>>
> >>>>> For example, in the case of name generation – i.e., ref unit –
> >>>>> our intuitions for which equations hold come from the concrete
> >>>>> model of nominal sets. But we don't know which of those
> >>>>> equations should hold in all models of name generation, and
> >>>>> which are "coincidental" to nominal sets.
> >>>>>
> >>>>> Another, more practical, example comes from the theory of
> >>>>> state. We all have the picture of memory as a big array which
> >>>>> is updated by assembly instructions a la the state monad. But
> >>>>> this model incorrectly models the behaviour of memory on modern
> >>>>> multicore systems. So a proper theory of state for this case
> >>>>> should have fewer equations than what the folk model of state
> >>>>> validates.
> >>>>>
> >>>>>
> >>>>> Best,
> >>>>> Neel
> >>>>>
> >>>>> On 19/05/2021 09:03, Sandro Stucki wrote:
> >>>>>> Talia: thanks for a thought-provoking question, and thanks
> >>>>>> everyone
> >>>> else
> >>>>>> for all the interesting answers so far!
> >>>>>>
> >>>>>> Neel: I love your explanation and all your examples!
> >>>>>>
> >>>>>> But you didn't really answer Talia's question, did you? I'd
> >>>>>> be
> >>> curious
> >>>>>> to know where and how HoTT without a computation rule for
> >>>>>> univalence would fit into your classification. It would
> >>>>>> certainly be a
> >>> language,
> >>>>>> and by your definition it has models (e.g. cubical ones)
> >>>>>> which, if I understand correctly, can be turned into an
> >>>>>> abstract machine
> >>> (either a
> >>>>>> rewriting system per your point 4 or whatever the Agda
> >>>>>> backends
> >>> compile
> >>>>>> to). So according to your definition of programming language
> >>>>>> (point
> >>> 3),
> >>>>>> this version of HoTT would be a programming language simply
> >>>>>> because there is, in principle, an abstract machine model for
> >>>>>> it? Is that
> >>> what
> >>>>>> you had in mind?
> >>>>>>
> >>>>>> Cheers
> >>>>>> /Sandro
> >>>>>>
> >>>>>>
> >>>>>> On Wed, May 19, 2021 at 6:21 AM Neel Krishnaswami
> >>>>>> <neelakantan.krishnaswami at gmail.com
> >>>>>> <mailto:neelakantan.krishnaswami at gmail.com>> wrote:
> >>>>>>
> >>>>>> [ The Types Forum,
> >>>>>> http://lists.seas.upenn.edu/mailman/listinfo/types-list
> >>>>>> <http://lists.seas.upenn.edu/mailman/listinfo/types-list>
> >>>>>> ]
> >>>>>>
> >>>>>> Dear Talia,
> >>>>>>
> >>>>>> Here's an imprecise but useful way of organising these
> >>>>>> ideas
> >>> that I
> >>>>>> found helpful.
> >>>>>>
> >>>>>> 1. A *language* is a (generalised) algebraic theory.
> >>>>>> Basically,
> >>>> think
> >>>>>> of a theory as a set of generators and relations in
> >>>>>> the
> >>> style
> >>>> of
> >>>>>> abstract algebra.
> >>>>>>
> >>>>>> You need to beef this up to handle variables (e.g.,
> >>>>>> see the
> >>>>> work of
> >>>>>> Fiore and Hamana) but (a) I promised to be
> >>>>>> imprecise, and
> >>> (b)
> >>>>> the
> >>>>>> core intuition that a language is a set of
> >>>>>> generators for
> >>>> terms,
> >>>>>> plus a set of equations these terms satisfy is
> >>>>>> already
> >>> totally
> >>>>>> visible in the basic case.
> >>>>>>
> >>>>>> For example:
> >>>>>>
> >>>>>> a) the simply-typed lambda calculus
> >>>>>> b) regular expressions
> >>>>>> c) relational algebra
> >>>>>>
> >>>>>> 2. A *model* of a a language is literally just any old
> >>> mathematical
> >>>>>> structure which supports the generators of the
> >>>>>> language and respects the equations.
> >>>>>>
> >>>>>> For example:
> >>>>>>
> >>>>>> a) you can model the typed lambda calculus using sets
> >>>>>> for types and mathematical functions for terms,
> >>>>>> b) you can model regular expressions as denoting
> >>>>>> particular languages (ie, sets of strings)
> >>>>>> c) you can model relational algebra expressions as
> >>>>>> sets of tuples
> >>>>>>
> >>>>>> 2. A *model of computation* or *machine model* is
> >>>>>> basically a description of an abstract machine that we think
> >>>>>> can be
> >>>>> implemented
> >>>>>> with physical hardware, at least in principle. So
> >>>>>> these are
> >>>>> things
> >>>>>> like finite state machines, Turing machines, Petri
> >>>>>> nets,
> >>>>> pushdown
> >>>>>> automata, register machines, circuits, and so on.
> >>> Basically,
> >>>>> think
> >>>>>> of models of computation as the things you study in
> >>>>>> a
> >>>>> computability
> >>>>>> class.
> >>>>>>
> >>>>>> The Church-Turing thesis bounds which abstract
> >>>>>> machines we
> >>>>> think it
> >>>>>> is possible to physically implement.
> >>>>>>
> >>>>>> 3. A language is a *programming language* when you can
> >>>>>> give at
> >>>> least
> >>>>>> one model of the language using some machine model.
> >>>>>>
> >>>>>> For example:
> >>>>>>
> >>>>>> a) the types of the lambda calculus can be viewed
> >>>>>> as
> >>> partial
> >>>>>> equivalence relations over Gödel codes for some
> >>> universal
> >>>>> turing
> >>>>>> machine, and the terms of a type can be assigned
> >>>>>> to
> >>>>> equivalence
> >>>>>> classes of the corresponding PER.
> >>>>>>
> >>>>>> b) Regular expressions can be interpreted into
> >>>>>> finite state machines
> >>>>>> quotiented by bisimulation.
> >>>>>>
> >>>>>> c) A set in relational algebra can be realised as
> >>> equivalence
> >>>>>> classes of B-trees, and relational algebra
> >>>>>> expressions
> >>> as
> >>>>> nested
> >>>>>> for-loops over them.
> >>>>>>
> >>>>>> Note that in all three cases we have to quotient the
> >>>>>> machine
> >>>>> model
> >>>>>> by a suitable equivalence relation to preserve the
> >>> equations of
> >>>>> the
> >>>>>> language's theory.
> >>>>>>
> >>>>>> This quotient is *very* important, and is the source
> >>>>>> of a
> >>> lot
> >>>> of
> >>>>>> confusion. It hides the equivalences the language
> >>>>>> theory
> >>> wants
> >>>> to
> >>>>>> deny, but that is not always what the programmer
> >>>>>> wants –
> >>> e.g.,
> >>>> is
> >>>>>> merge sort equal to bubble sort? As mathematical
> >>>>>> functions,
> >>>> they
> >>>>>> surely are, but if you consider them as operations
> >>>>>> running
> >>> on
> >>>> an
> >>>>>> actual computer, then we will have strong preferences!
> >>>>>>
> >>>>>> 4. A common source of confusion arises from the fact that
> >>>>>> if you
> >>>> have
> >>>>>> a nice type-theoretic language (like the STLC), then:
> >>>>>>
> >>>>>> a) the term model of this theory will be the initial
> >>>>>> model
> >>> in
> >>>>> the
> >>>>>> category of models, and
> >>>>>> b) you can turn the terms into a machine
> >>>>>> model by orienting some of the equations the
> >>> lambda-theory
> >>>>>> satisfies and using them as rewrites.
> >>>>>>
> >>>>>> As a result we abuse language to talk about the
> >>>>>> theory of
> >>> the
> >>>>>> simply-typed calculus as "being" a programming
> >>>>>> language.
> >>> This
> >>>> is
> >>>>>> also where operational semantics comes from, at
> >>>>>> least for
> >>>> purely
> >>>>>> functional languages.
> >>>>>>
> >>>>>> Best,
> >>>>>> Neel
> >>>>>>
> >>>>>> On 18/05/2021 20:42, Talia Ringer wrote:
> >>>>>> > [ The Types Forum,
> >>>>>> http://lists.seas.upenn.edu/mailman/listinfo/types-list
> >>>>>> <http://lists.seas.upenn.edu/mailman/listinfo/types-list>
> >>>>>> ]
> >>>>>> >
> >>>>>> > Hi friends,
> >>>>>> >
> >>>>>> > I have a strange discussion I'd like to start.
> >>>>>> > Recently I was
> >>>>>> debating with
> >>>>>> > someone whether Curry-Howard extends to arbitrary
> >>>>>> > logical
> >>>>>> systems---whether
> >>>>>> > all proofs are programs in some sense. I argued yes,
> >>>>>> > he
> >>> argued
> >>>>>> no. But
> >>>>>> > after a while of arguing, we realized that we had
> >>>>>> > different
> >>>>>> axioms if you
> >>>>>> > will modeling what a "program" is. Is any term that
> >>>>>> > can be
> >>> typed
> >>>>>> a program?
> >>>>>> > I assumed yes, he assumed no.
> >>>>>> >
> >>>>>> > So then I took to Twitter, and I asked the following
> >>> questions
> >>>>> (some
> >>>>>> > informal language here, since audience was Twitter):
> >>>>>> >
> >>>>>> > 1. If you're working in a language in which not all
> >>>>>> > terms
> >>>> compute
> >>>>>> (say,
> >>>>>> > HoTT without a computational interpretation of
> >>>>>> > univalence, so
> >>>> not
> >>>>>> cubical),
> >>>>>> > would you still call terms that mostly compute but
> >>>>>> > rely on
> >>>> axioms
> >>>>>> > "programs"?
> >>>>>> >
> >>>>>> > 2. If you answered no, would you call a term that does
> >>>>>> > fully
> >>>>>> compute in the
> >>>>>> > same language a "program"?
> >>>>>> >
> >>>>>> > People actually really disagreed here; there was
> >>>>>> > nothing
> >>>>> resembling
> >>>>>> > consensus. Is a term a program if it calls out to an
> >>>>>> > oracle?
> >>>>>> Relies on an
> >>>>>> > unrealizable axiom? Relies on an axiom that is
> >>>>>> > realizable,
> >>> but
> >>>>>> not yet
> >>>>>> > realized, like univalence before cubical existed? (I
> >>>>>> > suppose
> >>>> some
> >>>>>> reliance
> >>>>>> > on axioms at some point is necessary, which makes this
> >>>>>> > even
> >>>>>> weirder to
> >>>>>> > me---what makes univalence different to people who do
> >>>>>> > not
> >>> view
> >>>>>> terms that
> >>>>>> > invoke it as an axiom as programs?)
> >>>>>> >
> >>>>>> > Anyways, it just feels strange to get to the last
> >>>>>> > three
> >>> weeks of
> >>>>> my
> >>>>>> > programming languages PhD, and realize I've never once
> >>>>>> > asked
> >>>> what
> >>>>>> makes a
> >>>>>> > term a program 😅. So it'd be interesting to hear your
> >>> thoughts.
> >>>>>> >
> >>>>>> > Talia
> >>>>>> >
> >>>>>>
> >>>>>
> >>>>
> >>>
> >>
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