Danko Ilik - academic web page
http://www.speleologic.net/
Updates to my home page since January 01, 2014The paper "An Intuitionistic Formula Hierarchy Based on High-School Identities" accepted to Mathematical Logic Quarterly
https://arxiv.org/abs/1601.04876
We revisit the notion of intuitionistic equivalence
and formal proof representations by adopting the view of
formulas as exponential polynomials.
After observing that most of the invertible proof rules of
intuitionistic (minimal) propositional sequent calculi are
formula (i.e. sequent) isomorphisms corresponding to the
high-school identities, we show that one can obtain a more
compact variant of a proof system, consisting of non-invertible
proof rules only, and where the invertible proof rules have been
replaced by a formula normalisation procedure.
Moreover, for certain proof systems such as the G4ip sequent
calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even
possible to see all of the non-invertible proof rules as strict
inequalities between exponential polynomials; a careful
combinatorial treatment is given in order to establish this
fact.
Finally, we extend the exponential polynomial analogy to the
first-order quantifiers, showing that it gives rise to an
intuitionistic hierarchy of formulas, resembling the classical
arithmetical hierarchy, and the first one that classifies
formulas while preserving isomorphism. Sat, 15 August 2018Talk at EUTypes 2018 Working Meeting in Nijmegen, The
Netherlands: Industrial use of proof assistants: Common Criteria
evaluations
https://eutypes.cs.ru.nl/Meetings/EUTYPES2018
Common Criteria [ISO/IEC 15408] is an international
standard for certifying the security of computer systems. When
high evaluation assurance levels (EAL6 and EAL7) are required of
a system, Common Criteria (CC) requires formal modelling and
mechanized proofs of security properties. In this talk, we will
first have an overall look at the schema for CC evaluations,
after which we will concentrate on the scientific aspects
relating to proofs and programming languages, and the special
role that proof assistants play during these
evaluations.Mon, 22 January 2018Talk at CHoCoLa, ENS Lyon
http://chocola.ens-lyon.fr/
Talk on the topic of my POPL 2017 paper.Thu, 9 February 2017"Exp-log normal form of types" accepted to POPL 2017
http://conf.researchr.org/home/POPL-2017
My paper on the exp-log normal form of types has
been accepted to POPL 2017.Sat, 8 October 2016New version of "The exp-log normal form of types"
http://arxiv.org/abs/1502.04634
Abstract: Lambda calculi with algebraic data types
lie at the core of functional programming languages and proof
assistants, but conceal at least two fundamental theoretical
problems already in the presence of the simplest non-trivial
data type, the sum type. First, we do not know of an explicit
and implemented algorithm for deciding the beta-eta-equality of
terms---and this in spite of the first decidability results
proven two decades ago. Second, it is not clear how to decide
when two types are essentially the same, i.e. isomorphic, in
spite of the meta-theoretic results on decidability of the
isomorphism. In this paper, we present the exp-log normal form
of types---derived from the representation of exponential
polynomials via the unary exponential and logarithmic
functions---that any type built from arrows, products, and sums,
can be isomorphically mapped to. The type normal form can be
used as a simple heuristic for deciding type isomorphism, thanks
to the fact that it is a systematic application of the
high-school identities. We then show that the type normal form
allows to reduce the standard beta-eta equational theory of the
lambda calculus to a specialized version of itself, while
preserving the completeness of equality on terms. We end by
describing an alternative representation of normal terms of the
lambda calculus with sums, together with a Coq-implemented
converter into/from our new term calculus. The difference with
the only other previously implemented heuristic for deciding
interesting instances of eta-equality by Balat, Di Cosmo, and
Fiore, is that we exploit the type information of terms
substantially and this often allows us to obtain a canonical
representation of terms without performing sophisticated term
analyses. Fri, 1 July 2016Book chapter: Perspectives for proof unwinding by programming languages techniques
http://arxiv.org/abs/1605.09177
Abstract: In this chapter, we propose some future
directions of work, potentially beneficial to Mathematics and
its foundations, based on the recent import of methodology from
the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as
well as the working mathematician, is not a survey of the field,
but rather a personal view of the author who hopes that it may
inspire future and fellow researchers. Tue, 7 June 2016Forthcoming talk at CiE 2016
http://arxiv.org/abs/1601.04876
Title: An Intuitionistic Formula Hierarchy Based on
High-School IdentitiesSun, 1 May 2016An Intuitionistic Formula Hierarchy Based on High-School
Identities (with Taus Brock-Nannestad)
http://arxiv.org/abs/1601.04876
Abstract: We revisit intuitionistic proof theory
from the point of view of the formula isomorphisms arising from
high-school identities. We first show how sequent calculi for
intuitionistic proposition logic, and in particular the G4ip
calculus of Vorob'ev, Hudelmaier, and Dyckhoff can be
represented as a complete proof calculus that nevertheless
contains no invertible proof rules, called the high-school (HS)
variant of G4ip. We then show that all the rules of G4ip and HS
admit an arithmetical interpretation, namely each such proof
rule can be reduced to an inequality between exponential
polynomials. Finally, we extend the exponential polynomial
analogy to first-order quantifiers, showing that it gives rise
to a simple intuitionistic hierarchy of formulas, the first one
that classifies formulas up to isomorphism, and proceeds along
the same equivalences that lead to the classical arithmetical
hierarchy. Wed, 20 Jan 2016The exp-log normal form of formulas (Oct 14, talk at Type Theory and Realizability Workgroup, Paris)
http://www.pps.univ-paris-diderot.fr/gdt-types-realisabilite/
Abstract: Logical incarnations of type isomorphism
include the notions of constructive cardinality of sets and
strong intuitionistic equivalence of formulas. These are
challenging to study in simultaneous presence of functions
(exponentials) and sums (disjoint unions, disjunction). In this
talk, I will present a quasi-normal form of types that arises
from the decomposition of binary exponentiation into the unary
exponentiation and logarithm. This normal form can be applied
for disentangling the equational theory (beta-eta) of the lambda
calculus with sums. By an extension of the normal form from
simple types (propositional logic) to quantifiers, one can also
retrieve an "arithmetical" hierarchy for intuitionistic first
order logic. Finally, this suggests a sequent calculus for
intuitionistic logic that uses the notation of exponential
polynomials and removes the need for most of the invertible
proof rules of usual sequent calculi. Wed, 07 Oct 2015Talk at General Proof Theory, Celebrating 50 Years of Dag Prawitz's "Natural Deduction", Tübingen
http://ls.informatik.uni-tuebingen.de/GPT/
Title: High-school sequent calculus and an intuitionistic formula hierarchy preserving identity of proofsFri, 21 Aug 2015In the exp-log normal form of types (new version)
http://arxiv.org/abs/1502.04634
A new arXiv version of my manuscript on sum types and lambda calculus.Wed, 24 Jun 2015Talk at Logic and Applications 2015 (LAP 2015), Dubrovnik
http://imft.ftn.uns.ac.rs/math/cms/LAP2015
Title: Computational interpretations of the classical Axiom of ChoiceTue, 23 Jun 2015Talk at Continuity, Computability, Constructivity – From Logic to Algorithms (CCC 2015), Kochel am See
http://www.cs.swan.ac.uk/ccc2015/
Title: On the indispensability of bar
recursionTue, 23 Jun 2015Keynote talk at CiiT 2015, Pelister National Park
http://ciit.finki.ukim.mk/
Title: Proof Assistants, marriage of Proof Theory and Programming LanguagesFri, 24 Apr 2015Talk at Deducteam seminar: The exp-log normal form of types and canonical forms of terms
http://www.cri.ensmp.fr/people/hermant/deducteam/seminars.html
In presence of sum types, the lambda calculus does
not enjoy uniqueness of eta-long normal forms. The canonicity
problem also appears for proof trees of intuitionistic logic. In
this talk, I will show how the problem becomes easier if, before
choosing a canonical representative from a class of
beta-eta-equal terms, one coerces the term into the exp-log
normal form of its type. This type normal form arises from the
one for exponential polynomials.Tue, 5 May 2015Classical polarizations yield double negation translations (with Zakaria Chihani and Dale Miller)
http://www.lix.polytechnique.fr/~danko/dneg.pdf
Double negation translations map formulas to
formulas in such a way that if a formula is a classical theorem
then its translation is an intuitionistic theorem. We shall go
beyond just examining provability by looking at correspondences
between inference rules in classical proofs and in
intuitionistic proofs of translated formulas. In order to make
this comparison interesting and precise, we will examine focused
versions of proofs in classical and intuitionistic logics using
the LKF and LJF proof systems. We shall show that for a number
of known double negation translations, one can get essentially
identical (focused) intuitionistic proofs as (focused) classical
proofs. We shall argue that the different ways one can define a
double negation translation corresponds precisely to the
different ways one can polarize classical
formulas.Fri, 17 Apr 2015The Exp-Log Normal Form of Types and Canonical Terms for Lambda Calculus with Sums
http://arxiv.org/abs/1502.04634
In the presence of sum types, the eta-long
beta-normal form of terms of lambda calculus is not
canonical. Natural deduction systems for intuitionistic logic
(with disjunction) suffer the same defect, thanks to the
Curry-Howard correspondence. This canonicity problem has been
open in Proof Theory since the 1960s, while it has been
addressed in Computer Science, since the 1990s, by a number of
authors using decision procedures: instead of deriving a notion
of syntactic canonical normal form, one gives a procedure based
on program analysis to decide when any two terms of the lambda
calculus with sum types are essentially the same one. In this
paper, we show the canonicity problem is difficult because it is
too specialized: rather then picking a canonical representative
out of a class of beta-eta-equal terms of a given type, one
should do so for the enlarged class of terms that are of a type
isomorphic to the given one. We isolate a type normal form, ENF,
generalizing the usual disjunctive normal form to handle
exponentials, and we show that the eta-long beta-normal form of
terms at ENF type is canonical, when the eta axiom for sums is
expressed via evaluation contexts. By coercing terms from a
given type to its isomorphic ENF type, our technique gives
unique canonical representatives for examples that had
previously been handled using program analysis.Tue, 17 Feb 2015Typos fixed for version 2 of arxiv.org/abs/1301.5089
http://www.lix.polytechnique.fr/~danko/shiftanalysis-ERRATA.txt
Errata for arxiv.org/abs/1301.5089, corrected version to appear on Arxiv soon.Tue, 27 Jan 2015(March 9) Proof theoretic results concerning sum types and control operators
http://gallium.inria.fr/seminar.html
On March 9, I am giving a talk at the Gallium seminar. Abstract: In this talk, I will present some recent results from Proof
Theory that might also be interesting to Programming Languages Theory.
First, I will present work on isomorphisms of types in presence of sums,
as well as the implications for canonicity of eta-long normal forms of
lambda calculus with sums. Second, I will show how one can use a
normalization-by-evaluation proof (written in CPS style) to compile away
control operators from System T; this could be used as a compiler
technique for a side-effect-free fragment of a program.Fri, 23 Jan 2015Eliminating control operators from classical realizability
http://www.pps.univ-paris-diderot.fr/gdt-types-realisabilite/
On January 21st, I am giving a talk at GdT Théorie des types et réalisabilité. Abstract: The traditional method to extract programs from proofs of classical Analysis (Peano Arithmetic + Axiom of Choice) is to use an extension of Gödel's System T with bar recursion. An alternative method is to use an approach based on computational side-effects (control operators, quote/clock instructions) like in the works of Krivine or Herbelin. By classic results of Kreisel and Schwichtenberg, for the fragment of Analysis that makes sense computationally, the Π₂-fragment, we know that bar recursion is essentially primitive recursive — leaving open the question of how to actually avoid using it. In this talk, I will present some recent work (arxiv.org/abs/1301.5089) showing how realizers of System T can be extracted directly from proofs of the Σ₂-fragment of classical Analysis. Control operators are essential, but only at the meta-theoretical level: control operators can be compiled away from System T, at any simple type.Mon, 19 Jan 2015An interpretation of the Sigma-2 fragment of classical Analysis in System T
http://arxiv.org/abs/1301.5089
I finally updated my arxiv.org/abs/1301.5089v1 paper. Resubmitted.Mon, 8 Dec 2014Distilled Tutorial during PPDP 2014
http://www.lix.polytechnique.fr/~danko/PPDP-2014-tutorial/
I am giving a distilled tutorial during the conference PPDP 2014. The topic is writing proofs in continuation passing style, with the xase study of normalization-by-evaluation for Gödel's System T extended with delimited control. Tutorial web page is up.Sat, 29 Mar 2014Paper on sum isomorphisms accepted for publication
http://arxiv.org/abs/1401.2567
My paper on type isomorphisms in the presence of sums was accepted to CSL-LICS 2014.Sat, 29 Mar 2014PC member for PPDP 2014
http://www.cs.kent.ac.uk/events/2014/ppdp-lopstr-14/
I will be serving on the program committee for the 16th International Symposium on Principles and Practice of Declarative Programming (PPDP) taking place in Canterbury, UK on September 8-10, 2014Sat, 15 Mar 2014Talk at LCR Seminar, Villetaneuse -- RESCHEDULED
http://lipn.univ-paris13.fr/en/lcr
The talk is rescheduled for March 14.Wed, 26 Feb 2014Open Induction paper accepted for publication
http://arxiv.org/abs/1209.2229
The paper "A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators", coauthored with Keiko Nakata, accepted for publication Mon, 17 Feb 2014Lecturing at Mathematical Structures of Computation, Lyon
http://smc2014.univ-lyon1.fr/doku.php?id=week1
I will give lectures on week 1 of the Mathematical Structures of Computation winter school in Lyon (January 13-17).Thu, 19 Dec 2013Invited talk at Seminar for General Proof Theory, Belgrade
http://www.mi.sanu.ac.rs/seminars/seminar18.htm
I will give a talk at Kosta Došen's General Proof Theory seminar at the Serbian Academy of Sciences and Arts (February 24).Thu, 19 Dec 2013Talk at LCR Seminar, Villetaneuse
http://lipn.univ-paris13.fr/en/lcr
I will give a talk at the seminar of the LCR team, Université Paris 13 (March 7).Tue, 7 Jan 2014Axioms and Decidability for Type Isomorphism in Presence of Sums
http://arxiv.org/abs/1401.2567
I submitted a new paper about type isomorphisms i.e. constructive set cardinality.Sat, 11 Jan 2014My web page for the MSC Lyon 2014 Lectures
http://www.lix.polytechnique.fr/~danko/MSC-lyon-2014-webpage/
I set up a web page with materials for the winter school Recent developments in Type Theory, Mathematical Structures of Computation - Lyon 2014.Mon, 13 Jan 2014Talk at Deducteam seminar, Paris
http://www.cri.ensmp.fr/people/hermant/deducteam/seminars.html
I will give a talk on type isomorphisms on the Deducteam seminar at INRIA Paris -- av. de Italie, on February 7, 2014, at 10:00.Tue, 14 Jan 2014New version of "A compact representation of terms and
extensional equality at the exp-log normal form of types"
http://arxiv.org/abs/1502.04634
Lambda calculi with algebraic data types appear at
the core of functional programming languages, but still pose
theoretical challenges today: for instance, even in the presence
of the simplest non-trivial data type, the sum type, we do not
know how to assign a unique canonical normal form to a class of
beta-eta-equal programs. In this paper, we present the exp-log
normal form of types---derived from the representation of
exponential polynomials via the unary exponential and
logarithmic functions---that any type built from arrows,
products, and sums, can be isomorphically mapped to, but that
systematically minimizes the number of necessary sums in the
type. We then reduce the standard beta-eta equational theory of
the lambda calculus to a specialized version of itself, while
preserving completeness of the equality of terms. Finally, we
describe an alternative, more canonical, representation of terms
of the lambda calculus with sums, together with a
Coq-implemented type-directed partial evaluator into/from our
new term calculus. This is the first heuristic for deciding
interesting cases of beta-eta-equality that relies only on
syntactic comparison of normal forms, and not on performing
program analysis of the involved terms.Tue, 10 May 2016