CCC 2019:Computability, Continuity, Constructivity -

from Logic to Algorithms

Abstracts

Ljubljana (Slovenia), 2-6 September, 2019

This project has received funding from the European Union’s Horizon 2020 research andinnovation programme under the Marie Skłodowska-Curie grant agreement No 731143

ContentsTutorial: Computational content of proofsHelmut Schwichtenberg 3

Invited talk: Completeness is OverratedHannes Diener 4

Invited talk: A Verified ODE Solver and Smale’s 14th ProblemFabian Immler 5

Invited talk: Self-modulating moduli and continuous machinesFlorian Steinberg 6

Invited talk: An Effective Spectral Theorem for Bounded Self Adjoint Operators within ComputableAnalysis (TTE)Thomas Streicher 8

Invited talk: Analytic ordinary differential equations - From computational complexity to efficient andverified algorithmsHolger Thies 9

Instance reducibility and extended Weihrauch degreesAndrej Bauer, Kazuto Yoshimura 10

Joint approximations for multivariate real functionsFranz Brauße, Margarita Korovina, Norbert Müller 13

A note on the spatiality of localic products of countably based sober spacesMatthew de Brecht 15

Equality of mathematical structuresMartin Escardo 17

Randomized Computation of Continuous Data: Is Brownian Motion Strongly Computable?Willem L. Fouché, Hyunwoo Lee, Donghyun Lim, Sewon Park, Matthias Schröder, Martin Ziegler 18

Computing the limit set of planar differential equationsDaniel S. Graça, Ning Zhong 20

The Scott Model of PCF in Univalent Type TheoryTom de Jong 21

Haare Measure in Synthetic TopologyDavorin Lešnik 23

The Hyperspace Dimension TheoremJack H. Lutz, Elvira Mayordomo 25

A constructive predicative realizability topos for inductively generated formal topologyMaria Emilia Maietti, Samuele Maschio 26

Axiomatic Reals in Type Theory for Porgram ExtractionSewon Park 28

Wadge Hierarchy in Quasi-Polish SpacesVictor Selivanov 30

The Compact Hyperspace Monad, a Constructive ApproachDieter Spreen 32

Tensors of Quantitative Truth and the Combination of Algebraic EffectsNiels Voorneveld 33

A Gentzen-style translation of Gödel’s System TChuangjie Xu 35

2

Tutorial:

Computational content of proofs

Helmut Schwichtenberg

LMU Munich, Germany

Proofs have two aspects: they guarantee correctness, and they may have computational content.The Brouwer-Heyting-Kolmogorov (BHK) interpretation aims at clarifying the latter. However, in itsoriginal form it suffers from not explaining (i) what is a proof of a prime formula, and (ii) what is a“method” transforming a proof of a premise A into a proof of a conclusion B. We discuss a setup where(i) computational content arises solely from prime formulas with inductive or coinductive predicates,as their construction or destruction trees, and (ii) methods are computable functionals operating onpartial continuous functionals in the Scott-Ershov model. The least- und greatest-fixed-point axiomsfor (co)inductive predicates then have the corresponding structural (co)recursion operators as theircontent. There is a soundness theorem stating that the term expressing the computational content ofa proof realizes (in Kreisel’s modified sense) the proven formula. Applications include the extraction ofalgorithms operating on stream-represented real numbers from existence proofs in constructive analysis.

3

Invited talk:

Completeness is Overrated

Hannes Diener

University of Canterbury, New Zealand

It is a common theme in constructive analysis that theorems which are not constructive often can bemade so by adding the assumption that the underlying space is complete. In Bishop style mathematicsthis is cryptically known as the “lambda technique”, which owes its name to the common pattern ofchoosing a binary sequence—traditionally labelled lambda—which in turn is used to construct a Cauchysequence.

Any such constructed sequence is automatically Cauchy, so if we are working in a complete spaceit converges. In this talk we are going to show that in most cases a weakened form of completeness isactually sufficient. We will show that there is a plethora of examples of known theorems to which thisgeneralisation applies. The most prominent examples will be the Kreisel-Lacome-Shoenfield theorem,which states that all (computable) real-valued functions on a complete metric space are continuous, andthe problem of basic path glueing, which fundamentally underlies the development of homotopy theory.

Of course, none of this would be interesting, if there were no good examples of spaces that arecomputably complete in the weakened sense but not complete. Indeed, we will show that there are manynatural spaces falling into this category.

4

Invited talk:

A Verified ODE Solver and Smale’s 14th Problem

Fabian Immler

Carnegie Mellon University, USA

Smale’s 14th Problem is a conjecture about chaos in a particular dynamical system, the Lorenzattractor. The problem was solved by Warwick Tucker with a combination of regular analysis and acomputer-assisted part. The computer-assisted part yields numerical bounds on solutions of the LorenzODE, which are required to certify chaos.

The computer-assisted part of Tuckers proof has been formally verified in the interactive theoremprover Isabelle/HOL. I will present Isabelle/HOL’s library of ODEs and verified numerical methods andhow they underpin the formal verification of the computer-assisted part.

5

Invited talk:

Self-modulating moduli and continuous machines.

Florian Steinberg

Inria Saclay, France

A represented space X = (X, δX) is a pair of a setX and a partial surjective function δX : ⊆ B → X.Here B is usually taken to be NN or AQ for very concretely encodable sets Q and A such as the rationalnumbers or intervals with dyadic endpoints. The canonical example is X := R with the representationδR : QQ → X defined by δR(ϕ) = x ⇐⇒ ∀ε > 0, |ϕ(ε) − x| ≤ ε. An element ϕ ∈ B that the represen-tation takes to x is called a name of x. For instance, according to the example above, a name of a realnumber is a function that returns rational approximations of arbitrary quality.

Xf // Y

BF//

δX

OO

B′δY

OO

Each represented space comes with a natural topological, computability andcomplexity structure as the corresponding notions for B can be pushed forwardthrough the representation. An element of a represented space is called computableresp. polytime if it has such a name. Moreover, an operator F : ⊆ B → B is said torealize a function f : X → Y if it translates names of an argument x to names ofits value f(x). The function f is then called continuous, computable resp. polytimeif F can be chosen such.

Computable analysis defines a represented space YX of continuous functions from X to Y [6, 4]. Afunction f : X → Y has a computable realizer if and only if it has a computable name with respect tothe function-space representation. That the underlying set of YX is the set of all continuous functionscan be read as “any continuous function has a name”. This correspondence does not carry over tocomplexity theory as it is typically the case that any computable function has a polytime or even linear-time computable name [3]. We demonstrate this behaviour for a concrete construction of YX andcompare to recent results from higher-order complexity [1]. The construction specifies a universal U(similar to Weihrauch’s η) that assigns to each continuous operator on B a set of associates and wherethe names of a function are the associates of its realizers.

Afterwards we turn to continuity considerations. The handling of continuous operators can be seenas a model for operating on computable functions with restricted access to the algorithmic contentsand this makes efficiency considerations interesting also in a continuity setting. Let Q, A, Q′ and A′

be countable sets and write B := AQ and B′ := A′Q′. An operator F : ⊆ B → B′ is continuous if for

all ϕ ∈ dom(F ) and q′ ∈ Q′ there exists a finite list K ∈ Q∗, called a certificate, such that for allψ ∈ dom(F ) with ϕ|K = ψ|K it holds that F (ϕ)(q′) = F (ψ)(q′). An operator µ : ⊆ B → Q∗Q

′is called

a modulus for F if for any ϕ ∈ dom(F ) and any q′ ∈ Q′ the list µ(ϕ)(q′) is a certificate. There existsa standard method to extract a modulus of continuity from an algorithm computing an operator, andthe extracted modulus is computable. Moreover, this modulus is self-modulating.

One way to see that the universal U assigns associates to each continuous operator is to use theminimal modulus with respect to an enumeration of Q. The existence of the minimal modulus isan inherently classical fact and this procedure non-constructive. Furthermore, the associate obtained isusually far from optimal with respect to access to the input function. We note that the minimal modulusis always self-moduluating, that this seems to be the key ingredient for finding an associate and thatone can get a self-modulating modulus of an operator from any of its associates. However, if the domain

6

of the operator is complicated, a self-modulating modulus alone is not sufficient information. Thus, weconsider a pair of a function M : B → (A′ + 1)N×Q

′such that if ϕ ∈ dom(F ), then a′ := F (ϕ)(q′) is the

only element of A′ such that there is an n ∈ N with M(ϕ)(n, q′) = a′ and a self-modulating modulusν of M . Appropriate pairs (M,ν) can be constructed from associates by truncating the universal andfrom each such pair, one can build an associate.

Finally we argue that any M of the above type may be considered to describe a multivalued operatorand illustrate the use of this concept by efficiently and comprehensibly implementing some neccessarilypartial realizers for functions on Cantor-space and the reals in Coq [5]. We conclude by drawing parallelsto how computability is modelled in type theories and to the oracle machine model of computation thatis commonly used in computable anlaysis [2]: Given an index e of such a machine one can consider thefunction that Me such that Me(ϕ)(n, q′) = a′ if the computation of e on oracle ϕ and input q′ terminatesand returns a′ within the first n steps and let Me return the error symbol otherwise. We show that ifM is continuous, then so is its time function.

References

[1] Hugo Feree. Game semantics approach to higher-order complexity. J. Comput. Syst. Sci., 87(C):1–15, August 2017.

[2] Akitoshi Kawamura and Stephen Cook. Complexity theory for operators in analysis. In STOC’10—Proceedings of the2010 ACM International Symposium on Theory of Computing, pages 495–502. ACM, New York, 2010.

[3] Akitoshi Kawamura and Arno Pauly. Function spaces for second-order polynomial time. In Conference on Computabilityin Europe, pages 245–254. Springer, 2014.

[4] Arno Pauly. On the topological aspects of the theory of represented spaces. Computability, 5(2):159–180, 2016.

[5] Florian Steinberg, Laurent Thery, and Holger Thies. Quantitative continuity and computable analysis in Coq. HALpreprint, April 2019.

[6] Klaus Weihrauch. Computable Analysis. Texts in Theoretical Computer Science. An EATCS Series. Springer BerlinHeidelberg, 2002.

7

Invited talk:

An Effective Spectral Theorem for Bounded Self Adjoint Operators

within Computable Analysis (TTE)

Thomas Streicher

TU Darmstadt, Germany

As shown by Bauer, Lietz, Schroder, Simpson et. al. the TTE approach to computable analysis maybe understood as doing analysis within the function realizabilty topos or its effective variant, the socalled Kleene-Vesley topos, where all maps are effective but operate also on non-computable data.

We exemplify the convenience of this paradigm by showing that the Spectral Theorem for BoundedSelf Adjoint Operators does hold in the above mentioned model(s). The emphasis is on computabilityand not on extracting concrete algorithms though the latter is possible in principle. Instead our emphasisis on showing that the required data types actually live within the above mentioned model(s) and thushave admissible representations.

8

Invited talk:

Analytic ordinary differential equations – From computational

complexity to efficient and verified algorithms

Holger Thies1

1Kyushu University, Fukuoka, Japan

An important problem in numerical computation is to find the solution y to an initial value problem for ordinarydifferential equations of the form

y(t) = F (y(t)), y(0) = y0

where F : Rd → Rd and y0 ∈ Rd. From the viewpoint of actual computation, it is most natural to consider theabove problem as an operator that maps the real function F and the initial value y0 to the solution function y.

To discuss computability and complexity theoretical properties of such an operator, computable analysis usesrepresentations to encode elements of the relevant function spaces by elements of the Baire space B := Σ∗Σ∗

.Computing an operator can then be reduced to defining a realizer, a function that manipulates said encodings andsecond-order complexity theory [KC96, KC12] can be used to describe the computational complexity.

It is well known in real number complexity theory that the solution of an initial value problem can be compu-tationally hard even when the right-hand side function F is polynomial-time computable and Lipschitz continuous[Kaw10]. On the other hand, if F is analytic and polynomial-time computable in a neighborhood of the originthen so is y (see e.g. [MM93]). Extending this result to uniform computation of the solution operator requires thedefinition of a representation for analytic functions. Representations for analytic functions and polynomial-timecomputable realizers for many important operations have for example been considered in [KMRZ15] and have beenextended to multidimensional functions and ODE solving in [KST18]. As realizers basically describe algorithmsit is not difficult to translate the results to actual implementations in exact real arithmetic. Further, the rigoroustreatment of correctness in computable analysis naturally goes well with formal verification. Indeed, some ofthe basic aspects of computable analysis have recently been formalized in a library for the Coq proof assistant[Ste19, STT19].

This talk introduces some of the results on the uniform polynomial-time computability of solving analyticinitial value problems and other operators on analytic functions and how they can translated to provably correctalgorithms that can be formally verified in a proof assistant. As a direct implementation of the underlying theoryusually does not result in efficient implementations, the talk also discusses how more sophisticated methods e.g.from interval analysis can be incorporated to make algorithms more feasible while preserving correctness.

References[Kaw10] Akitoshi Kawamura. Lipschitz continuous ordinary differential equations are polynomial-space complete. Computational

Complexity, 19(2):305–332, 2010.

[KC96] Bruce M. Kapron and Stephen A. Cook. A new characterization of type-2 feasibility. SIAM J. Comput., 25:117–132, 1996.

[KC12] Akitoshi Kawamura and Stephen Cook. Complexity theory for operators in analysis. ACM Transactions in ComputationTheory, 4(2):Article 5, 2012.

[KMRZ15] Akitoshi Kawamura, Norbert Th. Muller, Carsten Rosnick, and Martin Ziegler. Computational Benefit of Smoothness.Journal of Complexity, 2015.

[KST18] Akitoshi Kawamura, Florian Steinberg, and Holger Thies. Parameterized complexity for uniform operators on multidimen-sional analytic functions and ODE solving. In International Workshop on Logic, Language, Information, and Computation,pages 223–236. Springer, 2018.

[MM93] Bernd Moiske and Norbert Th. Muller. Solving initial value problems in polynomial time. In Proc. 22 JAIIO-PANEL,volume 93, pages 283–293, 1993.

[Ste19] Florian Steinberg. The Incone library. https://github.com/FlorianSteinberg/incone, 2019. release v1.0.

[STT19] Florian Steinberg, Laurent Thery, and Holger Thies. Quantitative continuity and computable analysis in Coq. arXivpreprint arXiv:1904.13203, 2019.

9

Instance reducibility and

extended Weihrauch degrees

Andrej Bauer Kazuto Yoshimura

1 Instance reducibilities

In constructive reverse mathematics we often prove implications of the form

(∀y ∈ B .ψ(y)) =⇒ (∀x ∈ A .φ(x)), (1)

usually between statements which are constructively undecided. There is a natural and typical way of provingsuch an implication: for every x ∈ A find a suitable (not necessarily unique) y ∈ B such that ψ(y) implies φ(x).For example, the usual proof of

(∀x ∈ R . x ≤ 0 ∨ x > 0) =⇒ (∀α ∈ 2N . (∀k . αk = 0) ∨ (∃k . αk = 1)),

takes a binary sequence α and constructs from it the real number x =∑∞

k=0 αk · 2−k, and asks whether x > 0 inorder to decide whether α attains the value 1.

This method of proof is so prevalent, throughout mathematics, that it deserves a name. We write φ ⊆ A toindicate that φ is a predicate on a set A.

Definition 1. A predicate φ ⊆ A is instance reducible to a predicate ψ ⊆ B, which we write as (φ,A) ≤I (ψ,B)or just φ ≤I ψ, when

∀x ∈ A .∃y ∈ B .ψ(y)⇒ φ(x). (2)

We say that φ ⊆ A and ψ ⊆ B are instance equivalent, written as (φ,A) ≡I (ψ,A) or just φ ≡I ψ, when φ ≤I ψand ψ ≤I φ. The equivalence class of a predicate φ with respect to ≡I is called its instance degree.

It is immediately clear that ≤I is a preorder (reflexive and transitive) in which harder problems are higherup. At the bottom is the empty predicate on the empty set, and at the top the empty predicate on the singletonset. In the presence of excluded middle there is just one more degree, namely the full predicate on the singleton.In constructive mathematics, however, the instance degrees can have a much richer structure, and are related toWeihrauch degrees [1], as explained in §3.

Some implications of the form (1) are proved by reduction of an instance in A to several instances in B. Thismethod of proof is accommodated by the following construction.

Definition 2. For any set I and a predicate φ ⊆ A, the I-parameterization of φ is the predicate φI ⊆ AI definedon the set AI of all functions from I to A by φI(f) ⇐⇒ ∀i ∈ I . φ(f(i)).

A reduction of φ ⊆ A to ψ ⊆ B which uses a fixed number n of instances of ψ for every instance of φ is justinstance reducibility of φ to ψ[n] where [n] = 1, . . . , n because φ ≤I ψ

[n] unfolds to

∀x ∈ A .∃y1, . . . , yn ∈ B .ψ(y1) ∧ · · · ∧ ψ(yn)⇒ φ(x).

This is not to be confused with the reduction of φ(x) to a variable finite number n of instances ψ(y1), . . . , ψ(yn),which is expressed as (where

∐denotes the supremum of instance degrees)

φ ≤I

∐n∈Nψ

[n],

as it unfolds to∀x ∈ A .∃n ∈ N .∃y1, . . . , yn ∈ B .ψ(y1) ∧ · · · ∧ ψ(yn)⇒ φ(x).

A third possibility is φ ≤I ψN which reduces an instance φ(x) to countably many instances ψ(y0), ψ(y1), ψ(y2), . . .

10

⊥I

>I

>1

Ω

f

•

•

•

¬¬∀

Figure 1: The lattice of instance degrees

2 The lattice structure of instance degrees

The instance degrees form a large preorder whose carrier is the proper class of all predicates on all sets. It hasthe structure of a lattice.

Theorem 3. Instance reducibilities form a bounded distributive lattice. Moreover, set-indexed suprema and infimaof instance degrees exist, and finite infima distribute over set-indexed suprema.

To be quite precise, the theorem should speak about a pre-lattice, since ≤I is a preorder, but this is a minortechnicality.

Figure 1 shows the structure of instance degrees. The poset of truth values Ω embeds in it twice, once asthe principal ideal of degrees below the full predicate on the singleton; and again in an anti-monotone fashionabove >1. Of special interest are the instance degrees represented by the ¬¬-dense predicates, which form a lowersubclass and are closed by set-indexed suprema, hence a sublattice of instance degrees.

3 The extended Weihrauch degrees

By locating various reasoning principles in the lattice of instance degrees—excluded middle and its derivatives,continuity principles, choice principles, the formal Church-Turing thesis, etc.—one can start painting a pictureof reverse constructive mathematics, akin to how the Weihrauch degrees display the relationships between non-computable problems in computable mathematics, in the sense of Type Two Effectivity [3]. In fact, the Weihrauchdegrees form a sublattice of the instance degrees in the Kleene-Vesley topos. We outline the argument that worksgenerally in any relative realizability topos RT(A,Aeff), see [2, §4.5]. The classic Weihrauch degrees are obtainedwhen we specialize the underlying partial combinatory algebras A and Aeff to the Baire space and the effectiveBaire space, respectively.

Every object in a realizability topos is the image of a partitioned assembly [2, §3.2.3], from which it followseasily that ever instance degree is represented by a predicate on a partitioned assembly. (Recall that an assemblyis partitioned when its existence predicate takes values in the singleton sets.) With a little bit of further work onecan deduce the following characterization of the instance degrees.

Given a binary relation R ⊆ X×Y , define ‖S‖ = x ∈ X | ∃y ∈ Y . (x, y) ∈ S and S[x] = y ∈ B | (x, y) ∈ S.Let 〈−,−〉 be a pairing in the underlying partial combinatory algebra A.

Theorem 4. The (large) lattice of instance degrees in RT(A,Aeff) is equivalent to the (set-sized) lattice P(A×P(A))ordered by the relation , which is defined by the stipulation that A B if, and only if, there exist k1, k2 ∈ Aeff

such that:

1. for all a ∈ ‖A‖, k1 a is defined and k1 a ∈ ‖B‖,

2. if a ∈ ‖A‖ and u ∈ U ∈ B[k1 a] then k2〈a, u〉 is defined and k2〈a, u〉 ∈ A[a].

We call the elements of P(A × P(A)) the extended Weihrauch degrees, in the relative realizability toposRT(A, Aeff). Let us compare them to the usual Weihrauch degrees. Recall that F ⊆ A×A is Weihrauch-reducibleto G ⊆ A× A, written F ≤W G, when there exist κ1, κ2 ∈ Aeff such that:

11

1. for all a ∈ ‖F‖, κ1 a is defined and κ1 a ∈ ‖G‖,

2. if a ∈ ‖F‖ and u ∈ G[κ1 a] then κ2〈a, u〉 is defined and κ2〈a, u〉 ∈ F [a].

The resemblance is not accidental. An ordinary degree F ⊆ A × A may be construed as an extended degreeF ⊆ A× P(A), defined by F = (a, F [a]) | a ∈ ‖F‖ because F G is the same thing as F ≤W G. We may alsocharacterize the ordinary Weihrauch degrees as a sublattice of the instance degrees.

Theorem 5. In RT(A,Aeff), the Weihrauch degrees form a sublattice of instance degrees, represented by the¬¬-dense predicates on the modest objects [2, Def. 3.2.23].

There are extended Weihrauch degrees that are not ordinary Weihrauch degrees, some of which are quiteinteresting from the point of view of computable mathematics. For instance, the predicate “is Turing-computable”on NN is not ¬¬-dense, and the extended Weihrauch degree corresponding to the law of excluded is not modest.

References[1] Vasco Brattka and Guido Gherardi. Weihrauch degrees, omniscience principles and weak computability. The Journal of Symbolic

Logic, 76(1):143–176, 2011.

[2] Jaap van Oosten. Realizability: An Introduction To Its Categorical Side, volume 152 of Studies in logic and the foundations ofmathematics. Elsevier, 2008.

[3] Klaus Weihrauch. Computable Analysis. Springer, Berlin, 2000.

12

Joint approximationsfor multivariate real functions ∗

Franz Brauße1, Margarita Korovina2, and Norbert Müller1

1Abteilung Informatikwissenschaften, Universität Trier, Germany2A.P. Ershov Institute of Informatics Systems, Novosibirsk, Russia

We discuss data structures and algorithms for approximations of vectors (or rather finite lists) ofmultivariate real functions f : ⊆Rk → R. From the viewpoint of TTE, these approximations can beviewed as building blocks for representations of real continuous functions f : ⊆Rk → R.

Restricted to compact sets A ⊆ Rk within the domain, the resulting representations of single functionsare computably equivalent to the standard ‘box’ representations [ρk → ρ]A of the continuous functionsC(A,R) [Wei00], so the derived notion of computability remains unchanged.

The joint approximations used in this paper, however, usually behave significantly better in realworld implementations, as they also take into account inner dependencies between several approximatedfunctions arising during a computation. To this end, we apply ideas from generalized interval arithmetic[Han75] and from Taylor models [MB01, Jol11], thus generalizing the approach taken in [BKM15]: Ondomains given as polytopes, functions are approximated by polynomials with interval coefficients, where‘error symbols’ reflect dependencies.

In [DFKT14], there has been a related approach of a ‘function interval arithmetic’, while in [KN17]fast evaluations are discussed; both still lacking the aspect of inner dependencies.

As an application we aim at the field of SMT solving, where many different techniques are applieddepending on the structure of the problems under consideration. In case of the satisfiability problem forsemilinear constraints with rational coefficients, e.g., the simplex algorithm or inner point methods canbe used. Due to their success in SAT solver competitions, nowadays algorithms based on CDCL (conflictdriven clause learning) [JdM12, KTV09] are preferred for this problem class. For polynomial inequalitiesmany implementations use symbolic approaches based on cylindrical decomposition, which in practice isonly applicable in low dimensional cases. Already for elementary transcendental functions or in cases ofhigher dimensions other techniques are needed.

Recently, algorithms based on the construction of linear bounds to multivariate nonlinear functionshave been developed [BKKM19, CGI+17, CGI+18]. They combine symbolical and numerical computation,where, from our point of view, TTE seems to be the most appropriate theoretical setting for the numericalpart.

We present a prototypical implementation tangentspace1 where a symbolically defined function fis numerically evaluated in a region containing a ‘conflict’ point p using our proposed approximationscheme. The result is a symbolical term describing a local linear upper or lower bound g for f . Thisbound g separates p from f . The implementation is suitable for the inclusion in CDCL-style SMT solvers.Experimental results are very promising and show better results than obtained via naive interval (‘box’)arithmetic.

∗The research leading to these results has received funding from the DFG grant WERA MU 1801/5-1 and theDFG/RFBR grant CAVER BE 1267/14-1 and 14-01-91334.

This project has received funding from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Skłodowska-Curie grant agreement No 731143.

1http://informatik.uni-trier.de/~mueller/Research/Tangentspace/

13

References[BKKM19] Franz Brauße, Konstantin Korovin, Margarita Korovina, and Norbert Th. Müller. A CDCL-style calculus for

solving non-linear constraints. FroCoS 2019, London, UK, to appear in LNAI/LNCS, 2019.[BKM15] Franz Brauße, Margarita Korovina, and Norbert Th. Müller. Towards using exact real arithmetic for initial

value problems. PSI: 10th Ershov Informatics Conference 25 - 27 August 2015, Innopolis, Kazan, Russia, toappear in Lecture Notes in Computer Science, 2015.

[CGI+17] Alessandro Cimatti, Alberto Griggio, Ahmed Irfan, Marco Roveri, and Roberto Sebastiani. Satisfiability modulotranscendental functions via incremental linearization. In Proceedings CADE 26, pages 95–113, 2017.

[CGI+18] Alessandro Cimatti, Alberto Griggio, Ahmed Irfan, Marco Roveri, and Roberto Sebastiani. Experimenting onsolving nonlinear integer arithmetic with incremental linearization. In Proceedings SAT 2018 - 21st InternationalConference, SAT 2018, pages 383–398, 2018.

[DFKT14] Jan Duracz, Amin Farjudian, Michal Konecný, and Walid Taha. Function interval arithmetic. In MathematicalSoftware - ICMS 2014 - 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings, pages677–684, 2014.

[Han75] E. R. Hansen. A generalized interval arithmetic. In Interval Mathemantics: Proceedings of the InternationalSymposium, Karlsruhe, West Germany, May 20-24, 1975, pages 7–18, 1975.

[JdM12] D. Jovanovic and L. de Moura. Solving non-linear arithmetic. In IJCAR’2012, LNCS v. 7364, pages 339–354,2012.

[Jol11] Mioara Joldes. Approximations polynomiales rigoureuses et applications. PhD thesis, École Normale Supérieurede Lyon, 2011.

[KN17] Michal Konecný and Eike Neumann. Representations and evaluation strategies for feasibly approximablefunctions. CoRR, abs/1710.03702, 2017.

[KTV09] K. Korovin, N. Tsiskaridze, and A. Voronkov. Conflict resolution. In CP’09, LNCS v.5732, pages 509–523, 2009.[MB01] K. Makino and M. Berz. Higher order verified inclusions of multidimensional systems by taylor models. Nonlinear

Analysis: Theory, Methods & Applications, 47(5):3503 – 3514, 2001. Proceedings of the Third World Congressof Nonlinear Analysts.

[Wei00] Klaus Weihrauch. Computable analysis: an introduction. Springer-Verlag New York, Inc., Secaucus, NJ, USA,2000.

14

A note on the spatiality of localic products of

countably based sober spaces

Matthew de Brecht∗

Graduate School of Human and Environmental Studies, Kyoto University, Japanmatthew@i.h.kyoto-u.ac.jp

Let Top be the category of topological spaces and continuous maps, Loc the category of locales,and Ω : Top → Loc the usual functor mapping spaces to locales. Ω preserves colimits (since it has aright adjoint pt : Loc→ Top), but Ω does not preserve finite products in general. The purpose of thisnote is to investigate subcategories of countably based sober spaces for which the restriction of Ω doespreserve finite products.

Let S0 be the countable space defined in [2]. The underlying set of S0 is N<N, the set of finitesequences of natural numbers. A subbasis for the open subsets of S0 is given by sets of the formτ ∈ N<N |σ 6 τ, where σ ∈ N<N and is the prefix relation. Note that S0 is a countably based soberspace, its specialization order is the reverse of the prefix relation, and that S0 has uncountably manydistinct open sets ([2]; Proposition 6.1).

We first show that the localic product Ω(S0)×lΩ(S0) is not spatial by describing a winning strategyfor Player I in the game G(S0, S0) defined by T. Plewe (see Theorem 1.1 in [5] and the paragraph aboveit for a definition of the game). The proof strategy for the following lemma is essentially the same asP. Johnstone’s proof that Ω(Q) ×l Ω(Q) is not spatial (see Proposition II-2.14 of [4]), but the gametheoretic approach allows us to hide the use of transfinite ordinals.

Lemma 1. The localic product Ω(S0)×l Ω(S0) is not spatial.

Proof. We denote the length of σ ∈ N<N by |σ|. The empty string is denoted as ε, and the stringconsisting of m zeros is written 0(m). The string obtained by appending n ∈ N to σ ∈ N<N is writtenσ n. We also write σ τ for the concatenation of strings. For σ, τ ∈ N<N, define

Fσ,τ = s n | n ∈ N & s ∈ N<N & s σ & s n 6 σ & n ≤ |σ|+ |τ |, and

Uσ,τ = t ∈ N<N | (∀s ∈ Fσ,τ ) s 6 t.

Then σ ∈ Uσ,τ , hence U = Uσ,τ × Uτ,σ | σ, τ ∈ N<N is an open cover of S0 × S0.Observe that if s ∈ Uσ,τ and every element of s is less than or equal to |σ| + |τ | then s σ. Also

note that if U ⊆ S0 is open and σ ∈ U , then there exist infinitely many n ∈ N such that every stringthat has σ n as a prefix is also in U . This is because there must be a finite F ⊆ S0 such that the basicopen set W = τ ∈ N<N | (∀s ∈ F ) s 6 τ satisfies σ ∈ W ⊆ U . Fix any n ∈ N that is strictly largerthan any element contained in any of the strings in F . Then for each s ∈ F , we have that s 6 σ n andσ n 6 s, hence no extension of σ n has s as a prefix. It follows that every extension of σ n must bein W and therefore also in U .

Player I initializes the game by playing S0×S0 and the open covering U of S0×S0. The game beginswith round 1. For convenience, define V0 = W0 = S0, and x0 = y0 = ε, and m0 = n0 = 0. Player I’sstrategy for the i-th round (i ≥ 1) proceeds as follows:

∗This work was supported by JSPS Core-to-Core Program, A. Advanced Research Networks and by JSPS KAKENHIGrant Number 18K11166.

15

1. Player I chooses mi ∈ N such that every sequence extending yi−1 mi is in Wi−1. Player I thenplays xi = xi−1 ni−1 0(mi).

2. Player II must respond with an open subset Vi ⊆ Vi−1 containing xi.

3. Next, Player I finds distinct ni and n′i in N such that any sequence that has either xi ni or xi n′ias a prefix is in Vi. Player I plays yi = yi−1 mi 0(ni+n

′i).

4. Player II must respond with an open subset Wi ⊆Wi−1 containing yi.

The game then continues on to round i+ 1.We show that at the end of each round i ≥ 1, the open rectangle Vi ×Wi chosen by Player II is not

a subset of any open rectangle in U . Fix any σ, τ ∈ N<N with xi ∈ Uσ,τ and yi ∈ Uτ,σ. Since yi−1 yiwe have yi−1 ∈ Uτ,σ, and an inductive argument (keep reading) yields yi−1 τ . Using the fact that|yi−1| ≥ ni−1 it can be shown that every element occurring in xi is less than or equal to |yi−1| ≤ |σ|+ |τ |,hence the assumption xi ∈ Uσ,τ and the observation at the top of the second paragraph of this proofimplies xi σ. Similarly, every element of yi is less than or equal to |xi| ≤ |τ | + |σ|, hence yi ∈ Uτ,σimplies yi τ (thereby completing the inductive argument). Either xi ni 6 σ or xi n′i 6 σ, andni, n

′i ≤ |yi| ≤ |σ| + |τ |, thus xi ni 6∈ Uσ,τ or xi n′i 6∈ Uσ,τ , but both xi ni and xi n′i are in Vi, so

we conclude that Vi ×Wi 6⊆ Uσ,τ × Uτ,σ. Therefore, the above strategy is winning for Player I, henceΩ(S0)×l Ω(S0) is not spatial. ut

Let ωSob be the category of countably based sober spaces, and let QPol be the category of quasi-Polish spaces [1]. ωSob and QPol are closed under countable limits (as defined in Top), and therestriction of Ω to QPol preserves all countable limits (Theorems 4.4 and 4.5 of [3]).

Theorem 1. Assume C is a full subcategory of ωSob satisfying:

(1) C is closed under finite limits (as defined in Top),

(2) the restriction of Ω to C preserves finite products,

(3) C contains P(N) (the powerset of N with the Scott-topology), and

(4) every space in C is co-analytic (i.e., homeomorphic to a Π11-subspace of P(N)).

Then C is a full subcategory of QPol.

Proof. Assume for a contradiction that there is some space X in C which is not quasi-Polish. X is co-analytic by (4), hence Theorem 7.2 of [2] implies there is a Π0

2-subspace Y of X which is homeomorphicto either S0 or Q (the candidates SD and S1 mentioned in [2] can be omitted because X is sober). SinceY is a Π0

2-subspace of X, it is the equalizer of a pair of continuous functions f, g : X → P(N) (see theconcluding section of [1]), hence Y is in C by (1), (3), and the assumption that C is a full subcategoryof ωSob. Lemma 1 and the fact that Ω(Q) ×l Ω(Q) is not spatial imply Ω(Y × Y ) 6= Ω(Y ) ×l Ω(Y ).Therefore, the restriction of Ω to C does not preserve products, which contradicts (2). ut

We conjecture that the above theorem still holds if (3) is omitted. It is consistent with ZFC to replace“co-analytic” in (4) with any level of the projective hierarchy, and we conjecture that it is consistentwith ZF+(Dependent Choice) if (4) is removed completely.

References

[1] M. de Brecht. Quasi-Polish spaces. Annals of Pure and Applied Logic, 164 (2013), 356–381.

[2] M. de Brecht. A generalization of a theorem of Hurewicz for quasi-Polish spaces. Logical Methods in Computer Science,14 (2018), 1–18.

[3] R. Heckmann. Spatiality of countably presentable locales (proved with the Baire category theorem). MathematicalStructures in Computer Science, 25 (2015), 1607–1625.

[4] P. Johnstone. Stone spaces, Cambridge University Press (1982).

[5] T. Plewe. Localic products of spaces. Proceedings of The London Mathematical Society, 73 (1996), 642–678.

16

Equality of mathematical structures

Martın Hotzel Escardo

School of Computer ScienceUniversity of Birmingham, UK

Independently of any choice of foundation, we regard two groups to be the same, for all mathematicalpurposes, if they are isomorphic. Likewise, we consider two topological spaces to be the same if they arehomeomorphic, two metric spaces to be the same if they are isometric, two categories to be the same ifthey are equivalent, and so on.

Do we choose these notions of sameness, motivated by particular mathematical applications, or arethese notions of sameness imposed upon us, independently of what we want to do with groups, topologicalspaces, metric spaces and categories? This may be regarded as a philosophical question. However, if

• we adopt Martin-Lof type theory as our mathematical foundation,

• take the notion of equality to be the identity type, and

• assume Voevodsky’s univalence axiom,

then

• we can prove that equality of groups is isomorphism, equality of topological spaces is homeomor-phism, equality of metric spaces is isometry, equality of categories is equivalence etc.

For a large class of algebraic structures, this was first proved by Coquand and Danielsson [2]. For objectsof categories, this was formulated by Aczel [4], and for categories themselves this was proved by Ahrens,Kapulkin and Shulman [1].

This talk will be mainly expository, based on our lecture notes [3]. A minor novelty will be a generaltheorem and various specializations that can be used to attack all the above characterizations of equality,and more, in a uniform and modular way.

References

[1] Benedikt Ahrens, Krzysztof Kapulkin, and Michael Shulman. Univalent categories and the Rezk completion. In Extendedabstracts Fall 2013—geometrical analysis, type theory, homotopy theory and univalent foundations, volume 3 of TrendsMath. Res. Perspect. CRM Barc., pages 75–76. Birkhauser/Springer, Cham, 2015.

[2] Thierry Coquand and Nils Anders Danielsson. Isomorphism is equality. Indag. Math. (N.S.), 24(4):1105–1120, 2013.

[3] Martın Hotzel Escardo. Introduction to univalent foundations of mathematics with Agda. https://www.cs.bham.ac.

uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/, March-July 2019.

[4] The Univalent Foundations Program. Homotopy type theory—univalent foundations of mathematics. The UnivalentFoundations Program, Princeton, NJ; Institute for Advanced Study (IAS), Princeton, NJ, 2013.

17

Randomized Computation of Continuous Data:

Is Brownian Motion Strongly Computable?∗

Willem L. Fouche1, Hyunwoo Lee2, Donghyun Lim2,Sewon Park2, Matthias Schroder3, Martin Ziegler2

1 University of South Africa 2 KAIST 3 University of Birmingham

Randomization is a powerful technique in classical (i.e. discrete) Computer Science: Many difficult prob-lems have turned out to admit simple solutions by algorithms that ‘roll dice’ and are efficient/correct/optimal with high probability. Indeed fair coin flips have been shown computationally universal [Wal77].Over continuous data, closely connected to topology [Wei00, §2.2+§3], notions of probabilistic compu-tation are more subtle [BGH15, Col15].

1 Representing Borel Probability Measures

Recall that a measure space is a triple (X,A, µ), where X is a non-empty set, A is a σ-algebra overX, and µ is a measure on (X,A). For measure spaces (X,A, µ) and (Y,B, ν) and a measurable partialmapping F :⊆ X → Y , ν is the pushforward measure of µ w.r.t. F if µ

(F−1[V ]

)is defined and equal

to ν(V ) for every V ∈ B. In this case we say F realizes ν on µ and write ν 4 µ. This notion is similarto, but not in danger of confusion with, [Wei00, Definition 2.3.2]. Note that realizability is transitive;and a realizer F must have dom(F ) ∈ A of measure ν(Y ). In the sequel we consider topological spaces,implicitly equipped with the Borel σ-algebra, and a Borel probability measure.

Example 1 a) Consider the real unit interval X = [0, 1] equipped with the σ-algebra A of Borelsubsets and the Lebesgues probability measure λ.

b) Consider Cantor space C = 0, 1N equipped with the σ-algebra B of Borel subsets and the canonical(=fair coin flip) probability measure γ: γ(~w C) = 2−|~w|, where |~w| = n denotes the length of~w = (w0, . . . , wn−1) ∈ 0, 1n.

c) The continuous total mapping ρb : C 3 b 7→∑j≥0 bj2

−j−1 ∈ [0, 1] realizes ([0, 1],A, λ) on (C,B, γ):([0, 1],A, λ) 4 (C,B, γ).

d) Consider the real line R equipped with (the Borel σ-algebra and) the standard Gaussian/normalprobability distribution, realized on λ via the partial mapping G : (0, 1) 3 t 7→ Φ−1(t) ∈ R for thecumulative distribution Φ : R 3 s 7→

∫ s−∞ exp(−t2/2)/

√2π dt ∈ [0, 1].

[SS06, Proposition 13] established that suitably and adaptively biased coin flips are universal:

Fact 2 To every 2nd countable T0 space X with Borel probability measure µ there exists a Borel proba-bility measure γ on C such that (X,µ) has a continuous partial realizer over (C, γ).

∗Supported by the National Research Foundation of Korea (grant NRF-2017R1E1A1A03071032), by the InternationalResearch & Development Program of the Korean Ministry of Science and ICT (grant NRF-2016K1A3A7A03950702), andby the European Union’s Horizon 2020 MSCA IRSES project 731143. Dedicated to the memory of Klaus Keimel whoin 2014 suggested to the last author to study the computability of cadlag functions. We thank Volker Betz and FrankAurzada for advice and assistance.

18

The metric case is treated in [HR09, Theorem 5.1.1]. Using transitivity, we show that the probabilitymeasure γ on C can in fact be chosen as the canonical ‘fair’ one:

Theorem 3 Every Borel probability measure γ on Cantor space C admits a continuous partial realizerover the ‘fair’ measure (C, γ). This realizer is defined on C with the exception of at most countably manypoints: One cannot hope for a total realizer in general.

2 Characterizing (Strong) Computability of Brownian Motion

Fix a Borel probability measure µ on X and a representation ξ :⊆ C X [Wei00, §3]. A ξ-realizer of µis a mapping G :⊆ C → dom(ξ) such that ξ G :⊆ C → X is a realizer of µ in the above sense. Call µ(strongly) ξ-computable if it has a computable ξ-realizer.

1D Brownian Motion [Gal16, Mal15] is a probability measure on the space X := C0[0, 1] of (i) con-tinuous functions W : [0, 1]→ R satisfying (ii) W (0) = 0 and characterized by the following properties:iii) For every 0 ≤ r < s < t ≤ 1, W (t)−W (s) is independent of W (r). iv) W (t)−W (s) is Gaussiannormally distributed with mean 0 and variance |t− s|.Here we approach the question of whether this measure is computable [Fou08, DF13] in the sense ofadmitting a computable [ρ→ρ]-realizer [Wei00, §6]: Does there exist a Turing machine producing, givena random sequence of fair coin flips, (I) the real values W (a/2n) on dyadic rationals a/2n, a, n ∈ N; and(II) a modulus of continuity) of some W : [0, 1]→ R distributed according to the Wiener Measure? Ourresult reduces the conjecture from the probability distribution on a function space to that of an ordinaryreal random variable:

Theorem 4 Let ω : [0, 1]×[1,∞)→ [0,∞) denote any computable (and thus continuous) one-parameterfamily of subadditive functions strictly increasing in both arguments with ω(0, C) ≡ 0. Suppose that toevery Wiener Process W (except for a subset of measure zero) there exists a (necessarily unique) leastC = C(W ) ≥ 1 such that ω( · , C) constitutes a modulus of continuity of W . Then the following areequivalent:

• The Wiener Process W is computable (formally: has a computable [ρ→ρ]-realizer).

• There exists a random variable c with computable probability distribution such that ω( · , c) is amodulus of continuity of W with probability 1.

References

[BGH15] Vasco Brattka, Guido Gherardi, and Rupert Holzl. Probabilistic computability and choice. Information andComputation, 242:249–286, 2015.

[Col15] Pieter Collins. Computable stochastic processes. arXiv, 1409.4667v2, 2015.

[DF13] George Davie and Willem L. Fouche. On the computability of a construction of Brownian motion. MathematicalStructures in Computer Science, 23:1257–1265, 12 2013.

[Fou08] Willem L. Fouche. Dynamics of a generic brownian motion: Recursive aspects. Theor. Comput. Sci., 394(3):175–186, 2008.

[Gal16] Jean-Francois Le Gall. Brownian Motion, Martingales, and Stochastic Calculus, volume 274 of GTM. Springer,2016.

[HR09] Mathieu Hoyrup and Cristobal Rojas. Computability of probability measures and Martin-Lof randomness overmetric spaces. Inform. and Comput., 207(7):830–847, 2009.

[Mal15] Adrian Maler. Effective Theory of Levy and Feller Processes. PhD thesis, Pennsylvania State University, 2015.

[SS06] Matthias Schroder and Alex Simpson. Representing probability measures using probabilistic processes. Journalof Complexity, 22(6):768–782, 2006.

[Wal77] Alastair J. Walker. An efficient method for generating discrete random variables with general distributions. ACMTrans. Math. Softw., 3(3):253–256, September 1977.

[Wei00] Klaus Weihrauch. Computable Analysis. Springer, Berlin, 2000.

19

Computing the limit set of planar differential equations

Daniel S. Graca

Universidade do Algarve, Portugal& SQIG/Instituto de Telecomunicacoes, Portugal

Ning Zhong

University of Cincinnati, U.S.A.

In many applications it is often necessary to study the long-term behavior of a given dynamicalsystem. This is in general a very challenge problem - it is still far from being completely understood,despite impressive progress in the last century.

Nevertheless for planar dynamical systems defined by autonomous ordinary differential equations(ODEs) their long-term behaviors are well understood qualitatively. A well-known result (the Poincare-Bendixson theorem. See e.g. [2] for a precise statement) shows that for a planar dynamical system,the limit set of each initial point either contains an equilibrium point or is a periodic orbit. Peixoto’stheorem (See e.g. [2]) gives further information by showing that structurally stable ODEs are “typical”in the sense that the set formed by such systems is dense in the class of C1 ODEs defined on the unitball, and that their limit sets consist only of a finite number of equilibrium points/periodic orbits.

The above results qualitatively characterize the asymptotic behavior of ODEs in the plane. However,in applications there is often a need for some quantitative data about the limit set, and such qualitativeresults usually may not yield much further insights in that respect. Therefore a natural question iswhether the limit set of a planar dynamical system is computable.

In [1] we have shown that the limit set of a planar dynamical system is not uniformly computable.However, in light of Peixoto’s theorem, we conjecture that the limit set of a structurally stable planardynamical system is computable:

Conjecture. The operator which maps each structurally stable system y′ = f(y) defined on the unit ballB(0, 1) ⊆ R2, where f is of class C1, to its limit set is computable.

We will present several techniques currently under our consideration aiming to prove the conjecture(work in progress).

References

[1] D. S. Graca and N. Zhong. Computability in planar dynamical systems. Natural Computing, 10:1295–1312, 2011.

[2] L. Perko. Differential Equations and Dynamical Systems. Springer, 3rd edition, 2001.

20

The Scott Model of PCF in Univalent Type Theory

Tom de Jong

University of Birmingham, United Kingdom

We report on the development of the Scott model of the programming language PCF in constructivepredicative univalent type theory. By tackling a well-known problem in theoretical computer science, westrengthen the position of univalent type theory as a feasible foundation for mathematics. Moreover, weshow that the Scott model can be developed in a constructive and predicative setting.

To constructively account for the non-termination in PCF, we work with the partial map classifiermonad (also known as the lifting monad) from topos theory [7], which has been extended to constructivetype theory by Reus and Streicher [8] and to univalent type theory by Knapp and Escardo [4, 6]. Ourresults show that lifting is a viable approach to partiality in univalent type theory. Other approaches[1, 3] to partiality either require some form of choice or higher inductive-inductive types. We show thatone can do without these extensions.

Capretta’s delay monad has been used to give a constructive approach to domain theory [2]. Thisapproach uses setoids, so that every object comes with an equivalence relation that maps must preserve.Instead, we use the identity type, as usual in univalent type mathematics. Moreover, we do not make useof Coq’s impredicative Prop universe and our treatment incorporates directed complete posets (dcpos)and not just ω-cpos.

Some of this material has been presented at TYPES 2019 in June.

Framework. We work in intensional Martin-Lof Type Theory with inductive types (including theempty 0, unit 1, natural numbers and identity types),

∑- and

∏-types, functional and propositional

extensionality and propositional truncation. We work predicatively, so we do not assume propositionalresizing. Although we only need univalence for propositions (propositional extensionality), we emphasisethe importance of the idea of h-levels, which is fundamental to univalent type theory.

PCF and the Scott model. PCF has a type ι for natural numbers and a function type σ ⇒ τfor every two PCF types σ and τ . Every natural number n is represented as the numeral n of type ι.Moreover, PCF has a fixed point operator. To model this, we work with directed complete posets (dcpos)with a least element.

The lifting L(X) of a type X is defined as∑

P :Ω(P → X), where Ω is a type universe of propositions(subsingletons). Note that we can embed X into L(X) by x 7→ (1, λt.x). If X is a set, then L(X) is adcpo and it has a least element given by (0, fromemptyX) [6].

We write JσK for the interpretation of a PCF type σ, and JtK : JσK for the interpretation of a PCFterm t : σ. In our model, JιK ≡ LN. The function type σ ⇒ τ is interpreted as the dcpo of continuousmaps from JσK to JτK. For a term s : σ ⇒ τ and a term t : σ, the application (st) : τ is a term, and isinterpreted as function application JsK(JtK).

The operational semantics of PCF induce a binary reduction relation .∗ on terms, where s .∗ tintuitively means that “s computes to t”. We show our Scott model to work well with the operationalsemantics through soundness and computational adequacy. Soundness means that if s.∗ t, then JsK = JtK.Computational adequacy states that for any PCF term t of type ι and natural number n, if JtK = JnK,then t .∗ n.

21

Characterising PCF propositions. Recall that PCF terms of type ι are interpreted as elements ofthe lifting of the natural numbers. Hence, the first projection yields a proposition for every such term.Soundness and computational adequacy allow us to characterise these propositions as those of the form∃n : N(t .∗ n), where t is a PCF term of type ι. Intuitively, these propositions are semidecidable, i.e.of the form ∃n1, . . . , nk : N(P (n1, . . . , nk)) where P is a decidable predicate on Nk. In proving this, weare led to study indexed W-types, a particular class of inductive types, and when they have decidableequality. Moreover, we provide some conditions on a relation for its k-step reflexive transitive closure tobe decidable.

Formalisation. We have fully formalised our results in Coq using the UniMath library [9]. The codemay be found at https://github.com/tomdjong/UniMath/tree/paper. We have also formalised thelifting monad and the model in Agda using Martın Escardo’s TypeTopology library [5] to rigorouslycheck universe levels, see https://www.cs.bham.ac.uk/~mhe/agda-new/ScottModelOfPCF.html. Thefull paper can be found at https://arxiv.org/abs/1904.09810.

Acknowledgements. I would like to thank Martın Escardo for suggesting and supervising this project.I have also benefited from Benedikt Ahrens’s support and his help with UniMath.

References

[1] Thorsten Altenkirch, Nils Anders Danielsson, and Nicolai Kraus. Partiality, revisited: The partiality monad as a quotientinductive-inductive type. In Javier Esparza and Andrzej S. Murawski, editors, Foundations of Software Science andComputation Structures, pages 534–549. Springer Berlin Heidelberg, 2017.

[2] Nick Benton, Andrew Kennedy, and Carsten Varming. Some domain theory and denotational semantics in coq. InLecture Notes in Computer Science, pages 115–130. Springer Berlin Heidelberg, 2009.

[3] James Chapman, Tarmo Uustalu, and Niccolo Veltri. Quotienting the delay monad by weak bisimilarity. MathematicalStructures in Computer Science, 29(1):67–92, 2017.

[4] Martın H. Escardo and Cory M. Knapp. Partial elements and recursion via dominances in univalent type theory. InValentin Goranko and Mads Dam, editors, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017),volume 82 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1–21:16. Schloss Dagstuhl–Leibniz-Zentrum fur Informatik, 2017.

[5] Martın Hotzel Escardo et al. TypeTopology — Various new theorems in constructive univalent mathematics written inAgda. https://github.com/martinescardo/TypeTopology.

[6] Cory Knapp. Partial Functions and Recursion in Univalent Type Theory. PhD thesis, School of Computer Science,University of Birmingham, June 2018.

[7] Anders Kock. Algebras for the partial map classifier monad. In Lecture Notes in Mathematics, pages 262–278. SpringerBerlin Heidelberg, 1991.

[8] B. Reus and Th. Streicher. General synthetic domain theory — a logical approach (extended abstract). In Eugenio Moggiand Giuseppe Rosolini, editors, Category Theory and Computer Science, pages 293–313. Springer Berlin Heidelberg,1997.

[9] Vladimir Voevodsky, Benedikt Ahrens, Daniel Grayson, et al. UniMath — a computer-checked library of univalentmathematics. https://github.com/UniMath/UniMath.

22

Haar Measure in Synthetic Topology

Davorin Lesnik

We construct the Haar measure in the framework of synthetic topology, and discuss measures in thissetting in general.

Classically, the Haar measure is a left-invariant non-trivial positive measure on a Hausdorff locallycompact topological group, unique up to a positive scalar factor.

1 Setting

We work in the usual setting of synthetic topology, ie. a topos with the subobject classifier (“set of truthvalues”) Ω and a chosen Σ ⊆ Ω, closed under finite ∧ (“set of open truth values”), which equips everyobjects with its intrinsic topology. Additionally we postulate:

• complements of open/closed subsets are closed/open,

• N exists, is overt and satisfies the Scott’s principle,

• the Cantor set 2N is compact.

2 Borel Reals

Definition 2.1. For Γ ⊆ Ω, closed under finite ∧, and a set X, define PΓ(X) to be the set of subsetsof X, classified by Γ. If X comes equipped with sufficiently nice relations ≤ and , define cuts on it:

left cuts:

LΓ(X) :=L ∈ PΓ(X)

∣∣ L inhabited ∧ L = ↓↓L ∧ ∀a, b ∈ L.∃x ∈ L. a, b ≤ x,

right cuts:

RΓ(X) :=R ∈ PΓ(X)

∣∣ R inhabited ∧R = ↑↑R ∧ ∀a, b ∈ R.∃x ∈ R. a, b ≥ x,

two-sided cuts:

TΓ(X) :=

(L,R) ∈ LΓ(X)×RΓ(X)∣∣ ∀a ∈ L.∀b ∈ R. a b

.

For (L,R) ∈ TΓ(X) define that it is right maximal when ∀(L′, R′) ∈ TΓ(X). L ⊆ L′ ⇒ R ⊇ R′. Defineleft maximal analogously. Then define Dedekind cuts as

DΓ(X) := (L,R) ∈ TΓ(X) | (L,R) is left and right maximal .In the usual models R = DΣ(Q). We also have the left and the right reals R−→ = LΣ(Q) and

R←− = RΣ(Q).Defining Dedekind cuts via maximality rather than locatedness allows us to construct further variants

of reals, including the Borel reals BR := DB(Q), where B is the set of Borel truth values (closure of Σunder negation and countable joins).

Every left Borel cut L ∈ LB(Q) can be completed to a Borel Dedekind cut (L, R), where L and L“have the same supremum”. Similarly for right cuts. Operations on BR generally require completions of

cuts, for example (L′, R′) + (L′′, R′′) = (L′ + L′′, R′ +R′′). The Borel reals form an ordered field (x ∈ BRis invertible ⇔ |x| > 0).

23

3 Haar Measure on a Compact Hausdorff Group

Let G be an overt compact Hausdorff group. Let O(G) denote the intrinsic topology of G and SCmp(G)the set of subcompact subsets of G (those K ⊆ G, for which the truth value of K ⊆ U is open for everyU ∈ O(G)) which are saturated (ie. K =

⋂ U ∈ O(G) | K ⊆ U).Assume we have an indexing N → O(G) × SCmp(G), i 7→ (Ui,Ki), such that each Ui is inhabited,

Ui ⊆ Ki, and the image of the indexing is dense in the following sense: for every U ∈ O(G) and inhabitedK ∈ SCmp(G), if K ⊆ U , then there exists i ∈ N such that K ⊆ Ui ⊆ Ki ⊆ U (it follows that (G,G)is indexed). Additionally, assume the pairs (Ui,Ki) are maximal: for all U ∈ O(G) and K ∈ SCmp(G)with U ⊆ K, if Ui ⊆ U , then Ki ⊆ K, and if K ⊆ Ki, then U ⊆ Ui. (The classical analogue of this is acompact Hausdorff countably based topological group.)

Every Borel subset S ⊆ G has its indicator function indS : G→ BR, given by

indS(x) :=(q ∈ Q

∣∣ q < 1 ∧ x ∈ S ∨ q < 0,q ∈ Q

∣∣ q > 0 ∧ x ∈ S ∨ q > 1).

If U ∈ O(G), then indU maps into R−→. If K ∈ SCmp(G), then indK maps into R←−.

For U ∈ O(G), K ∈ SCmp(G) and i ∈ N define

N−→i(U) :=q ∈ Q

∣∣∣ ∃(λ, a) ∈ (Q×G)∗. q <∑

k

λk ∧ ∀x ∈ G.∑

k

λkindakKi(x) < indU (x),

N←−i(K) :=q ∈ Q

∣∣∣ ∃(λ, a) ∈ (Q×G)∗. q >∑

k

λk ∧ ∀x ∈ G.∑

k

λkindakUi(x) > indK(x),

κ−→(U) :=q ∈ Q

∣∣∣ ∃i ∈ N. q <N−→i(U)

N←−i(G)

, κ←−(K) :=

q ∈ Q

∣∣∣ ∃i ∈ N. q >N←−i(K)

N−→i(G)

.

We have N−→i(U), κ−→(U) ∈ R−→ and N←−i(K), κ←−(K) ∈ R←−.

For any i ∈ N define κ(i) :=(κ−→(Ui), κ←−(Ki)

). Expand κ linearly to (Q × N)∗, then to the Borel

Dedekind completion L1(G) := DB((Q× N)∗

). Here (Q× N)∗ is a vector lattice over Q, equipped with

order relations(q′, i′) ≤ (q′′, i′′) := ∀x ∈ G.

∑

k

λkindUi′k

(x) ≤∑

k

λkindUi′′k

(x),

(q′, i′) (q′′, i′′) := ∀x ∈ G.∑

k

λkindKi′k

(x) <∑

k

λkindUi′′k

(x),

and L1(G) is a vector lattice over BR (to define addition, one needs to complete the cuts, like on BR).

Theorem 3.1. The above gives a well-defined left invariant BR-linear functional κ : L1(G)→ BR. The cutrepresenting the indicator function of G gets mapped to 1; in this sense κ is a probability measure on G.It restricts to a left invariant R-linear functional RG → R.

4 Generalizations

We discuss some generalizations: Haar measure not probabilistic, G not compact (only locally compactin a suitable sense), the indexing (Ui,Ki) not indexed by N (ie. G not necessarily countably based),weaker assumptions on the synthetic topological model.

24

The Hyperspace Dimension Theorem∗

Jack H. Lutz1 and Elvira Mayordomo2

1Department of Computer Science, Iowa State University, Ames, IA 50011, USA2Departamento de Informatica e Ingenierıa de Sistemas, Instituto de Investigacion en Ingenierıa de

Aragon, Universidad de Zaragoza, 50018 Zaragoza, Spain

We prove a general hyperspace dimension theorem. Let X be a separable metric space, and let K(X)be the hyperspace of X, i.e., the set of all nonempty compact subsets of X endowed with the Hausdorffmetric. For each gauge family (Hausdorff family of gauge functions [7]) ϕ, we define a jump ϕ of ϕthat is also a gauge family. Our hyperspace dimension theorem says that, for every subset E of X, theϕ-gauged Hausdorff dimension of K(E) in K(X) is at most the ϕ-gauged packing dimension of E in X.A few very special cases of this theorem were previously known [6, 1].

The logical structure of our proof is of particular interest. We first extend two algorithmic fractaldimensions – computability-theoretic versions of classical Hausdorff and packing dimensions that assigndimensions dim(x) and Dim(x) to individual points x ∈ X – to arbitrary separable metric spaces andto arbitrary gauge families. We then extend the point-to-set principle of J. Lutz and N. Lutz [2]) inthese same two ways. The resulting point-to-set principle says that, for every separable metric space X,every gauge family ϕ, and every subset E of X, the ϕ-gauged Hausdorff dimension of E is the minimum,for all oracles A ⊆ N, of the supremum, for all points x ∈ E, of the ϕ-gauged dimension of x relativeto the oracle A. Finally we use this principle to prove our hyperspace dimension theorem. This isone of a handful of cases – all very recent [4, 3, 5] and all using the point-to-set principle – in whichcomputability-theoretic methods have been used to prove new theorems in classical geometric measuretheory, theorems whose statements do not involve computability theory or logic.

References

[1] Manav Das. The upper entropy index of a set and the Hausdorff dimension of its hyperspace. Monatshefte furMathematik, 166:371–378, 2012.

[2] Jack H. Lutz and Neil Lutz. Algorithmic information, plane Kakeya sets, and conditional dimension. ACM Transactionson Computation Theory, 10(2):7:1–7:22, 2018.

[3] Neil Lutz. Fractal intersections and products via algorithmic dimension. In 42nd International Symposium on Math-ematical Foundations of Computer Science, MFCS 2017, August 21-25, 2017 - Aalborg, Denmark, pages 58:1–58:12,2017.

[4] Neil Lutz and Donald M. Stull. Bounding the dimension of points on a line. In Theory and Applications of Modelsof Computation - 14th Annual Conference, TAMC 2017, Bern, Switzerland, April 20-22, 2017, Proceedings, pages425–439, 2017.

[5] Neil Lutz and Donald M. Stull. Projection theorems using effective dimension. In 43rd International Symposium onMathematical Foundations of Computer Science, MFCS 2018, August 27-31, 2018, Liverpool, UK, pages 71:1–71:15,2018.

[6] Mark McClure. The Hausdorff dimension of the hyperspace of compact sets. Real Analysis Exchange, 22:611–625,1996/1997.

[7] C. A. Rogers. Hausdorff Measures. Cambridge University Press, 1970.

∗We thank the National University of Singapore Institute for Mathematical Sciences for their support during the programEquidistribution: Arithmetic, Computational and Probabilistic Aspects, where part of this research was carried out. Thefirst author’s research was supported in part by National Science Foundation grants 1247051 and 1545028.

25

A constructive predicative realizability topos for inductively generated

formal topology

Maria Emilia Maietti and Samuele Maschio

In previous work in [MM] we produced a predicative version of Hyland’s effective topos ([Hyl82]),called pEff , formalizable in Feferman’s classical predicative theory of non-iterated fixpoints ID1 ([Fef79]).

This construction was based on the elementary quotient completion in [MR12] performed on a pred-icative version of a tripos in [HJP80] built out of the realizability semantics in [IMMS18]. Such asemantics was introduced to make evident how proofs in the Minimalist Foundation in [MS05, Mai09]can be turned into programs. Indeed it validates the intensional level of the Minimalist FoundationmTT in [Mai09] extended with the axiom of choice and formal Church’s thesis in such a way that theinterpretation of arithmetics coincides with Kleene realizability.

Here we adopt the same categorical constructions to produce a constructive predicative ver-sion of Hyland’s Effective Topos, also capable of formalizing the extensional version of the MinimalistFoundation extended with inductively generated formal topologies in [CSSV03], called MFind.

To this purpose we build a predicative realizability topos cpEff by applying the elementary quotientcompletion construction to a predicative tripos built out of the realizability semantics in [MMR], whichextends that in [IMMS18] but also validates some inductive definitions sufficient to formalize inductivelygenerated formal topologies in [CSSV03].

The novelty of such a semantics is that it is formalizable in a constructive metatheory as Aczel’sconstructive set theory CZF ([AR01]) extended with the Regular Extension Axiom REA. However, theconstruction of cpEff is not strictly predicative a la Feferman, contrary to that of pEff .

A main application of the constructive nature of cpEff is that not only we can embed it into Eff butalso in an instance of the construction of an effective Heyting pretopos with small maps in [vdBM11]when it is performed on the syntactic category of CZF + REA.

Another crucial application of cpEff is that of including a representation of Joyal’s point-free topol-ogy of Dedekind real numbers and hence the possibility of extracting programs from point-free construc-tive analysis within MFind.

References

[AR01] P. Aczel and M. Rathjen. Notes on constructive set theory. Mittag-Leffler Technical Report No.40, 2001.

[CSSV03] T. Coquand, G. Sambin, J. Smith, and S. Valentini. Inductively generated formal topologies. Annals of Pureand Applied Logic, 124(1-3):71–106, 2003.

[Fef79] S. Feferman. Constructive theories of functions and classes. In Logic Colloquium ’78 (Mons, 1978), Stud. LogicFoundations Math., pages 159–224, Amsterdam-New York, 1979. North-Holland.

[HJP80] J. M. E. Hyland, P. T. Johnstone, and A. M. Pitts. Tripos theory. Bull. Austral. Math. Soc., 88:205–232, 1980.

[Hyl82] J. M. E. Hyland. The effective topos. In The L.E.J. Brouwer Centenary Symposium (Noordwijkerhout, 1981),volume 110 of Stud. Logic Foundations Math., pages 165–216. North-Holland, Amsterdam-New York,, 1982.

[IMMS18] H. Ishihara, M.E. Maietti, S. Maschio, and T. Streicher. Consistency of the intensional level of the minimalistfoundation with church’s thesis and axiom of choice. Arch. Math. Log., 57(7-8):873–888, 2018.

[Mai09] M. E. Maietti. A minimalist two-level foundation for constructive mathematics. Annals of Pure and AppliedLogic, 160(3):319–354, 2009.

[MM] M. E. Maietti and S. Maschio. A predicative variant of Hyland’s effective topos. Submitted. arXiv:1806.08519.

26

[MMR] M. E. Maietti, S. Maschio, and M. Rathjen. A realizability semantics for inductive formal topologies, Church’sthesis and axiom of choice. Submitted. arXiv:1905.11966.

[MR12] M. E. Maietti and G. Rosolini. Elementary quotient completion. Theory and Applications of Categories, 27,2012.

[MS05] M. E. Maietti and G. Sambin. Toward a minimalist foundation for constructive mathematics. In L. Crosilla andP. Schuster, editor, From Sets and Types to Topology and Analysis: Practicable Foundations for ConstructiveMathematics, number 48 in Oxford Logic Guides, pages 91–114. Oxford University Press, 2005.

[vdBM11] Benno van den Berg and Ieke Moerdijk. Aspects of predicative algebraic set theory, II: Realizability. Theoret.Comput. Sci., 412(20):1916–1940, 2011.

27

Axiomatic Reals in Type Theory for Program Extraction ∗

Sewon Park †

School of Computing, KAIST

There are various ways of introducing real numbers in programming languages; one is to constructreals as sequences of approximations and another is to add a datatype for reals and their arithmeticas primitives in the languages. The later approach can be found in [5], [9], [6], [2], and [3]. Takingthe later approach, making the primitives’ semantics computable becomes essential: testing inequalitiesbeing partial and multivalued branching being provided.

The goal of this work is to extend type theories thus that partiality and multivaluedness can beexpressed; hence, alternative axioms for reals whose corresponding primitives, considering Curry-Howardcorrespondence, are computable can be added. At the end, certified programs using a primitive datatypefor reals get extracted from proofs in the type theories.

Whereas real numbers are usually constructed co-inductively in type theories (e.g., see [4]), Coq’sstandard library for real numbers [8] is one example of the axiomatic approach; a type for reals, relatedoperations and properties are assumed in Coq. However, using the library is not suitable since it assumesaxioms whose corresponding primitives are not computable.

In order to do classical reasoning for reals, we want weak disjunction ∨ and weak existence ∃, separatefrom constructive ∨ and ∃. For example, when R is the type for reals, ∀x y:R x > y ∨ x = y ∨ y > x isinhabited but ∀x y:R x > y ∨ x = y ∨ y > x is not. We start with a type theory with either truncation [1]or Coq’s Prop universe with proof irrelevance.

sign(x, ε) := tt if x > −ε | ff if ε > x is one example of multivalued branching in a program usinga primitive datatype for reals. A naive approach is to make the type ∀x,ε:R ε > 0 → x > −ε ∨ ε > xinhabited 1. But the attempt fails since it reduces multivaluedness. This gives us a reason for introducingseparate multivalued types:

Axiom 1 (multivalued types). Given any type T , its multivalued type is mv T . Given any t : T wehave ι t : mv T . For any function f : A → mv B we can lift its domain lift f : mv A → mv B such thatlift f (ι t) :≡ f t. mv (mv T ) is isomorphic to mv T so that there is isoT : mv (mv T ) → mv T whereisoT (ι t) :≡ t for all t : mv T .

We define multivalued existence ∃x:AB(x) by mv(∃x:AB(x)

)and multivalued disjunction A ∨ B by

mv (A ∨B). Now, we derive p ∨ q from p ∨ q for certain p and q.

Axiom 2 (partiality type). S is a type with two known terms ↓, ↑: S . Let s ↓:= s =↓ as an abbreviation.The following is assumed. select : ∀s1,s2:S s1 ↓ ∨ s2 ↓ → s1 ↓ ∨ s2 ↓

What select says is that for any given two partialities s1 and s2, when we weakly know at least oneof the two is defined, we can multivalued-ly select one that is defined.

Axiom 3 (real numbers). R is a type with two known terms 0,1 : R. It is equipped with the followingfunctions: add,mul, d : R → R → R, sub : R → R, div : ∀x:Rx 6= 0 → R and gt : R → R → Prop. We use

∗This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government(MSIT) (No. NRF-2016K1A3A7A03950702 & No. NRF-2017R1E1A1A03071032) and the grant funded by the Koreagovernment (MOE) (No. NRF-2017R1D1A1B05031658) .†Email address: swelite@kaist.ac.kr1this is true when R is constructed and can be derived from the Approximate Splitting Principle in [7].

28

the following abbreviations: x+ y := add x y, x ∗ y := mul x y, x− y := x+ sub y, x÷p y := x ∗ div p y,x > y := gt x y, and |x| := d x 0. Moreover, let us abbreviate a sequential addition of 1 for n times forany natural number n > 0 as n.

Classical axioms are assumed in Prop and yield n 6= 0 for any n. Hence, let x/n denote x÷p n wherethe proof irrelevance helps. Inductively dividing 1 by 2, we define the precision embedding prec : N → Rwhere N is the type of natural numbers. The followings are assumed:

– semi : ∀x y:R ∃s:S s ↓↔ x > y

– lim : ∀p:R→Prop(∃!z:R p z)→ (∀n:N∃x:R∃z:R p z ∧ prec n > d x z)→ ∃z:R p zExample 4 (Multivalued Approximate Splitting Principle). From classical axioms, we can prove x >y− ε ∨ y > x− ε when ε > 0. Using semi and select, we get x > y− ε ∨ y > x− ε. Hence, we can definethe term comp : ∀x y ε:R ε > 0→ x > y − ε ∨ y > x− ε.Example 5 (Maximum of two real numbers). Define M(x, y, z) := x > y → z = x ∧ x = z → z =x ∧ y > x→ z = y. Assume any x y : R and n : N. Then, ∃!z:R M(x, y, z) can be proven using classicalaxioms. When x > y − prec n, prove x approximates the maximum and when y > x − prec n, prove ydoes. Induction on x > y − prec n ∨ y > x − prec n with ι yields a function f : (x > y − prec n ∨ y >x− prec n)→ ∃z:R∃!u:RM(x, y, u) ∧ prec n < d z u. Hence, applying lift f with comp and the uniquenesson lim yields max : ∀x y :R∃z:R M(x, y, z).

Example 6 (Intermediate Value Theorem). Define uniq(f, x, y) := ‘continuous real function f weaklyhas an unique root in (x; y) and f y > 0 > f x.’ From classical axioms, prove weak IVT. Let a :=(2 ∗ x + y)/3 and b := (x + 2 ∗ y)/3. Assuming uniq(f, x, y), prove f y > 0 > f a → uniq(f, a, y) andf b > 0 > f x → uniq(f, x, b). Prove f y > 0 > f a ∨ f b > 0 > f x using classical axioms, semiand select. Together with lift and induction on N, we have multivalued root refinement. Since the rootis unique, applying the fact and the multivalued root refinement to lim yields IVT : ∀f :R→Runiq(f, 0, 1)→∃z f z = 0 ∧ 1 > z > 0.

By erasing terms of types in Prop and mapping assumed symbols properly, we get a program writtenin a programming language with primitive datatypes for reals. Note that mv A should be mapped to themapping of A since multivaluedness will not be distinguished in programming languages. For example,comp will yield a program computing multivalued comparison, max will yield a program computing themaximum of two real numbers and, IVT will yield a program computing the root of a real function usingtrisection algorithm.

References

[1] Awodey, S., Bauer, A.: Propositions as [types]. Journal of logic and computation 14(4), 447–471 (2004)

[2] Bauer, A., Park, S., Simpson, A.: Command-like expressions for real infinite-precision calculations. In: Muller, N.T.,Rump, S.M., Weihrauch, K., Ziegler, M. (eds.) Reliable Computation and Complexity on the Reals (Dagstuhl Seminar17481). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018)

[3] Brauße, F., Collins, P., Kanig, J., Kim, S., Konecny, M., Lee, G., Muller, N., Neumann, E., Park, S., Preining, N.,et al.: Semantics, logic, and verification of” exact real computation”. arXiv preprint arXiv:1608.05787 (2016)

[4] Cruz-Filipe, L., Geuvers, H., Wiedijk, F.: C-corn, the constructive coq repository at nijmegen. In: InternationalConference on Mathematical Knowledge Management. pp. 88–103. Springer (2004)

[5] Escardo, M.H.: Pcf extended with real numbers. Theoretical Computer Science 162(1), 79–115 (1996)

[6] Escardo, M.H., Simpson, A.: Abstract datatypes for real numbers in type theory. In: Rewriting and Typed LambdaCalculi, pp. 208–223. Springer (2014)

[7] Schwichtenberg, H.: Constructive analysis with witnesses. Proof Technology and Computation. Natio Science Series pp.323–354 (2006)

[8] development team, T.C.: The Coq proof assistant reference manual. LogiCal Project (2004), http://coq.inria.fr,version 8.0

[9] Tucker, J.V., Zucker, J.I.: Abstract versus concrete computation on metric partial algebras. ACM Transactions onComputational Logic (TOCL) 5(4), 611–668 (2004)

29

Wadge Hierarchy in Quasi-Polish Spaces

Victor Selivanov

A.P. Ershov Institute of Informatics Systems SB RASand Kazan Federal University

vseliv@iis.nsk.su

The classical Borel, Luzin, and Hausdorff hierarchies in Polish spaces, which are defined usingset-theoretic operations, play an important role in descriptive set theory (DST) which exists formore than a century already. Recently, these hierarchies were extended and shown to have niceproperties also for quasi-Polish spaces [1] which include many non-Hausdorff spaces of interestfor several branches of mathematics and theoretical computer science.

The Wadge hierarchy, introduced in [6], is non-classical in the sense that it is based on a notionof reducibility that was not recognized in the classical DST, and on using ingenious versions ofGale-Stewart games rather than the set-theoretic operations. For subsets A,B of the Baire spaceN = ωω, A is Wadge reducible to B (A ≤W B), if A = f−1(B) for some continuous function f onN . The quotient-poset of the preorder (P (N );≤W ) under the induced equivalence relation ≡Won the power-set of N is called the structure of Wadge degrees in N . W. Wadge [6] characterisedthe structure of Wadge degrees of Borel sets (i.e., the quotient-poset of (B(N );≤W )) up toisomorphism. In particular, this quotient-poset is semi-well-ordered, hence it is well-founded andhas no 3 pairwise incomparable elements.

This gives rise to the Wadge hierarchy Σα(N )α<ν (for a rather large ordinal ν) in N whichis a great refinement of the Borel hierarchy. The Wadge hierarchy was originally defined only forthe Baire space. Here we attempt to find the “correct” extension of the Wadge hierarchy fromthe Baire space to arbitrary quasi-Polish spaces.

A straightforward approach to this problem would be to show that Wadge reducibility in suchspaces behaves similarly to its behaviour in the Baire space, i.e. it is a semi-well-order. Unfortu-nately, this is not the case: for many natural quasi-Polish spaces X the structure (B(X);≤W ))in not well-founded and has antichains with more than 2 elements.

The second approach to the problem was independently suggested in [4, 5]. The approach isbased on the characterization of quasi-Polish spaces as the countably based T0-spaces X such thatthere is a continuous open surjection ξ from N onto X [1]. Namely, one can define the Wadgehierarchy Σα(X)α<ν in X by Σα(X) = A ⊆ X | ξ−1(A) ∈ Σα(N ). One easily checks that thedefinition of Σα(X) does not depend on the choice of ξ,

⋃α<ν Σα(X) = B(X), Σα(X) ⊆∆β(X)

for all α < β < ν, and any Σα(X) is downward closed under the Wadge reducibility on X. Thisdefinition is short and elegant but it gives no real understanding of how the levels look like.

The third possible approach (traditional to the classical DST) proposed in [5] is to apply a re-finement process according to which one starts with the Borel hierarchy and subsequently definessuitable “natural” refinements of the hierarchies already available. At the first step of this processwe obtain the Hausdorff hierarchies over each level of the Borel hierarchy thorouphly investigatedin [1]. Further refinements may be done using more sophisticated set-theoretic operations whichextend and modify some operations introduced in [6] for the Baire space. In this way we described

30

in [5] an increasing sequence of pointclasses Σα(X)α<λ, λ = supω1, ωω11 , ω

ωω11

1 , . . . which ex-haust the sets of finite Borel rank, and we claimed their coincidence with the corresponding classesfrom the second approach.

Here we not only prove this claim but also extend it to the whole Wadge hierarchy Σα(X)α<υin quasi-Polish spaces. Our characterisation of this hierarchy is based on a new characterizationof the Wadge hierarchy in the Baire space space from [3].

The characterization in [3] concerns not only hierarchies of sets discussed so far but also moregeneral hierarchy of functions A : N → Q from the Baire space to an arbitrary countable bqo Q[2]. We identify such functions with Q-partitions of N of the form A−1(q)q∈Q in order to stresstheir close relation to k-partitions (obtained when Q = k = 0, . . . , k − 1 is an antichain withk-elements) studied by several authors.

In fact, our set-theoretical characterisation of all the classes Σα(X)α<υ from the secondapproach holds not only for sets but also for k-partitions, k ≥ 2. Sets correspond to 2-partitions.

References

[1] de Brecht, M.: Quasi-Polish spaces, Annals of pure and applied logic, 164, (2013), 356–381.

[2] van Engelen, F., Miller, A., Steel, J.: Rigid Borel sets and better quasiorder theory. Contemporary mathematics,65 (1987), 199–222.

[3] Kihara, T., Montalban, A.: On the structure of the Wadge degrees of BQO-valued Borel functions. ArXiv:1705.07802 v1 [Math.LO] 22 May 2017.

[4] Pequignot, Y.: A Wadge hierarchy for second countable spaces. Archive for Mathematical Logic 54.5-6 (2015), pp.659–683. doi: 10.1007/s00153-015-0434-y.

[5] Selivanov, V.L.: Towards a descriptive theory of cb0-spaces. Mathematical Structures in Computer Science. v.28 (2017), issue 8, 1553–1580. DOI: http://dx.doi.org/10.1017/S0960129516000177. Earlier version in: ArXiv:1406.3942v1 [Math.GN] 16 June 2014.

[6] Wadge, W.: Reducibility and Determinateness in the Baire Space. PhD thesis, University of California, Berkely,1984.

31

The Compact Hyperspace Monad, a Constructive Approach∗

Dieter Spreen

Department of Mathematics, University of Siegen57068 Siegen, Germany

As is well known, the collection of all non-empty compact subsets of a compact Hausdorff space is a compactHausdorff space again with respect to the Vietoris topology. Moreover, the functor mapping a compact Hausdorffspace to its hyperspace is a monad.

In this talk a constructive version of the result will be presented. We work in intuitionistic logic extended byinductive and co-inductive definitions (cf. Berger [1, 2]). As in Berger/Spreen [3], only compact Hausdorff spacesare considered that come equipped with a distinguished finite set of continuous endo-functions the ranges of whichcover the underlying space. Classically, they can be characterised co-inductively in a natural way. We only workwith the co-inductive characterisations.

In [2] a co-inductive inductive characterisation of the uniformly continuous maps f : r0, 1sn Ñ r0, 1s has beengiven. It can be generalised to the more general case considered in the talk and is used to define the hom-sets ofthe category under consideration.

The result in this paper is part of an ongoing research programme started by U. Berger. The central observationis that by constructively reasoning on the basis of co-inductive and/or inductive definitions computational contentis derived. Reducibility facilitates the extraction of algorithms from the corresponding proof. The frameworkpresented here in particular allows to deal with compact-valued maps and their selection functions. Maps of thiskind abundantly occur in applied mathematics. They have applications in areas such as optimal control andmathematical economics, to mention a few. In addition, they are used to model non-determinism.

References[1] U Berger, Realisability for induction and coinduction with applications to constructive analysis, Journal of Universal Computer

Science 16 (18) (2010) 2535–2555.

[2] U Berger, From coinductive proofs to exact real arithmetic: theory and applications, Logical Methods in Computer Science 7(1)(2011) 1–24, doi: 10.2168/LMCS7(1:8)2011.

[3] U Berger, D Spreen, A coinductive approach to computing with compact sets, Journal Logic & Analysis 8 (3) (2016) 1–35, doi:10.4115/jla.2016.8.3.

∗This project has received funding from the European Unions Horizon 2020 research and innovation programme under the MarieSklodowska-Curie grant agreement No 731143.

32

Tensors of Quantitative Truth

and the Combination of Algebraic Effects

Niels Voorneveld∗

University of Ljubljana

Functional programming provides a paradigmatic example of computations with infinite data (namelyfunctions). Furthermore, real world functional programming languages with effects provide a usefulprogramming style for programming with infinite data structures such as streams.

Program equivalence is a useful notion for functional programming languages, as it allows us toidentify if two programs have the same behaviour, and can thus be exchanged without consequence.Following the programme of [4], such an equivalence can be specified by a logic of behavioural properties,where two programs are equivalent if they satisfy the same properties. However, for some (combinationsof) effects it is more natural, and sometimes even necessary, to generalise the logic to a quantitativelogic [5].

We will look at a functional programming language with general recursion and algebraic effects, whereeffects are represented by algebraic operators op : αn → α. For instance, in the case of probability, wehave an operator or : α2 → α where or(M,N) is the computation which has equal probability ofevaluating M as it has for evaluating N . For each individual effect, a natural quantitative truth spacecan be chosen to reflect program behaviour. Such a truth space can often be chosen to be a distributivecomplete lattice, where the order gives a notion of implication, and suprema and infima give quantitativenotions of disjunctions and conjunctions.

Using a truth space A, a program M will satisfy some formula φ to a certain degree (M |= φ) ∈ A. Inthe case of probability, the truth space can be given by the real number interval [0, 1] with the standardordering, consisting of probabilities of satisfaction. If a program interacts with a global store, whosestates are given by a set S, a natural truth space is P(S) with inclusion order, containing sets of states(e.g. preconditions and postcondintions) for satisfaction. One may also model cost, where properties aresatisfied under certain prices from truth space N ∪ ∞. We give this space the reverse ordering, sincecheaper is better.

Though [5] features some examples of logics for combinations of effects, the truth spaces were con-structed separately for each case. The main contribution of this talk is a uniform way of combining effects,based on combining the truth spaces with he following tensor construction (featured e.g. in [1, 3, 6]):

Definition 1 For A and B two complete lattices, there is a complete lattice A⊗B given byS ⊆ A×B | ∀a ⊂ A, b ⊂ B, a× b ⊆ S ⇔ (

∨a,∨b) ∈ S with inclusion order.

The operation ⊗ on complete lattices is a symmetric monoidal product, and the Booleans are a unit forthis operation. It also holds that the 2-linear (or bilinear) functions of space A×B → C correspond tothe 1-linear (or linear) functions of space A⊗B → C, where in general f : A1×· · ·×An → B is n-linearif suprema are preserved in each of the n arguments separately.

In the case of probability with global store, [0, 1]⊗P(S) is equivalent to the function space S → [0, 1]with pointwise order. An element f in this space gives for each starting state s ∈ S, an expectation f(s)

∗Supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0326, and by EU-MSCA-RISE project 731143 (CID).

33

that a property is satisfied. In general, we can see elements of A ⊗ B as a set of pairs of truth valueswhich can be implied by the program, where more truth in one direction might result in less truth inthe other. For instance, for probability with cost, you might have to pay more to get a higher likelihoodof termination.

In the logic of [5], quantitative modalities q are used to lift quantitative formulas φ on values toquantitative formulas q(φ) on computations. This is done by way of a function JqK : TA → A whichtransforms an effect tree, whose nodes are effect operators and leaves are truth values from A, into atruth value. Sufficiently nice modalities q can be specified locally by giving for each effect operatorop : αn → α an n-linear function qop : An → A. For instance, for probability with an expectationmodality E, Eor : [0, 1]2 → [0, 1] is given by the function (a, b) 7→ (a + b)/2 computing the averageprobability of the two possible continuations.

When combining effects, such modalities can be combined as well. For example, for probability withmodality E and global store with modality G (identifying correct starting states), the combined modalityE⊗ G can locally at the or operator be described as the function (E⊗ G)or : (S → [0, 1])2 → (S → [0, 1])defined by post-composition with the function (a, b) 7→ (a + b)/2. In general, each modality fromone theory can be combined with each modality from the other theory, following an elegant definitionexploiting the universal property of the tensor construction.

In summary, using this tensor construction on truth spaces gives us a convenient uniform way ofcombining effects and their behavioural properties. In this talk, we will explore the naturality of thisconstruction in more detail and see how it can be applied to a variety of examples. We will also seehow it relates to the logic from [5], which allows us to establish that the quantitative logic specifies acongruent (substitution invariant) program equivalence. A comparison to a classic account of combiningeffects as defined in [2] is a subject for future research.

References

[1] Marcel Erne and Jorge Picado. Tensor products and relation quantales. Algebra universalis, 78(4):461–487, Dec 2017.

[2] Martin Hyland, Gordon Plotkin, and John Power. Combining effects: Sum and tensor. Theor. Comput. Sci., 357(1):70–99, July 2006.

[3] Bart Jacobs. Semantics of weakening and contraction. Annals of Pure and Applied Logic, 69(1):73 – 106, 1994.

[4] Alex Simpson and Niels Voorneveld. Behavioural equivalence via modalities for algebraic effects. In ESOP 27, pages300–326, 2018.

[5] Niels Voorneveld. Quantitative logics for equivalence of effectful programs. In MFPS XXXV, 2019.

[6] Rudolf Wille. Tensorial decomposition of concept lattices. Order, 2(1):81, 1985.

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A Gentzen-style translation of Godel’s System T

Chuangjie Xu

Mathematisches Institut, Ludwig-Maximilians-Universitat MunchenTheresienstr. 39, 80333 Munich, Germany

xu@math.lmu.de

The author introduces in [5] a syntactic translation to reveal the continuity structure of functionsNN → N that are definable in Godel’s System T1. The idea is to translate natural numbers into functionalsfrom the Baire space NN, while function spaces are translated recursively. This is exactly in the spiritof Gentzen’s negative translation [1] of classical logic into minimal logic.

Negative translations have been generalized by replacing ¬¬ by a nucleus [4]. Similarly, we introducea notion of nucleus relative to T to generalize our syntactic translation. A nucleus is an endofunction Jon types equipped with terms

η : ρ→ Jρ (·)κ : (σ → Jρ)→ Jσ → Jρ

satisfying the equations of Kleisli extension; thus J is a strong monad. Now we are ready to translate Textended with products and coproducts in Gentzen’s style: Types are translated as follows

ιJ :≡ Jι for base type ι(A→ B)J :≡ AJ → BJ

(A×B)J :≡ AJ ×BJ

(A+B)J :≡ J(AJ +BJ)

which corresponds to Gentzen’s translation of propositions. And our term translation (t : ρ) 7→ (tJ : ρJ)corresponds to Genten’s soundness proof.

Working with different nuclei, we reveal various structures of functions that are definable in T. In thetalk, I will present some examples of nuclei and their applications, including those for majorizability [2],(uniform) continuity [5] and bar recursion [3].

References

[1] Hajime Ishihara, A note on the Godel-Gentzen Translation., Mathematical Logic Quarterly 46 (2000), no. 1, 135–137.

[2] Ulrich Kohlenbach, Applied Proof Theory: Proof Interpretations and their Use in Mathematics, Springer Monographsin Mathematics, 2008.

[3] Paulo Oliva and Silvia Steila, A direct proof of Schwichtenberg’s bar recursion closure theorem, The Journal of SymbolicLogic 83 (2018), no. 1, 70–83.

[4] Benno van den Berg, A Kuroda-style j-translation, Archive for Mathematical Logic 58 (2019), no. 5–6, 627–634.

[5] Chuangjie Xu, A syntactic approach to continuity of T-definable functionals, 2019, arXiv:1904.09794 [math.LO].

1This result was presented in the previous CCC workshop in Faro.

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