Thursday, 23 November 2023, Aula G (Ca’ Vignal 2).

17:00

Rolf Hennicker (LMU Munich)

*Epistemic Ensembles*

in cooperation with A. Knapp (Univ. of Augsburg) and M. Wirsing (LMU Munich)

An ensemble is formed by a collection of agents which run concurrently to accomplish (together) a certain task. For that purpose agents must collaborate in some way, for instance by explicit interaction via message passing. In this talk we present an epistemic approach where collaboration is based on the knowledge that agents have about themselves, about other agents and about their environment. Any change of knowledge caused by an action of one agent may influence the behaviour of other agents. Hence, interaction is implicit.

For specifying properties of epistemic ensembles we propose a dynamic logic style such that necessary and possible ensemble behaviours are expressed by modalities on (complex) epistemic actions. Our semantic models are labelled transition systems with ensemble actions as labels. Such transitions model two aspects, (i) the control flow of an ensemble and (ii) changes of epistemic information caused by the epistemic effect of an agent action.

Epistemic ensembles are implemented by epistemic processes, one for each ensemble agent, which are composed in parallel to form an ensemble realisations. We provide a structured operational semantics for such processes which allows us to introduce a (formal) correctness notion: An ensemble realisation is correct w. r. t. an ensemble specification if its semantics is an epistemic model of the specification. Two ensemble realisations are equivalent, if their models are epistemically bisimilar. We show that this can be checked by checking bisimilarity of their single processes in the usual sense of process algebra.

Thursday, 26 October 2023, Sala Verde (Ca’ Vignal 3 - La Piramide).

10:30

Samuele Maschio (Università degli Studi di Padova)

*Implicative models of set theory*

Implicative algebras were introduced by A.Miquel to provide a common foundation for Heyting/Boolean-valued logic and realizability. In this talk I will show how one can produce models for intuitionistic and classical Zermelo-Fraenkel set theory (I)ZF using implicative algebras, generalizing forcing and realizability models for set theory, both intuitionistic and classical . This result has an application to the theory of toposes, since it entails that every topos which is obtained from Set by means of the tripos-to-topos construction contains a model of (I)ZF, provided one assumes a large enough inaccessible cardinal exists. This talk is based on a joint work with A.Miquel.

**Lecturer and Examiner:** Dr. Iosif Petrakis

**Period:** Nov-Dec 2023, Start: Tuesday 07.11.2023

**Type of course:** Hybrid

**Zoom-links:** Please send an e-mail to Dr. Iosif Petrakis or to Dr. Giulio Fellin

**Venue:** Department of Computer Science, University of Verona

**Rooms:** Auletta Atrio - Borgo Roma - Ca’ Vignal 1 - Strada Le Grazie 15, Verona.

For the lecture of Friday 01.12.2023 only: Aula G - Borgo Roma - Ca’ Vignal 2 - Strada Le Grazie 15, Verona.

**Time:** Tuesday 14:00-17:00 and Friday 10:00-13:00

**Contact:** Iosif Petrakis

**Number of academic hours:** 35

**Assessment method:** Written exam

**Intended audience:** Computer scientists and mathematicians (PhD-level and advanced Master’s-level)

**Abstract:** Martin-Löf’s type theory (MLTT) is a predicative modification of lambda-calculus with many applications to the theory of programming languages. The recent extension of MLTT with the axiom of univalence (UA) by the late Fields medalist Voevodsky reveals surprising connections between homotopy theory and logic. Homotopy Type Theory (HoTT) can be described as MLTT + UA + Higher Inductive Types. HoTT and Category Theory (CaT), are currently among the most actively studied mathematical frameworks for the logical foundations of mathematics and theoretical computer science. The following topics are planned to be covered: introduction to MLTT, function-extensionality axiom, the groupoid model of Hofmann and Streicher, Voevodsky’s axiom of univalence, homotopy n-types, higher inductive types, the fundamental group of the higher circle, relations between HoTT and constructive mathematics, relations between HoTT and CaT.

**No prior knowledge of homotopy theory and type theory is required.**

**Lecture notes will be available.**

**References:**

E. Rijke: Introduction to Homotopy Type Theory, https://arxiv.org/abs/2212.11082, 2022, pre-publication version, which will be published by Cambridge University Press.

I. Petrakis: Logic in Computer Science, Lecture notes, 2022. https://www.mathematik.uni-muenchen.de/~petrakis/LCS.pdf

The Univalent Foundations Program Homotopy Type Theory: Univalent Foundations of Mathematics, Institute for Advanced Study, Princeton, 2013. https://homotopytypetheory.org/book/

25-26 September, 2023

Marco Benini (Università degli Studi dell’Insubria)

*1. Homotopy Type Theory: A Gentle Introduction*: Monday, September 25th, 16:00, Aula G (Ca’ Vignal 2).

**Abstract:**
Homotopy Type Theory (HoTT) is, syntactically, Martin-Löf Type Theory
(MLTT) plus the axiom called “univalence”.
Hence, it is, at the same time, an abstract functional programming
language, a logical system, and a synthetic way to describe homotopy
spaces.
The seminar introduces the type system, focusing on the homotopy
interpretation, univalence, and higher inductive types, i.e., the main
novelties of HoTT with respect to MLTT.
The seminar aims at providing the fundamental ideas of HoTT to
mathematicians and computer scientists willing to understand what is
HoTT without the burden of a formal, in-depth presentation.

*2. Homotopy Type Theory: Equality as Equality*: Tuesday, September 26th, 16:00, Aula I (Ca’ Vignal 2).

**Abstract:**
The seminar aims at discussing the logical properties of judgemental
and propositional equalities in HoTT. It shows the fundamental
reasoning techniques of HoTT, and a few small novel results.
Also, the material provides a critical view of the current stage of
development of the theory, showing to what extent a systematic
approach is needed.
This talk is intended for mathematicians and computer scientists
willing to see more of the formal and technical side of HoTT, and some
minimal
knowledge about type theories, algebraic topology, and constructive
mathematics is presumed.

The talks will be held both in presence and on Zoom. In order to get the Zoom link, please contact Giulio Fellin.

Tuesday, 9 May 2023

14:30, Aula E (Ca’ Vignal 1).

Ingo Blechschmidt (Universität Augsburg)

*Synthetic scheme theory: a simpler framework for algebraic geometry*

**Note:** This talk has been postponed to a later date.

**Abstract:**
The modern framework for algebraic geometry put forward by Grothendieck
and his school has been enormously successful, providing the basis for
many deep cornerstone results of the subject. Not least, we owe to this
basis the proof of Fermat’s Last Theorem.
Despite these successes, Grothendieck himself expressed discontent with his framework,
and especially in recent years, concerns about the limitations and
technical difficulties of the modern theory of algebraic geometry arose.^{1}

With the benefit of hindsight, now that the mathematical content of the then-revolutionary new approach to algebraic geometry is well-understood, we propose an update to the foundations of algebraic geometry, called synthetic scheme theory, built on three postulates. These postulates capture essential geometric properties and allow us to reason constructively, avoiding the use of transfinite principles and other highly abstract concepts. Our hope is that this approach will allow for a clearer and more intuitive expression of the central notions and insights of algebraic geometry, requiring less technical machinery, will facilitate integrated developments, and promote computer-assisted proofs in the subject.

Crucially, our approach rests on the greater axiomatic freedom provided by constructive mathematics: The three postulates are inconsistent with classical logic.

16:30, Aula Gino Tessari (Ca’ Vignal 2).

Hans Leiß (Universität München)

_Algebraic Representation of the Fixed-Point Closure of *-continuous Kleene Algebras_

**Abstract:**
The theorem of Chomsky and Schützenberger says that each context-free language L over a finite alphabet X is the image of the intersection
of a regular language R over X+Y and the context-free Dyck-language
D over X+Y of strings with balanced brackets from Y under the
bracket-erasing homomorphism from the monoid of strings over X+Y to
that of strings over X.

Within Hopkins’ algebraization of formal language theory we show that the algebra C(X) of context-free languages over X has an isomorphic copy in a suitable tensor product of the algebra R(X) of regular languages over X with a quotient of R(Y) by a congruence describing bracket matches and mismatches. It follows that all context-free languages over X can be defined by regular(!) expressions over X+Y where the letters of X commute with the brackets of Y, thereby providing a substitute for a fixed-point-operator.

This generalizes the theorem of Chomsky and Schützenberger and leads to an algebraic representation of the fixed-point closure of *-continuous Kleene algebras.

Perhaps the most pressing are that the currently framework heavily relies on the axiom of choice and other classical principles–even though high-level reasoning in algebraic geometry is often constructive–and that the framework references and requires gadgets of little geometric significance, such as injective or flabby resolutions. These foundational issues block integrated developments of algorithms in algebraic geometry, hindering us from extracting certified algorithms directly from proofs, and render computer formalization of algebraic geometry particularly challenging.↩︎

Tuesday, 21 March 2023, Aula I (Ca’ Vignal 2).

15:00

Matteo Tesi (Scuola Normale Superiore, Pisa)

*Quantificatori moltiplicativi, esponenziali e cut-elimination*
(j.w.w. Carlo Nicolai e Mario Piazza)

**Abstract:**
L’aggiunta di un predicato di verità governato da regole ingenue conduce spesso a sistemi contraddittori. Un metodo per recuperare la consistenza consiste nell’adottare una logica priva di contrazione: questa strada è stata seguita - tra gli altri - da Grishin e Cantini. Successivamente un ulteriore approccio basato su quantificatori moltiplicativi è stato proposto da Zardini, ma il sistema si è rivelato inconsistente. Nel presente talk proponiamo un’analisi della logica dei quantificatori moltiplicativi. In primo luogo offriamo una presentazione consistente di un sistema con predicato di verità decitazionale con quantificatori additivi. In seguito mostriamo come gli esponenziali (! e ?) possano essere interpretati in modo corretto e fedele mediante l’uso dei quantificatori moltiplicativi. Infine proponiamo una dimostrazione sintattica di eliminazione del taglio per un calcolo con quantificatori moltiplicativi. L’analisi della strategia porterà alla luce la presenza di contrazione implicita nelle regole dei quantificatori.

Monday, 23 January 2023, Aula F (Ca’ Vignal 1).

15:00

Tatsuji Kawai (Japan Advanced Institute of Science and Technology)

*Predicative theory of stably locally compact locales*

We give a predicative presentation of stably locally compact locales, the class of locales which includes locally compact regular locales (e.g., localic reals) as its subclass. In our setting, a stably locally compact locale is presented as a quasi-proximity lattice, a quasi-bounded distributive lattice (distributive lattice without top) together with a certain idempotent relation on it. Using this structure, we construct a coreflection from the category of locally compact regular locales and cobounded maps to that of stably locally compact locales and perfect maps. The construction of this coreflection generalizes Dedekind’s construction of real numbers as pairs of a lower and an upper cut.

16:30

Hajime Ishihara (Japan Advanced Institute of Science and Technology)

*Reflexive combinatory algebras*

We introduce the notion of reflexivity for combinatory algebras. Reflexivity can be thought of as an equational counterpart of the Meyer-Scott axiom of combinatory models, which indeed allows us to characterise an equationally definable counterpart of combinatory models. This new structure, called strongly reflexive combinatory algebra, admits a finite axiomatisation with seven closed equations, and the structure is shown to be exactly the retract of combinatory models. Lambda algebras can be characterised as strongly reflexive combinatory algebras which are stable.

This is a joint work with Marlou M. Gijzen and Tatsuji Kawai.

**On the Algebra of Logic**

Tommaso Moraschini (University of Barcelona)

The minicourse On the Algebra of Logic [2 ETCS Mat/01] by Tommaso Moraschini, University of Barcelona, is being videoregistered from now on.

Attendees are expected to have verified, refreshed or acquired adequate basic knowledge of universal algebra before following the videolectures by working through chapters 1 and 2 of the course notes.

Members of the University of Verona are kindly asked to register for the course mailing list by writing to Giulio Fellin. Students of the University of Verona who want to obtain credits for this course also in the future should contact prof. Peter Schuster.

**Friday, June 4th, 2021, 11:00, on Zoom.**

Roberta Bonacina (Tübingen)

*Two-level type theory*

*Abstract:*

Homotopy type theory (HoTT) extends Martin-Löf type theory with the univalence axiom, which establishes a tight connection between types and homotopy spaces and allows to identify isomorphic objects. Univalence has very useful applications, but it has also drawbacks: properties that are not invariant under homotopy cannot be expressed internally. An important case is the concept of semisimplicial types, whose definition is so far elusive in HoTT. Two-level type theory (2LTT) is a formal theory which allows constructions that require access to non-homotopy-invariant notions. It is composed of two separate levels of types related by a conversion function that preserves context extensions; the outer level is Martin-Löf type theory plus the uniqueness of identity proofs, and the inner level is HoTT. In this talk we will discuss the consequences of univalence, and introduce 2LTT as a way to overcome some difficulties. Then, we will introduce the notion of model for this system, and define the syntactical category as a first step to prove an initiality result for the syntax of 2LTT. The last part of the talk is based on a joint work-in-progress with Benedikt Ahrens and Nicolai Kraus.

The seminar will be held on Zoom. In order to get access to the Zoom meeting, please contact Peter Schuster or Giulio Fellin.

**Wednesday, April 14th, 2021, 16:30, on Zoom.**

Gianluca Amato (Chieti-Pescara)

*Universal Algebra in UniMath* (joint work with Marco Maggesi and Cosimo Perini Brogi)

*Abstract:*

In this talk we will report on recent experiments in implementing the basic notions of universal algebra in the Unimath library. Unimath is a replacement of the standard library of the Coq proof assistant based on the univalent foundations of mathematics. Since the use of general higher order inductive types is forbidden in Unimath, a relevant part of this work has been the implementation of terms as lists of function symbols in such a way that the operations on terms are computable by the standard Coq evaluation mechanism.

More information can be found in the preprint.

The seminar will be held on Zoom. In order to get access to the Zoom meeting, please contact Peter Schuster or Giulio Fellin.