From 11 June 2024 to 13 June 2024, by Ingo Blechschmidt.

Tuesday, 9 April 2024, Sala Verde (Ca’ Vignal 3 - La Piramide) from 16:30 to 18:30

16:30

Andreas Weiermann (Universiteit Gent)

*Monadic second order limit laws for natural well orderings*

We prove monadic second order limit laws for ordinals stemming from segments of some prominent proof-theoretic ordinals like ωω , ε0, Γ0, . . . . The results are based on a combination of automata theoretic results, tree enumeration theory and Tauberian methods. We believe that our results will hold in very general contexts.

Some results have been obtained jointly with Alan R. Woods (who unfortunately passed away in 2011).

17:30

Fedor Pakhomov (Universiteit Gent)

*Reverse Mathematics of WQO’s of Transfinite Sequences with Finite Range*

In this talk I will present the result about the provability in ATR_0 of the theorem of Nash-Williams that sequences with finite range over a wqo form a wqo. For this we develop a framework where we work with α-wqo’s over primitive recursive set theory extended with a global enumeration function and then transfer the results obtained there to ATR_0.

The talk is based on a research project joint with Giovanni Soldà.

Tuesday, 19 March 2024, Aula H (Ca’ Vignal 2).

15:00

Hugo Herbelin (INRIA, Rocquencourt-Paris)

*On the logical structure of some maximality and well-foundedness principles equivalent to choice principles: the contrapositive case of Teichmüller-Tukey lemma and Berger’s update induction*

**Workshop on Proof, Argumentation, Computation, Modalities And Negation (PACM∧N)**

From Wednesday, 20 March, to Friday, 22 March 2024 in Sala Verde (Ca’ Vignal 3 - La Piramide).

More details will be available on the workshop website.

Thursday, 18 January 2024, Sala Riunioni (Ca’ Vignal 2) from 9:30 to 13:00

9:30

Paul Gorbow (Stockholms universitet)

*The Intentum and Intentic Models*

I will present the main ideas of a research proposal in the making. It aims to develop a dynamic (potentialist) account of an agent’s first-order intentional attitudes as representative models of (possible) reality. The account introduces the new notion of the intentum, and formally develops intentic models consisting of such intenta. These are meant to be structures that an agent can dynamically construct mentally, and which consequently help us philosophically explain and reason about beliefs, wishes, imaginations, etc. Here follow some promising applications of this framework:

It offers a novel theory of propositions and intentionality, somewhat in the spirit of Frege, but with distinct advantages compared to the two classical accounts of Frege and Russell.

It provides a foundational approach of value to the fragmented field of theoretical research on AI agenthood.

It enables a modular account of the notoriously complex notion of reference.

It provides a lens for approaching philosophical questions about identity.

It helps to philosophically explain knowledge of mathematics.

It provides a modern formal basis for the Kantian theory of categories, with a particular application to mathematical induction (as “synthetic a priori” knowledge).

It sheds light on arbitrary objects.

In my talk, I will explain the basic ideas of the framework and explain a select few of these applications.

10:30

Matteo Tesi (Technischen Universität Wien)

*Subintuitionistic logics and their modal companions: a nested approach*

In the present talk we deal with subintuitionistic logics and their modal companions. In particular, we introduce nested calculi for subintuitionistic systems and for modal logics in the S5 modal cube ranging from K to S4. The latter calculi differ from standard nested systems, as there are multiple rules handling the modal operator. As an upshot, we get a purely syntactic proof of the Gödel–McKinsey–Tarski embedding which preserves the structure and the height of the derivations. Finally, we obtain a conservativity result for classical logic over a weak subintuitionistic system.

12:00

Giulio Fellin (Università degli Studi di Verona)

*Modal logic for induction: from chain conditions to arithmetic*

j.w.w. S. Negri and P. Schuster

We use modal logic to obtain syntactical, proof-theoretic versions of transfinite induction as axioms or rules within an appropriate labelled sequent calculus. While transfinite induction proper, also known as Noetherian induction, can be represented by a rule, the variant in which induction is done up to an arbitrary but fixed level happens to correspond to the Gödel–Löb axiom of provability logic. To verify the practicability of our approach in actual practice, we sketch a fairly universal pattern for proof transformation and test its use in several cases. Among other things, we give a direct and elementary syntactical proof of Segerberg’s theorem that the Gödel–Löb axiom characterises precisely the (converse) well-founded and transitive Kripke frames. We then show that, by adding appropriate rules, this approach allows us to obtain a calculus for a bounded version of Peano Arithmetic. Finally, we observe that, by slightly modifying the rules, we can do the same for arithmetic in ordinals other than ω.

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.↩︎