# We are hosting Proof Society Summer School

Proof Society Summer School is going to take place at our Computational Foundry this September. For more information and to apply, please click on the image below:

Proof Society Summer School is going to take place at our Computational Foundry this September. For more information and to apply, please click on the image below:

Philipp Schlicht is visiting us today from Bristol. He’ll give us an introduction to automatic structures in the theory seminar today (2pm, CoFo 201). Abstract: Automatic structures were first studied by Khoussainov and Nerode in 1995. I will first give an introduction with a focus on automatic ordinals and groups and then talk about tree-automatic structures. In particular, I will mention some recent partial results with Jain, Khoussainov and Stephan on the isomorphism problem for tree-automatic ordinals.

This week we are glad to welcome two visitors here at Swansea, namely Hideki Tsuiki from Kyoto University and Kristijonas Čyra from the Imperial College of London. Hideki will give a talk on “Infinite Adequacy Theorem through Coinductive Definitions” today at 14:00 as a part of our Theory seminar series and Kristijonas will speak on “Argumentation-enabled Explainable AI Applications” this Thursday at 15:00 at the CoFo.

Today Martin Caminada (Cardiff University) is visiting Swansea. He will give a talk about “A Logical Account of Dishonesty” as a part of out Theory seminar series. Abstract: Although formal logic is usually applied to reason about truth, in the current resentation we apply it to reason about dishonesty. Based on work in philosophy, we provide a formalisation of different forms of dishonesty, including lies, half-truth and bullshit, and discuss their formal properties. We also provide maxims for dishonest communication, that agents should ideally try to satisfy, both for moral and self-interested reasons.

To book an event, please visit click here and login using your Swansea credentials.

Dongseong Seon (선동성) from KAIST, Korea is visiting Swansea. This Thursday he will give a talk on computing Haar averages as a part of our Theory seminar series. For more infromation on this topic, please click here.

As a part of our Theory Seminars, today we welcomed Ryota Akiyoshi from Waseda University, who gave a talk on Takeuti’s finitism. Abstract: In this talk, we address several mathematical and philosophical issues of Gaisi Takeuti’s proof theory, who is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He furthered the realization of Hilbert’s program by formulating Gentzen’s sequent calculus for higher-oder logics, conjecturing that the cut-elimination holds for it (Takeuti’s conjecture), and obtaining several stunning results in the 1950-60’s towards the solution of his conjecture. This talk consists of two parts. (1) To summarize Takeuti’s background and the argument of the well-ordering proof of ordinals up to ε0 , (2) To evaluate it on philosophical grounds. Also, we will explain several mathematical and philosophical issues to be solved. This is joint work with Andrew Arana.

Today Erisa Karafili from the Imperial College London has given a talk on “Helping Forensic Analysts to Analyze and Attribute Cyber-Attacks” as a part of our Theory seminars. Abstract: The frequency and harmfulness of cyber-attacks are increasing every day, and with them also the amount of data that the cyber-forensics analysts need to collect and analyze. Analyzing and discovering who performed an attack or from where it originated would permit to put in act targeted mitigative and preventive measures. In my talk, I will present two techniques that help the forensics analyst to analyze and attribute cyber-attacks. The first technique is a formal analysis process that allows an analyst to filter the enormous amount of evidence collected and either identify crucial information about the attack (e.g., when it occurred, its culprit, its target) or, at the very least, perform a pre-analysis to reduce the complexity of the problem in order to then draw conclusions more swiftly and efficiently. The second technique is a novel argumentation-based reasoner (ABR) for analyzing and attributing cyber-attacks that includes in its reasoning technical and social evidence.

Ulrich and Olga have attended the annual CID project meeting and are going to give talks tomorrow at the Continuity, Comparability, Constructivity conference in Faro.

Tomorrow we are holding our end-of-summer logic minisymposium. Speakers: Åsa Hirvonen Ken-etsu Fujita Masahiko Sato Schedule: 2:00 Åsa Hirvonen (Helsinki University): On continuous logic 2:40 Coffee & Cake 3:00 Ken-etsu Fujita (Gunma University): A constructive proof of the Church—Rosser theorem 3:50 Masahiko Sato (Kyoto University): Unification of the Lambda-Calcululs and Combinatory Logic On continuous logic, Åsa Hirvonen (Helsinki University) Continuous first order logic in its current form was developed at the beginning of this millennium to offer a language for the model theoretic study of metric structures, such as Banach spaces and probability spaces. It is a many-valued logic but has a different motivation and semantic than, e.g., Lukasiewicz logic. Many model theoretical properties of first order logic generalise to it, such as compactness and Löwenheim-Skolem theorems. This talk is a general introduction to the syntax and semantics as well as some basic properties of continuous logic. A constructive proof of the Church—Rosser theorem, Ken-etsu Fujita (Gunma University) My motivation behind this talk comes from a quantitative analysis of reduction systems based on the two viewpoints, computational cost and computational orbit. In the first part, we show that an upper bound function for the Church—Rosser theorem of type-free lambda-calculus with beta-reduction must be in the fourth level of the Grzegorczyk hierarchy. That is, the number of reduction steps to arrive at a common reduct is bounded by a function in the smallest Grzegorczyk class properly extending that of elementary functions. At this level we also find common reducts for the confluence property. The proof method developed here can be applied not only to type-free lambda-calculus with beta-eta-reduction but also to typed lambda-calculi such as Pure Types Systems. In the second part, we propose a formal system of reduction paths for parallel reduction, wherein reduction paths are generated from a quiver by means of three path-operators, concatenation, monotonicity, and cofinality. Next, we introduce an equational theory and reduction rules for the reduction paths, and show that the rules on paths are terminating and confluent so that normal paths are obtained. Following the notion of normal paths, a graphical representation of reduction paths is provided, based on which unique path and universal common-reduct properties are established. Finally, transformation rules from a conversion sequence to a reduction path leading to the universal common-reduct are given, and path matrices are also defined as block matrices of adjacency matrices in order to count reduction orbits. Unification of the Lambda-Calcululs and Combinatory Logic, Masahiko Sato (Kyoto University): The Lambda calculus and combinatory logic have been studied as two closely related but distinct systems of logic and computation. In this talk, however, we will argue that they are in fact one and the same calculus. We substantiate our argument by introducing what we will call the external syntax and the internal syntax for the two calculi. The external syntax will be given as a natural common extension of syntax for the \lambda-calculus and combinatory logic. The terms defined by the internal syntax will be characterized as the closed alpha eta normal terms of the external syntax. Thus, the external syntax provides us with human readable syntax containing all the traditional \lambda-terms and combinatory-terms, and the internal syntax is suitable for the infrastructure of a proof assistant. This is an ongoing joint work with Takafumi Sakurai and Helmut Schwichtenberg.

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