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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Ulrich, Monika and Olga attended Hausdorff Trimester Program Types, Sets and Constructions in Bonn, Germany. During this research trip they have participated in the Constructive Mathematics workshop and had a chance to collaborate with partners from the past and existing projects, including COMPUTAL, CONRCON and CID.
Irina and Markus attended SPIN symposium that took place in Malaga, Spain, where they gave a tutorial on Software Model Checking for Mobile Security.
Professor Moshe Vardi from Rice University, Texas, will give a talk on The Automated-Reasoning Revolution: From Theory to Practice and Back Wednesday 27th June, 12-1pm, Computational Foundry Seminar Room (Talbot 909). Abstract: For the past 40 years computer scientists generally believed that NP-complete problems are intractable. In particular, Boolean satisfiability (SAT), as a paradigmatic automated-reasoning problem, has been considered to be intractable. Over the past 20 years, however, there has been a quiet, but dramatic, revolution, and very large SAT instances are now being solved routinely as part of software and hardware design. In this talk I will review this amazing development and show how automated reasoning is now an industrial reality. I will then describe how we can leverage SAT solving to accomplish other automated-reasoning tasks. Counting the number of satisfying truth assignments of a given Boolean formula or sampling such assignments uniformly at random are fundamental computational problems in computer science with applications in software testing, software synthesis, machine learning, personalized learning, and more. While the theory of these problems has been thoroughly investigated since the 1980s, approximation algorithms developed by theoreticians do not scale up to industrial-sized instances. Algorithms used by the industry offer better scalability, but give up certain correctness guarantees to achieve scalability. We describe a novel approach, based on universal hashing and Satisfiability Modulo Theory, that scales to formulas with hundreds of thousands of variables without giving up correctness guarantees. Read more.
Magne Haveraaen from the University of Bergen is visiting our Department in May-July 2018.
Ulrich and Olga visited Japan as part of the CID project in April 2018.
Amir Tabatabai and Rahele Jalali, both PhD students at the Institute of Mathematics of the Czech Academy of Sciences under the supervision of Pavel Pudlak, are visiting Swansea University 13 Nov – 6 Dec 2017. Amir will give a talk on Computational Flows in Arithmetic on 16 November. More information: A computational flow is a pair consisting of a sequence of computational problems of a certain sort and a sequence of computational reductions among them. In this talk we will explain the basics of the theory of computational flows and how they make a sound and complete interpretation for bounded theories of arithmetic. This property helps us to decompose a first order arithmetical proof to a sequence of computational reductions by which we can extract the computational content of the low complexity statements in some bounded theories of arithmetic such as .
We would like to welcome Arno Pauly, who has recently joined the computer science department here in Swansea. Today he is going to give a talk on Noncomputability in analysis as a part of our Computational Foundry Seminar Series. More information: Many theorems in analysis state the existence of a certain object depending on some parameter. Each such theorem has an associated computational task: Compute the object from the parameter. From the viewpoint of a constructivist, these tasks are intricately linked to the meaningful truth of the theorems. From a pragmatic perspective, the applicability of a theorem to fields like physics or economics is tied to the solvability of the associated computational task.