Nominal String Diagrams

We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This requires us to take nominal sets as a monoidal category, not with the cartesian product, but with the separated product. To this end, we develop the beginnings of a theory of monoidal categories internal in a symmetric monoidal category. As an instance, we obtain a notion of a nominal PROP as a PROP internal in nominal sets. A 2-dimensional calculus of simultaneous substitutions is an application

Completeness of Nominal PROPs

We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams.Comment: arXiv admin note: text overlap with arXiv:1904.0753

Completeness of Nominal PROPs

We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams

Tool support for reasoning in display calculi

We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. Second, we provide embeddings of the calculus in the theorem prover Isabelle for formalising proofs about D.EAK. As a case study we show that the solution of the muddy children puzzle is derivable for any number of muddy children. Third, there is a set of meta-tools, that allows us to adapt the tool for a wide variety of user defined calculi

Display calculi and nominal string diagrams

This thesis is divided into two sections, encompassing two main topics: display calculi and nominal string diagrams. In the first section of the thesis, we introduce display calculi and present their advantages and drawbacks compared to sequent calculi. The rest of the section presents the calculus toolbox, a meta-tool for formalising display calculi. The tool includes a tree editor and a type-checker, which aid the user in exploring display calculi more efficiently. Section two grew out of an attempt to build a calculus of simultaneous substitutions for a display version of first order logic. This section explores the topic of string diagrams, in particular, we present two categorical formalisations of nominal string diagrams, along with a formal translation of ordinary string diagrams into nominal string diagrams (and vice versa).</div

Software Tool Support for Modular Reasoning in Modal Logics of Actions

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Software Tool Support for Modular Reasoning in Modal Logics of Actions

We present a software tool for reasoning in and about propositional sequent calculi for modal logics of actions. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. The tool generates embeddings of the calculus in the theorem prover Isabelle/HOL for formalising proofs about D.EAK. Integrating propositional reasoning in D.EAK with inductive reasoning in Isabelle/HOL, we verify the solution of the muddy children puzzle for any number of muddy children. There also is a set of meta-tools that allows us to adapt the software for a wide variety of user defined calculi

Software Tool Support for Modular Reasoning in Modal Logics of Actions

We present a software tool for reasoning in and about propositional sequent calculi for modal logics of actions. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. The tool generates embeddings of the calculus in the theorem prover Isabelle/HOL for formalising proofs about D.EAK. Integrating propositional reasoning in D.EAK with inductive reasoning in Isabelle/HOL, we verify the solution of the muddy children puzzle for any number of muddy children. There also is a set of meta-tools that allows us to adapt the software for a wide variety of user defined calculi

Interlaboratory comparison of cosmogenic 21 Ne in quartz

International audienceWe performed an interlaboratory comparison study with the aim to determine the accuracy of cosmogenic 21Ne measurements in quartz. CREU-1 is a natural quartz standard prepared from amalgamated vein clasts which were crushed, thoroughly mixed, and sieved into 125–250 μm and 250–500 μm size fractions. 50 aliquots of CREU-1 were analyzed by five laboratories employing six different noble gas mass spectrometers. The released gas contained a mixture of 16–30% atmospheric and 70–84% non-atmospheric (predominantly cosmogenic) 21Ne, defining a linear array on the 22Ne/20Ne-21Ne/20Ne three isotope diagram with a slope of 1.108 ± 0.014. The internal reproducibility of the measurements is in good agreement with the formal analytical precision for all participating labs. The external reproducibility of the 21Ne concentrations between labs, however, is significantly overdispersed with respect to the reported analytical precision. We report an average reference concentration for CREU-1 of 348 ± 10 × 106 at [21Ne]/g[SiO2], and suggest that the 7.1% (2σ) overdispersion of our measurements may be representative of the current accuracy of cosmogenic 21Ne in quartz. CREU-1 was tied to CRONUS-A, which is a second reference material prepared from a sample of Antarctic sandstone. We propose a reference value of 320 ± 11 × 106 at/g for CRONUS-A. The CREU-1 and CRONUS-A intercalibration materials may be used to improve the consistency of cosmogenic 21Ne to the level of the analytical precision