{"title":"Topology","authors":"S. Friedrich","doi":"10.1002/9781119563549.ch5","DOIUrl":"https://doi.org/10.1002/9781119563549.ch5","url":null,"abstract":"The study of anatomy based on regions or divisions of the body and emphasizing the relations between various structures (muscles and nerves and arteries etc.) in region.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131271041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complete Non-Orders and Fixed Points","authors":"A. Yamada, Jérémy Dubut","doi":"10.4230/LIPIcs.ITP.2019.30","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITP.2019.30","url":null,"abstract":"In this paper, we develop an Isabelle/HOL library of order-theoretic concepts, such as various completeness conditions and fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often without any property of ordering, thus complete non-orders. In particular, we generalize the Knaster–Tarski theorem so that we ensure the existence of a quasi-fixed point of monotone maps over complete non-orders, and show that the set of quasi-fixed points is complete under a mild condition – attractivity – which is implied by either antisymmetry or transitivity. This result generalizes and strengthens a result by Stauti and Maaden. Finally, we recover Kleene’s fixed-point theorem for omega-complete non-orders, again using attractivity to prove that Kleene’s fixed points are least quasi-fixed points.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129033896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Parrow, J. Borgström, L. Eriksson, Ramunas Gutkovas, Tjark Weber
{"title":"Modal Logics for Nominal Transition Systems","authors":"J. Parrow, J. Borgström, L. Eriksson, Ramunas Gutkovas, Tjark Weber","doi":"10.23638/LMCS-17(1:6)2021","DOIUrl":"https://doi.org/10.23638/LMCS-17(1:6)2021","url":null,"abstract":"A concurrent system is a computer system with components that run in parallel and interact with each other. Such systems are ubiquitous and are notably responsible for supporting the infrastructure for transport, commerce and entertainment. They are very difficult to design and implement correctly: many different modeling languages and verification techniques have been devised to reason about them and verifying their correctness. However, existing languages and techniques can only express a limited range of systems and properties.In this dissertation, we address some of the shortcomings of established models and theories in four ways: by introducing a general modal logic, extending a modelling language with types and a more general operation, providing an automated tool support, and adapting an established behavioural type theory to specify and verify systems with unreliable communication.A modal logic for transition systems is a way of specifying properties of concurrent system abstractly. We have developed a modal logic for nominal transition systems. Such systems are common and include the pi-calculus and psi-calculi. The logic is adequate for many process calculi with regard to their behavioural equivalence even for those that no logic has been considered, for example, CCS, the pi-calculus, psi-calculi, the spi-calculus, and the fusion calculus.The psi-calculi framework is a parametric process calculi framework that subsumes many existing process calculi. We extend psi-calculi with a type system, called sorts, and a more general notion of pattern matching in an input process. This gives additional expressive power allowing us to capture directly even more process calculi than was previously possible. We have reestablished the main results of psi-calculi to show that the extensions are consistent.We have developed a tool that is based on the psi-calculi, called the psi-calculi workbench. It provides automation for executing the psi-calculi processes and generating a witness for a behavioural equivalence between processes. The tool can be used both as a library and as an interactive application.Lastly, we developed a process calculus for unreliable broadcast systems and equipped it with a binary session type system. The process calculus captures the operations of scatter and gather in wireless sensor and ad-hoc networks. The type system enjoys the usual property of subject reduction, meaning that well-typed processes reduce to well-typed processes. To cope with unreliability, we also introduce a notion of process recovery that does not involve communication. This is the first session type system for a model with unreliable communication.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115601739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Matroids","authors":"Jonas Keinholz","doi":"10.1002/9781118033104.ch10","DOIUrl":"https://doi.org/10.1002/9781118033104.ch10","url":null,"abstract":"One of the primary goals of pure mathematics is to identify common patterns that occur in disparate circumstances, and to create unifying abstractions which identify commonalities and provide a useful framework for further theorems. For example the pattern of an associative operation with inverses and an identity occurs frequently, and gives rise to the notion of an abstract group. On top of the basic axioms of a group, a vast theoretical framework can be built up, investigating the classification of groups, their internal structure, and the relationships and operations on groups. Matroids similarly provide a useful linking abstraction. They were first discovered independently by Hassler Whitney [10] and B. L. van der Waerden in the mid 1930’s [11]. Whitney had developed a notion of independence and rank in the context of graph theory, and noted similarities with the concepts of linear independence and dimension from linear algebra. By identifying the properties of abstract ‘independence’ which made these commonalities occur, he introduced the concept of a matroid, whose definition has proven immensely fruitful. Similarly, van der Waerden was interested in generalizing the notion of ‘independence’ from the examples of linear independence and algebraic independence. Shortly after the initial work by Whitney and van der Waerden, Birkhoff [2] noted that matroids were connected with a certain type of semimodular lattice that he had been studying. Thus, matroids provide a link between graph theory, linear algebra, transcendence theory, and semimodular lattices. Several decades later, Jack Edmonds noted the importance of matroids for the field of combinatorial optimization. This connection is due to two fundamental breakthroughs. First of all, Edmonds and a number of other researchers discovered a new type of matroid arising from the combinatorial theory of transversals. Second, Rado and Edmonds noted that matroids were intrinsically connected with the notion of a greedy algorithm (more historical details are in [11] and [3]). These developments have made matroids a mainstay of the field of combinatorial optimization.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125963600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Analysing and Comparing Encodability Criteria for Process Calculi","authors":"Kirstin Peters, R. V. Glabbeek","doi":"10.4204/EPTCS.190.4","DOIUrl":"https://doi.org/10.4204/EPTCS.190.4","url":null,"abstract":"Encodings or the proof of their absence are the main way to compare process calculi. To analyse the quality of encodings and to rule out trivial or meaningless encodings, they are augmented with quality criteria. There exists a bunch of different criteria and different variants of criteria in order to reason in different settings. This leads to incomparable results. Moreover it is not always clear whether the criteria used to obtain a result in a particular setting do indeed fit to this setting. We show how to formally reason about and compare encodability criteria by mapping them on requirements on a relation between source and target terms that is induced by the encoding function. In particular we analyse the common criteria full abstraction, operational correspondence, divergence reflection, success sensitiveness, and respect of barbs; e.g. we analyse the exact nature of the simulation relation (coupled simulation versus bisimulation) that is induced by different variants of operational correspondence. This way we reduce the problem of analysing or comparing encodability criteria to the better understood problem of comparing relations on processes.","PeriodicalId":280633,"journal":{"name":"Arch. Formal Proofs","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129665250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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