Wyniki wyszukiwania - Biblioteka Nauki

1

Filozoficznie prowokująca teoria kategorii

100%

Heller M.

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nr 65

PL

Recenzja książki: Elaine Landry (red.), Categories for the Working Philosopher, Oxford University Press, Oxford, 2017, ss. xiv+471.

2

Timed Delay Bisimulation is an Equivalence Relation for Timed Transition Systems

71%

Gribovskaya N. , Virbitskaite I.

Fundamenta Informaticae

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2009

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tom Vol. 93, nr 1-3

127-142

EN

Timed transition systems are a widely studied model for real-time systems. The intention of the paper is to show the applicability of the general categorical framework of openmaps in order to prove that timed delay equivalence is indeed an equivalence relation in the setting of timed transition systems with invariants. In particular, we define a category of the model under consideration and an accompanying (sub)category of observations to which the corresponding notion of open maps is developed. We then use the open maps framework to obtain an abstract equivalence relation which is established to coincide with timed delay bisimulation.

3

An Algebraic Semantics for QVT-Relations Check-only Transformations

71%

Guerra E. , de Lara J.

Fundamenta Informaticae

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2012

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tom Vol. 114, nr 1

73-101

EN

QVT is the standard for model transformation defined by the OMG in the context of the Model-Driven Architecture. It is made of several transformation languages. Among them, QVTRelations is the one with the highest level of abstraction, as it permits developing bidirectional transformations in a declarative, relational style. Unfortunately, the standard only provides a semiformal description of its semantics, which hinders analysis and has given rise to ambiguities in existing tool implementations. In order to improve this situation, we propose a formal, algebraic semantics for QVT-Relations check-only transformations, defining a notion of satisfaction of QVT-Relations specifications by models.

4

Category Theory in the hands of physicists, mathematicians, and philosophers

71%

Stopa M.

Zagadnienia Filozoficzne w Nauce

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2020

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nr 69

283-293

EN

Book review: Category Theory in Physics, Mathematics, and Philosophy, Kuś M., Skowron B. (eds.), Springer Proc. Phys. 235, 2019, pp.xii+134.

5

“Is logic a physical variable?” Introduction to the Special Issue

71%

Eckstein M. , Skowron B.

Zagadnienia Filozoficzne w Nauce

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2020

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nr 69

7-13

EN

“Is logic a physical variable?” This thought-provoking question was put forward by Michael Heller during the public lecture “Category Theory and Mathematical Structures of the Universe” delivered on 30th March 2017 at the National Quantum Information Center in Sopot. It touches upon the intimate relationship between the foundations of physics, mathematics and philosophy. To address this question one needs a conceptual framework, which is on the one hand rigorous and, on the other hand capacious enough to grasp the diversity of modern theoretical physics. Category theory is here a natural choice. It is not only an independent, well-developed and very advanced mathematical theory, but also a holistic, process-oriented way of thinking.

6

Category Free Category Theory and Its Philosophical Implications

71%

Heller M.

Logic and Logical Philosophy

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2016

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tom 25

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nr 4

447-459

EN

There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely eliminated. Category theory seems to be the correct mathematical theory for clarifying conceptual possibilities in this respect. In this theory, objects acquire their identity either by definition, when in defining category we postulate the existence of objects, or formally by the existence of identity morphisms. We show that it is perfectly possible to get rid of the identity of objects by definition, but the formal identity of objects remains as an essential element of the theory. This can be achieved by defining category exclusively in terms of morphisms and identity morphisms (objectless, or object free, category) and, analogously, by defining category theory entirely in terms of functors and identity functors (categoryless, or category free, category theory). With objects and categories eliminated, we focus on the “philosophy of arrows” and the roles various identities play in it (identities as such, identities up to isomorphism, identities up to natural isomorphism ...). This perspective elucidates a contrast between “set ontology” and “categorical ontology”.

7

A Categorical View on Algebraic Lattices in Formal Concept Analysis

62%

Hitzler P. , Krötzsch M. , Zhang G.-Q.

Fundamenta Informaticae

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2006

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tom Vol. 74, nr 2,3

301-328

EN

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. To this end, we build on the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.

8

Category theory in geography?

62%

Arlinghaus S. L. , Kerski J.

Quaestiones Geographicae

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2015

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tom 34

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nr 4

9

Category Theory and Set Theory as Theories about Complementary Types of Universals

62%

Ellerman D.

Logic and Logical Philosophy

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2017

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tom 26

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nr 2

145–162

EN

Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_F = {x | F(x)} for a property F(.) could never be self-predicative in the sense of u_F \in u_F . But the mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals - which can be seen as forming the “other bookend” to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato’s Theory of Forms as well as for the idea of a “concrete universal” in Hegel and similar ideas of paradigmatic exemplars in ordinary thought.

10

Adhesivity with Partial Maps instead of Spans

62%

Heindel T.

Fundamenta Informaticae

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2012

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tom Vol. 118, nr 1-2

1-33

EN

The introduction of adhesive categories revived interest in the study of properties of pushouts with respect to pullbacks that started over thirty years ago for the category of graphs. Adhesive categories - of which graphs are the 'archetypal' example - are defined by a single property of pushouts along monos that implies essential lemmas and central theorems of double pushout rewriting such as the local Church-Rosser Theorem. The present paper shows that a strictly weaker condition on pushouts suffices to obtain essentially the same results: it suffices to require pushouts to be hereditary, i.e. they have to remain pushouts when they are embedded into the associated category of partial maps. This fact however is not the only reason to introduce partial map adhesive categories as categories with pushouts along monos (of a certain stable class) that are hereditary. There are two equally important motivations: first, there is an application relevant example category that cannot be captured by the more established variations of adhesive categories; second, partial map adhesive categories are 'conceptually similar' to adhesive categories as the latter can be characterized as those categories with pushout along monos that remain bi-pushouts when they are embedded into the associated bi-category of spans. Thus, adhesivity with partial maps instead of spans appears to be a natural candidate for a general rewriting framework.

11

Does God Think the Same Way We Do? On the Logical Apophatism of Michał Heller

54%

Grygiel W.

Verbum Vitae

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2023

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tom 41

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nr 3

751-765

EN

Apophatic theology is an approach in theology that emphasizes the limitation of human language and concepts in describing the nature of the Divine. Rooted in ancient religious traditions, apophatic theology has gained attention in contemporary discourse for its potential convergence with the scientific method. This paper expands on a novel application of this approach in which the formal methods of science such as logic and mathematics are engaged to inquire into how God thinks and to what degree the modes of human reasoning can be projected on the nature of the Divine mind. This application has been proposed by Michał Heller and is referred to as the logical apophatism. In the course of the analysis carried out in this paper more in-depth understanding of the logical apophatism has been obtained by contrasting it with classical approaches to negative theology such as the Triplex Via and supplementing with recent advances within the cognitive sciences. It is concluded that Heller’s use of the apophatic approach is manifestly non-standard and its novelty consists in the shift of emphasis from the negative character of the language of theology to the logic of the Divine mind and the logic that underpins the workings of the Universe.

12

A critical analysis of the philosophical motivations and development of the concept of the field of rationality as a representation of the fundamental ontology of the physical reality

54%

Grygiel W.

Zagadnienia Filozoficzne w Nauce

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2022

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nr 72

87-108

EN

The unusual applicability of mathematics to the description of the physical reality still remains a major investigative task for philosophers, physicists, mathematicians and cognitive scientists. The presented article offers a critical analysis of the philosophical motivations and development of a major attempt to resolve this task put forward by two prominent Polish philosophers: Józef Życiński and Michał Heller. In order to explain this particular property of mathematics Życiński has first introduced the concept of the field of rationality together with the field of potentiality to be followed by Heller’s formal field and the field of categories. It turns out that these concepts are fully intelligible once located within philosophical stances on the relations between mathematics and physical reality. It will be argued that in order to achieve more extended conceptual clarification of the precise meaning of the field of rationality, further advancements in the understanding of the nature of the human mind are required.

13

Teoria kategorii i niektóre jej logiczne aspekty

54%

Stopa M.

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nr 64

7-58

PL

This article is intended for philosophers and logicians as a short partial introduction to category theory (CT) and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in toposes, which, in general, is an intuitionistic logic. We next present two families of toposes whose tautologies are identical with those of classical propositional logic. The relatively extensive bibliography is given in order to support further studies.

14

Low-Cost Dynamic Constraint Checking for the JVM

54%

Grzanek K.

Journal of Applied Computer Science Methods

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2016

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tom Vol. 8 No. 2

115--136

EN

Using formal methods for software verification slowly becomes a standard in the industry. Overall it is a good idea to integrate as many checks as possible with the programming language. This is a major cause of the apparent success of strong typing in software, either performed on the compile time or dynamically, on runtime. Unfortunately, only some of the properties of software may be expressed in the type system of event the most sophisticated programming languages. Many of them must be performed dynamically. This paper presents a flexible library for the dynamically typed, functional programming language running in the JVM environment. This library offers its users a close to zero run-time overhead and strong mathematical background in category theory.

15

ChR: Dynamic Functional Constraints Checking in R

54%

Grzanek K.

Journal of Applied Computer Science Methods

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2017

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tom Vol. 9 No. 1

65--78

EN

Dynamic typing of R programming language may issue some quality problems in large scale data-science and machine-learning projects for which the language is used. Following our efforts on providing gradual typing library for Clojure we come with a package chR - a library that offers functionality of run-time type-related checks in R. The solution is not only a dynamic type checker, it also helps to systematize thinking about types in the language, at the same time offering high expressivenes and full adherence to functional programming style.

16

Modular Construction of Finite and Complete Prefixes of Petri net Unfoldings

54%

Madalinski A. , Fabre E.

Fundamenta Informaticae

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2009

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tom Vol. 95, nr 1

219-244

EN

This paper considers distributed systems, defined as a collection of components interacting through interfaces. Components, interfaces and distributed systems are modeled as Petri nets. It is well known that the unfolding of such a distributed system factorises, in the sense that it can be expressed as the composition of unfoldings of its components. This factorised form of the unfolding generally provides a more compact representation of the system runs, because each component does not need to represent the possible choices (conflicts) appearing in the other components. Moreover, the unfolding factorisation makes it possible to analyse the system by parts. The paper focuses on the derivation of a finite and complete prefix (FCP) in the unfolding of a distributed system. Specifically, one would like to directly obtain such a FCP in factorised form, without computing first a FCP of the global distributed system and then factorising it. The construction of such a “modular FCP” is based on deriving summaries of component behaviours w.r.t. their interfaces, that are then communicated to the neighbouring components. The latter combine these summaries with their local behaviours, and prepare interface summaries for the next components. This globally takes the form of a message passing algorithm, where the global system is never considered.

17

The Logic of God

45%

Heller M.

Edukacja Filozoficzna

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2019

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nr 68

227-244

EN

When speaking on the “logic of God”, we can understand the logic of our reasoning about God or the logic as it is supposedly employed by God. It is rather obvious, at least for believers in God, that we can infer something about God’s logic in the latter meaning, from how the logic operates in the world created by Him. In the present essay, my strategy is to use this narrow window through which we can grasp some glimpses of “God’s ways of thinking”. There are strong reasons to believe that it is category theory that best displays the role of logic in the system of our mathematical and physical knowledge. It gives us a refreshingly new perspective on logic and its various applications, and could be a good starting point for our speculations concerning the “logic of God”. A quick look at category theory and its applications to physics shows that logic can change from theory to theory, or from level to meta-level. This poses the question of the existence of “superlogic” to which all other logics would somehow be subordinated. The fact that this question remains unanswered forces us to face the problem of plurality of logics. Usually, it is tacitly assumed that the role of “superlogic” is played by classical logic with its non-contradiction law as the most obvious tautology. We briefly discuss paraconsistent logic as an example of a logical system in which contradictions are allowed, albeit under the condition that they do not make the system to explode, i.e. that they do not spill over the whole system. Such logic is an internal logic in categories called cotopoi (or complement topoi). I refer to some theological discussions, both present and from the past, that associate “God’s logic” with classical logic, in particular with the non-contradiction principle. However, we argue that this principle should not be absolutized. The only thing we can, with some certainty, assert on “God’s logic” is that it is not an exploding logic, i.e. that it is not an “anything goes logic”. God is a Source-of-All-Rationality but His rationality need not to conform to our standards of what is rational. This “principle of logical apophaticism” is formulated and briefly discussed. In the history of theology at least one attempt is known to reconstruct the “process of God’s thinking”, namely Leibniz’s idea of God’s selecting the best world to be created from among all possible worlds. Some modifications are suggested which we believe Leibniz would have introduced in his reconstruction, if he knew present developments in categorical logic.

PL

Mówiąc o „logice Boga”, możemy mieć na myśli logikę naszego myślenia o Bogu albo logikę, jaką w naszym wyobrażeniu posługuje się Bóg. Jest dość oczywiste, przynajmniej dla wierzących, że to i owo na temat logiki Boga w tym drugim znaczeniu możemy wywnioskować z logiki obowiązującej w stworzonym przez Niego świecie. W tym eseju stosuję właśnie tę strategię, by zidentyfikować niektóre przebłyski „Boskiego sposobu myślenia”. Istnieją dobre powody, by przyjąć, że rolę logiki w systemie naszej matematycznej i fizycznej wiedzy najlepiej obrazuje teoria kategorii. Daje nam ona nowe, świeże spojrzenie na logikę i jej różne zastosowania, i może być dobrym punktem wyjścia dla badań nad „logiką Boga”. Pobieżne nawet przyjrzenie się zastosowaniom teorii kategorii w fizyce pozwala się przekonać, że logika może się zmieniać przy przejściu od teorii do teorii i z jednego poziomu ogólności na drugi. Nasuwa się pytanie o istnienie „superlogiki”, której w jakimś sensie podlegałyby wszystkie logiki niższego rzędu. Fakt, że nie potrafimy na to pytanie odpowiedzieć, stawia nas w obliczu problemu logicznego pluralizmu. Zwykle przyjmuje się milcząco, że rolę „superlogiki” odgrywa logika klasyczna z zasadą niesprzeczności jako najbardziej oczywistą tautologią. Artykuł omawia krótko logikę parakonsystentną jako przykład logiki, w której sprzeczności są dozwolone, choć jedynie pod warunkiem, że nie eksplodują, tzn., nie rozlewają się na całość systemu. Taka logika jest wewnętrzną logiką w kategoriach zwanych ko-toposami (complement topoi). Przywołuję niektóre teologiczne dyskusje, zarówno dawne, jak i współczesne, w których „logikę Boga” utożsamiano z logiką klasyczną, a zwłaszcza z zasadą niesprzeczności, starając się pokazać, że zasady tej jednak nie powinno się absolutyzować. Jedyne, co z jakąś pewnością możemy powiedzieć o „logice Boga”, to że nie jest to logika bez żadnych reguł, w której wszystko jest dozwolone. Bóg jest Źródłem-Wszelkiej-Racjonalności, ale Jego racjonalność nie musi spełniać naszych standardów. Formułuję zatem i krótko omawiam tę „zasadę logicznej apofatyczności”. Historia teologii zna przynajmniej jedną próbę zrekonstruowania „procesu Boskiego namysłu” – wizję Leibniza, u którego Bóg ze wszystkich światów, które mógłby stworzyć, wybiera najlepszy. Sugeruję na koniec kilka zmian, które moim zdaniem Leibniz wprowadziłby do swojej rekonstrukcji, gdyby znał współczesną logikę kategorialną.

18

On the validity of the definition of a complement-classifier

45%

Stopa M.

Zagadnienia Filozoficzne w Nauce

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2020

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nr 69

111-128

EN

It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes (co-toposes), which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain two definitions of a topos, the concept of a complement-classifier (and thus of a co-topos as well) is, at least in general and within the conceptual framework of category theory, not appropriately defined. For this purpose, I first analyze the standard notion of a subobject classifier, show its connection with the representability of the functor Sub via the Yoneda lemma, recall some other properties of the internal structure of a topos and, based on these, I critically comment on the notion of a complement-classifier (and thus of a co-topos as well).

19

Special Issue on point-free geometry and topology. An introduction

45%

Coppola C. , Gerla G.

Logic and Logical Philosophy

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2013

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tom 22

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nr 2

139-143

EN

In the first section we briefly describe the methodological assumptions of point-free geometry and topology. We also outline the history of geometrical theories based on the notion of a region. The second section is devoted to a concise presentation of the content of the LLP special issue on point-free theories of space.

20

Pattern-based Rewriting through Abstraction

45%

Bottoni P. , Guerra E. , de Lara J.

Fundamenta Informaticae

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2016

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tom Vol. 144, nr 2

109--160

EN

Model-based development relies on models in different phases for different purposes, with modelling patterns being used to document and gather knowledge about good practices in specific domains, to analyse the quality of existing designs, and to guide the construction and refactoring of models. Providing a formal basis for the use of patterns would also support their integration with existing approaches to model transformation. To this end, we turn to the commonly used, in this context, machinery of graph transformations and provide an algebraic-categorical formalization of modelling patterns, which can express variability and required/forbidden application contexts. This allows the definition of transformation rules having patterns in left and right-hand sides, which can be used to express refactorings towards patterns, change the use of one pattern by a different one, or switch between pattern variants. A key element in our proposal is the use of operations to abstract models into patterns, so that they can be manipulated by pattern rules, thus leading to a rewriting mechanism for classes of graphs described by patterns and not just individual graphs. The proposal is illustrated with examples in object-oriented software design patterns and enterprise architecture patterns, but can be applied to any other domain where patterns are used for modelling.