Coherence and valid isomorphism in closed categories applications of proof theory to category theory in a computer sclentist perspective

2005 ◽  
pp. 1-4 ◽  
Author(s):  
Guiseppe Longo
Keyword(s):  
Category Theory ◽  
Proof Theory

Categories for the Working Philosopher

2018 ◽  
Keyword(s):  
Mathematical Modeling ◽  
Computer Science ◽  
Category Theory ◽  
Proof Theory ◽  
Scientific Theories ◽  
General Audience ◽  
Seminal Work ◽  
The World

Borrowing from the title of Saunders Mac Lane’s seminal work Categories for the Working Mathematician, this book aims to bring the concepts of category theory to philosophers working in areas ranging from mathematics to proof theory to computer science to ontology, from physics to biology to cognition, from mathematical modeling to the structure of scientific theories to the structure of the world. Moreover, it aims to do this in a way that is accessible to a general audience. Each chapter is written by either a category-theorist or a philosopher working in one of the represented areas, and in a way that is accessible and is intended to build on the concepts already familiar to those philosophers working in these areas.

ETA-RULES IN MARTIN-LÖF TYPE THEORY

2019 ◽  
Vol 25 (03) ◽  
pp. 333-359
Author(s):  
ANSTEN KLEV
Keyword(s):  
Lower Order ◽  
Category Theory ◽  
Type Theory ◽  
Proof Theory ◽  
Arbitrary Element ◽  
Higher Order

AbstractThe eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher order eta rule is part of that type theory. The main aim of this article is to clarify this somewhat puzzling situation. It will be argued that lower order eta rules do not, whereas the higher order eta rule does, accord with the understanding of judgemental identity as definitional identity. A subsidiary aim is to clarify precisely what an eta rule is. This will involve showing how such rules relate to various other notions of type theory, proof theory, and category theory.

scholarly journals Identity of Proofs Based on Normalization and Generality

2003 ◽  
Vol 9 (4) ◽  
pp. 477-503 ◽  
Author(s):  
Kosta Došen
Keyword(s):  
Equivalence Relation ◽  
Category Theory ◽  
Classical Logic ◽  
Proof Theory ◽  
Natural Deduction ◽  
Lambda Calculus ◽  
Free Form ◽  
Low Dimensional Topology ◽  
Sequent Systems ◽  
Low Dimensional

AbstractSome thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal formin natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic.The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well.The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.

A logical framework combining model and proof theory

2013 ◽  
Vol 23 (5) ◽  
pp. 945-1001 ◽  
Author(s):  
FLORIAN RABE
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Type Theory ◽  
Proof Theory ◽  
Software Verification ◽  
Logical Framework ◽  
Dependent Type Theory ◽  
Logical Frameworks ◽  
Balanced Approach ◽  
Dependent Type

Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics.Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap – for example in applications to software verification – they have fundamentally different perspectives on logic.In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations.

Structural Proof Theory

2001 ◽  
Author(s):  
Sara Negri ◽  
Jan von Plato ◽  
Aarne Ranta
Keyword(s):  
Proof Theory ◽  
Structural Proof Theory

Basic Proof Theory

2000 ◽  
Author(s):  
A. S. Troelstra ◽  
H. Schwichtenberg
Keyword(s):  
Proof Theory

不定自然変換理論に基づく比喩理解モデルの計算論的実装の試み

2020 ◽  
Author(s):  
Shunsuke Ikeda ◽  
Miho Fuyama ◽  
Hayato Saigo ◽  
Tatsuji Takahashi
Keyword(s):  
Category Theory ◽  
Analogical Reasoning ◽  
Machine Learning Techniques ◽  
Huge Amount ◽  
Metaphor Comprehension ◽  
Weighted Directed Graph ◽  
Learning Techniques ◽  
Descriptive Validity ◽  
Association Data ◽  
Human Participants

Machine learning techniques have realized some principal cognitive functionalities such as nonlinear generalization and causal model construction, as far as huge amount of data are available. A next frontier for cognitive modelling would be the ability of humans to transfer past knowledge to novel, ongoing experience, making analogies from the known to the unknown. Novel metaphor comprehension may be considered as an example of such transfer learning and analogical reasoning that can be empirically tested in a relatively straightforward way. Based on some concepts inherent in category theory, we implement a model of metaphor comprehension called the theory of indeterminate natural transformation (TINT), and test its descriptive validity of humans' metaphor comprehension. We simulate metaphor comprehension with two models: one being structure-ignoring, and the other being structure-respecting. The former is a sub-TINT model, while the latter is the minimal-TINT model. As the required input to the TINT models, we gathered the association data from human participants to construct the ``latent category'' for TINT, which is a complete weighted directed graph. To test the validity of metaphor comprehension by the TINT models, we conducted an experiment that examines how humans comprehend a metaphor. While the sub-TINT does not show any significant correlation, the minimal-TINT shows significant correlations with the human data. It suggests that we can capture metaphor comprehension processes in a quite bottom-up manner realized by TINT.

Category Theory and Foundations

2018 ◽  
Author(s):  
Michael Ernst
Keyword(s):  
Category Theory ◽  
Mathematical Thinking ◽  
Mathematical Practice ◽  
Foundations Of Mathematics ◽  
Ongoing Debate ◽  
Technical Adequacy ◽  
Theoretical Foundations

In the foundations of mathematics there has been an ongoing debate about whether categorical foundations can replace set-theoretical foundations. The primary goal of this chapter is to provide a condensed summary of that debate. It addresses the two primary points of contention: technical adequacy and autonomy. Finally, it calls attention to a neglected feature of the debate, the claim that categorical foundations are more natural and readily useable, and how deeper investigation of that claim could prove fruitful for our understanding of mathematical thinking and mathematical practice.

Proof Theory of the Cut Rule

2018 ◽  
Author(s):  
J. R. B. Cockett ◽  
R. A. G. Seely
Keyword(s):  
Proof Theory ◽  
Graphical Representation ◽  
Basic Component ◽  
Graphical Representations ◽  
Mathematical Language ◽  
Basic Logic ◽  
Cut Rule ◽  
Structural Rules ◽  
The One ◽  
Categorical Semantics

This chapter describes the categorical proof theory of the cut rule, a very basic component of any sequent-style presentation of a logic, assuming a minimum of structural rules and connectives, in fact, starting with none. It is shown how logical features can be added to this basic logic in a modular fashion, at each stage showing the appropriate corresponding categorical semantics of the proof theory, starting with multicategories, and moving to linearly distributive categories and *-autonomous categories. A key tool is the use of graphical representations of proofs (“proof circuits”) to represent formal derivations in these logics. This is a powerful symbolism, which on the one hand is a formal mathematical language, but crucially, at the same time, has an intuitive graphical representation.

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