IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Some More Results on Intuitionistic Semi Open SetsIJERA Editor
The purpose of this paper is to define and study the intuitionistic semi open sets in intuitionistic topological
space. Also some properties of this set are discussed.
Dialectica Categories Surprising Application: mapping cardinal invariantsValeria de Paiva
Talk at 2nd Set Theory and General Topology Week in Salvador, Bahia, Brazil, March 2012
Abstract: Goethe famously said that "Mathematicians are like Frenchmen:
whatever you say to them they translate into their own language and forthwith it is something entirely different." True. Even more true of category theorists. Following this great tradition of appropriating other people's work, I want to tell you how I learned about "cardinalities of the continuum" from Blass and Morgan and da Silva
and how I want to rock their boat, just a little, in the direction of my kind of mathematics.
Date: March 9, 2016
Course: UiS DAT911 - Foundations of Computer Science (fall 2016)
Please cite, link to or credit this presentation when using it or part of it in your work.
Memetic Algorithms have become one of the key methodologies behind solvers that are capable of tackling very large, real-world, optimisation problems. They are being actively investigated in research institutions as well as broadly applied in industry. In this talk we provide a pragmatic guide on the key design issues underpinning Memetic Algorithms (MA) engineering. We begin with a brief contextual introduction to Memetic Algorithms and then move on to define a Pattern Language for MAs. For each pattern, an associated design issue is tackled and illustrated with examples from the literature. We then fast forward to the future and mention what, in our mind, are the key challenges that scientistis and practitioner will need to face if Memetic Algorithms are to remain a relevant technology in the next 20 years.
Some More Results on Intuitionistic Semi Open SetsIJERA Editor
The purpose of this paper is to define and study the intuitionistic semi open sets in intuitionistic topological
space. Also some properties of this set are discussed.
Dialectica Categories Surprising Application: mapping cardinal invariantsValeria de Paiva
Talk at 2nd Set Theory and General Topology Week in Salvador, Bahia, Brazil, March 2012
Abstract: Goethe famously said that "Mathematicians are like Frenchmen:
whatever you say to them they translate into their own language and forthwith it is something entirely different." True. Even more true of category theorists. Following this great tradition of appropriating other people's work, I want to tell you how I learned about "cardinalities of the continuum" from Blass and Morgan and da Silva
and how I want to rock their boat, just a little, in the direction of my kind of mathematics.
Date: March 9, 2016
Course: UiS DAT911 - Foundations of Computer Science (fall 2016)
Please cite, link to or credit this presentation when using it or part of it in your work.
Memetic Algorithms have become one of the key methodologies behind solvers that are capable of tackling very large, real-world, optimisation problems. They are being actively investigated in research institutions as well as broadly applied in industry. In this talk we provide a pragmatic guide on the key design issues underpinning Memetic Algorithms (MA) engineering. We begin with a brief contextual introduction to Memetic Algorithms and then move on to define a Pattern Language for MAs. For each pattern, an associated design issue is tackled and illustrated with examples from the literature. We then fast forward to the future and mention what, in our mind, are the key challenges that scientistis and practitioner will need to face if Memetic Algorithms are to remain a relevant technology in the next 20 years.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Talk at the Melbourne Logic Seminar
on Shawn Standefer invitation
Dialectica categories for the Lambek calculus
Valeria de Paiva
The Topos Institute, Berkeley, CA
Abstract:
Dialectica categorical models of the Lambek Calculus were first presented in the Amsterdam Colloquium a long time ago. Following Lambek's lead, we approached the Lambek Calculus from the perspective of Linear Logic and adapted the Dialectica categorical models for Linear Logic to Lambek's non-commutative calculus. The old work took for granted the syntax of the Lambek calculus and only discussed the exciting possibilities of new models for the modalities that Linear Logic introduced. Many years later we find that the work on dialectica models of the Lambek calculus is still interesting and that it might inform some of the most recent work on the relationship between Categorial Grammars and notions of Distributional Semantics.
Thus we revisited the old work, making sure that the syntax details that were sketchy on the first version got completed and verified, using automated tools such as Agda and Ott. Ultimately we are interested in the applicability of the original systems to their intended uses in the construction of semantics of Natural Language. But before we can discuss it, we need to make sure that the mathematical properties that make the Lambek calculus attractive are all properly modeled and this is the main aim of this paper.
We recall the Lambek calculus with its Curry-Howard isomorphic term assignment system. We extend it with a $\kappa$ modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality $!$, as Girard did, re-introducing weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present algebraic semantics and categorical semantics, proved sound and complete for the whole system. Finally, we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Le nuove frontiere dell'AI nell'RPA con UiPath Autopilot™UiPathCommunity
In questo evento online gratuito, organizzato dalla Community Italiana di UiPath, potrai esplorare le nuove funzionalità di Autopilot, il tool che integra l'Intelligenza Artificiale nei processi di sviluppo e utilizzo delle Automazioni.
📕 Vedremo insieme alcuni esempi dell'utilizzo di Autopilot in diversi tool della Suite UiPath:
Autopilot per Studio Web
Autopilot per Studio
Autopilot per Apps
Clipboard AI
GenAI applicata alla Document Understanding
👨🏫👨💻 Speakers:
Stefano Negro, UiPath MVPx3, RPA Tech Lead @ BSP Consultant
Flavio Martinelli, UiPath MVP 2023, Technical Account Manager @UiPath
Andrei Tasca, RPA Solutions Team Lead @NTT Data
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
By Design, not by Accident - Agile Venture Bolzano 2024
Fibrational Versions of Dialectica Categories
1. Fibrational Versions of Dialectica Categories
Valeria de Paiva
Cuil, Inc.
Stanford Logic Seminar
May 2010
2. Outline
Introduction
Original Dialectica Categories (de Paiva, 1988)
Original Fibrational Dialectica (Hyland, 2002)
Cartesian Closed Dialectica categories (Biering, 2008)
Discussion
3. Introduction: What, why?
(Prove existence and describe) Cartesian Closed Dialectica
categories
Chapter 4 of Bodil Biering’s PhD thesis “Dialectica
Interpretations: A Categorical Analysis”, 2008
¨
Why? Categorical understanding of Godel’s Dialectica
Interpretation
What for?
¨
For Godel, the interpretation was a way of proving
consistency of arithmetic, an extension of Hilbert’s
programme
For me (20 years ago) a way of producing models of Linear
Logic from a proven way of understanding logic
For Biering: a way of unifying categorical structures,
brought back to proof theory?...
4. Biering’s Thesis Abstract
The Dialectica interpretations are remarkable syntactic
constructions
Use these constructions to develop new mathematical
structures: Dialectica categories, the Dialectica- and
Diller-Nahm triposes, and the Dialectica- and Diller-Nahm
toposes
The mathematical structures created from the functional
interpretations provide us with new models for type
theories and programming logics
Studying the mathematical structures we gain new insights
into the syntactical constructions.
Product: Biering et al “Copenhagen interpretation”
5. Biering’s Thesis Table of Contents
A collection of articles, want to discuss chapter 4...
Introduction
Topos Theoretic Versions of Dialectica Interpretations
A Unified View on the Dialectica Triposes
Cartesian Closed Dialectica Categories
The Copenhagen interpretation
(BI Hyperdoctrines and Higher Order Separation Logic)
6. Functional Interpretations
¨
Starting with Godel’s Dialectica interpretation a series of
”translation functions” between theories
Avigad and Feferman on the Handbook of Proof Theory:
¨
This approach usually follows Godel’s original
example: first, one reduces a classical theory C to
a variant I based on intuitionistic logic; then one
reduces the theory I to a quantifier-free functional
theory F.
Examples of functional interpretations:
Kleene’s realizability
Kreisel’s modified realizability
Kreisel’s No-CounterExample interpretation
Dialectica interpretation
Diller-Nahm interpretation
7. ¨
Godel’s Dialectica Interpretation
For each formula A of HA we associate a formula of the form
AD = ∃u∀xAD (u, x) (where AD is a quantifier-free formula of
¨
Godel’s system T) inductively as follows: when Aat is an atomic
formula, then its interpretation is itself.
Assume we have already defined AD = ∃u∀x.AD (u, x) and
B D = ∃v ∀y .BD (v , y ).
We then define:
(A ∧ B)D = ∃u, v ∀x, y.(AD ∧ BD )
(A → B)D = ∃f : U → V , F : U × X → Y , ∀u, y.
( AD (u, F (u, y )) → BD (fu; y))
(∀zA)D (z) = ∃f : Z → U∀z, x.AD (z, f (z), x)
(∃zA)D (z) = ∃z, u∀x.AD (z, u, x)
The intuition here is that if u realizes ∃u∀x.AD (u, x) then f (u)
realizes ∃v ∀y.BD (v , y) and at the same time, if y is a
counterexample to ∃v ∀y .BD (v , y ), then F (u, y) is a
counterexample to ∀x.AD (u, x).
8. Categorical Dialectica Interpretation
Main references:
The ‘Dialectica’ Interpretation and Categories (P. J. Scott,
Zeit. fur Math Logik und Grund. der Math. 24, 1978)
The Dialectica Categories, (de Paiva, AMS vol 92, 1989)
Proof Theory in the Abstract, (Hyland, APAL 2002)
Dialectica categories are naturally symmetric monoidal
closed, but not cartesian closed categories.
We would like them to be cartesian closed.
Why?
Can we make them Cartesian closed?
Yes, in different ways.
9. Plan of Biering’s ‘Cartesian Closed Dialectica
Categories’
Recall the definitions and basic properties of original
dialectica categories
Discuss three approaches to classes of Cartesian closed
Dialectica categories
Preferred way explained, leads to a generalisation of
construction
Show monads and comonads in generalisation
Example of non-Girardian comonad that produces weak
exponentials
Example of extensional version of Dialectica
Conclusions
10. Original Dialectica Categories (V. de Paiva, 1987)
Suppose that we have a category C, with finite limits,
interpreting some type theory.
The category Dial(C) has as objects triples A = (U, X , α),
where U, X are objects of C and α is a sub-object of U × X ,
that is a monic in Sub(U × X ).
A map from A = (U, X , α) to B = (V , Y , β ) is a pair of maps
(f , F ) in C, f : U → V , F : U × X → Y such that
α(u, F (u, y )) ≤ β (f (u), y)
The predicate α is not symmetric: read (U, X , α) as
∃u.∀x.α(u, x), a proposition in the image of the Dialectica
interpretation. The functionals f and F correspond to the
dialectica interpretation of implication.
11. Original Dialectica Categories
My thesis has four chapters, four main definitions and four main
theorems. The first two chapters are about the “original”
dialectica categories.
Theorem (V de Paiva, 1987)
If C is a ccc with stable, disjoint coproducts, then Dial(C) has
products, tensor products, units and a linear function space
(1, ×, ⊗, I, →) and Dial(C) is symmetric monoidal closed.
This means that Dial(C) models Intuitionistic Linear Logic (ILL)
without modalities. How to get modalities? Need to define a
special comonad and lots of work to prove theorem 2...
12. Original Dialectica Categories
!A must satisfy !A →!A⊗!A, !A ⊗ B →!A, !A → A and !A →!!A,
together with several equations relating them.
The point is to define a comonad such that its coalgebras are
commutative comonoids and the coalgebra and the comonoid
structure interact nicely.
Theorem (V de Paiva, 1987)
Given C a cartesian closed category with free monoids
(satisfying certain conditions), we can define a comonad T on
Dial(C) such that its Kleisli category Dial(C)T is cartesian
closed.
Define T by saying A = (U, X , α) goes to (U, X ∗ , α ∗ ) where X ∗
is the free commutative monoid on X and α ∗ is the multiset
version of α.
Loads of calculations prove that the linear logic modalitiy ! is
well-defined and we obtain a full model of ILL and IL, a
posteriori of CLL.
Construction generalized in many ways, cf. dePaiva, TAC, 2006.
13. Fibrational Dialectica (M. Hyland, 2002)
Given C a category with finite limits and a pre-ordered fibration
p : E → C with for each I in C, a preordered collection of
predicates E(I) = (E(I), ), construct a category Dial(p). The
objects A of Dial(p) are triples (U, X , α) where U, X are objects
in C and α is in E(U × X ). Maps in Dial(p) are pairs of maps in
C,(f , F ) in C, f : U → V , F : U × X → Y such that
α(u, F (u, y)) β (f (u), y)
Theorem (Hyland, 2002)
If C is a CCC and p : E → C is (pre-ordered) fibered cartesian
closed then Dial(p) is symmetric monoidal closed.
∼
If, moreover, C has finite, distributive coproducts and E(0) = 1
and the injections X → X + Y and Y → X + Y induce an
equivalence E(X + Y ) ≡ E(X ) × E(Y ) then Dial(p) has finite
products.
This is a generalization of the original Dialectica categories, where
the fibration is the subobject fibration.
14. Categorical Logic in slogans
“categorical model theory” or “categorical proof theory”?
Categorical model theory: model theory, where models are
categories, instead of sets.
Categorical proof theory: models are categories,
propositions are objects, derivations are arrows in the
category. concept of two different proofs being ‘the same’
Logic connectives correspond to structure in the
categories, Lawvere and Lambek in the 60’s.
Textbook: Lambek and Scott “Introduction to higher order
categorical logic”, 1986.
Main example: Intuitionistic propositional logic modeled as
Cartesian Closed Categories (CCCs).
Works for first-order intuitionistic logic too, models are
hyperdoctrines
15. Fibrations for Dummies
Fibrations can be used for modeling several kinds of type
theories. Too complicated, perhaps?
Intuition: given a set I, consider the I-indexed family of sets
(Xi )i∈I or X (i), for each i in I.
Want something similar where instead of indexing sets, we
have a base category B, creating a “family of objects of a
category indexed by objects I in the base category B”
There are two “equivalent” ways of doing this, using
“indexed categories” or “fibrations”.
A fibration is a structure p : E → B, where p is a functor
and E, B are categories, satisfying certain axioms.
Think of B as Sets and E as Fam(Sets) where Fam(Sets)
is the category where objects are families of sets {Xi }i∈I , I
and Xi are sets. Maps from f : {Xi }i∈I → {Yj }j∈J consist of
a function φ : I → J (a re-indexing function) together with a
family of maps {fi : Xi → Yφ (i) }i∈I
16. Fibrations for Dummies 2
Which certain axioms? Given a functor p : E → B when
can we think of each object X in E as a family (Xi )i∈I
indexed by I = pX in B?
There is a functor p : Fam(Sets) → (Sets) that takes the
family {Xi }i∈I to I.
Call a map f : {Xi }i∈I → {Yj }j∈J , vertical if pf = 1I – no
re-indexing going on.
Call a map f : {Xi }i∈I → {Yj }j∈J , cartesian if each fi is an
isomorphism, pure re-indexing, fi s don’t do any work.
Given a family of sets {Xi }i∈I any map α : K → I induces a
(cartesian) map f : {Xα(k) }k∈K → {Xi }i∈I where each fi is
the identity.
cartesian maps have a universal property: an arbitrary
map f factors through a cartesian map g, when φ = pf
factors through α = pg and the factorization is uniquely
determined.
17. Fibrations for Dummies 3
Definition
A map g : X → X in E is called cartesian if given any map
f : Y → X , each factorization of φ = pf through α = pg uniquely
determines a factorization of f through g.
[insert picture]
Definition
A functor p : E → B is a fibration, if for all X in E and maps
α : K → I = pX there exists an object X and a cartesian map
g : X → X such that pg = α.
Cartesian liftings are unique up to iso, so: If p : E → B is a
fibration, a cleavage for this fibration is a particular choice of
cartesian liftings. A fibration equipped with a particular
cleavage is called a cloven fibration. (usually to show a
functor is a fibration, we produce a cleavage)
18. Proof Theory in the Abstract (Troelstra Fest)
A destilation of the program of ”proof theory in the abstract” as
developed since the late 60s. Structure of paper in 2002:
Background
Dialectica
Diller-Nahm
Classical logic
Highly recommended for philosophy and clarity of explanation
of the problems with classical logic.
19. Proof Theory in the Abstract (Troelstra Fest)
For Dialectica main theorems are:
Theorem (Hyland Thm 2.3, p.8, 2002)
If T is a CCC interpreting some type theory and p : P → T is
(pre-ordered) fibered cartesian closed then Dial = Dial(p) as
defined is symmetric monoidal closed.
Natural propositional structure in place
Theorem (Hyland Thm 2.5, p.9, 2002)
The fibration q : Dial → T has left and right adjoints to
re-indexing along product projections. These satisfy the
Beck-Chevalley conditions.
Predicate structure in place.
But to really interpret Dialectica we need conjunctions and
disjunctions. Here the maths turns out not be so pretty.
20. Proof Theory in the Abstract (Troelstra Fest)
For conjunction, to cope with diagonals A → A ⊗ A need ‘weak
cases definition’. To cope with projections A ⊗ B → A need
inhabited types. For disjunction we need a weak coproduct as
well as a weak initial object and a codiagonal. The structure
can be made to work, but it ain’t good category theory. (These
problems and way-outs were known from the non-fibrational case).
Theorem (Hyland, Thm 2.6 p.14, 2002)
The poset reflection of our indexed category Dial → T of proofs is a
first-order hyperdoctrine: we get indexed Heyting algebras and good
quantification.
Diller-Nahm has better structure!
Hyland suggests an abstract analysis of the interpretation of set
theory studied by Burr.
Jacobs, Streicher and de Paiva first version of 2.3 in note in 1995 (Jacob’s book exercise). Streicher’ note “A
¨
Semantic Version of the Diller-Nahm Variant of Godel’s Dialectica Interpretation”, 2000.
21. Cartesian Closed Dialectica Categories?
Several people took up more (categorical) analysis of
Dialectica, e.g. Streicher, Rosolini, Birkedal. etc...
The natural structure of Dial(p) is smcc with finite products.
Biering: How can we make it cartesian closed?
Make the tensor product, a cartesian product
Hyland’s ‘hackery’ above
Get a Girardian comonad on the Dialectica category
Diller-Nahm variants (VdP88), (Strei?), (Burr98)?, ...
Add enough structure to define a weak function space –
without making the tensor a product, (Biering06)
22. Cloven Dialectica Categories
Hyland defined the category Dial(p) for pre-ordered fibrations.
Biering generalized it, using fibrations into usual categories,
requiring p to be a cloven fibration. This is needed to obtain
associativity of composition in Dial(p). She defines Dial(p) and
prove it’s a category. With appropriate conditions it has some of
the structure we want.
Theorem (Biering, 2008)
Let p : E → T be a cloven fibration. If T has finite, distributive
coproducts and products, and the injections X → X + Y and
Y → X + Y induce an equivalence µ : E(X ) × E(Y ) ≡ E(X + Y ),
natural in X and Y , then Dial(p) has binary products.
∼
Moreover, if E(0) = 1, then Dial(p) has a terminal object.
The proof is a direct generalization of Hyland’s proof.
How can one make the category Dial(p) above cartesian
closed? It has products and a terminal object, but no function
spaces. Biering will define ‘weak’ function spaces for a
particular example of fibration, the codomain fibration.
23. Codomain Dialectica Categories?
What is the codomain fibration?
For any category C, there is a category Arr (C), whose objects
are the arrows of C, say α : X → I.
A morphism in Arr (C) from α : X → I to β : Y → J consists of a
pair of morphisms, f : X → Y and g : I → J.
There is a functor cod : Arr (C) → C which sends an object in
Arr (C), say α : X → I to its codomain I, and a morphism, the
square, (f : X → Y , g : I → J) to g, the ‘lower’ edge.
The functor cod : Arr (C) → C is a fibration.
Theorem (Biering, Prop 3.7 p6, 2008)
Let C be a category with finite limits and finite coproducts, and
assume that the coproducts are stable and disjoint, then
Dial(cod(C)) has finite products.
Can we make Dial(cod(C)) cartesian closed? Almost...
24. A comonad in Dial(p)
Definition
Let C be a category with finite products and stable, disjoint
coproducts. The functor − + 1 : C → C together with families of
maps ι : X → X + 1 and µ : (X + 1) + 1 → X + 1 is a monad on
C. Define a comonad L+ on the subobject fibration
Dial(Sub(C)) using the monad − + 1 as follows. Let α be a
subobject of U × X in C , then L + (α, U, X ) = (α+, U, (X + 1))
where α+ is reindexing of α along the arrow
∼
U × (X + 1) = U × X + U → (U × X ) + 1.
This means that α + (u, x) = α(u, x), if x is in X, if x is in 1.
This comonad simply makes the second coordinate
well-pointed.
25. A comonad in Dial(p)
The comonad L+ just defined is not Girardian, that is we do not
have the isomorphism
∼
!(A × B) =!A⊗!B
Had this iso being satisfied, then the Kleisli category of the
comonad would be cartesian closed. As it is, the best we can
do is to have weak function spaces, ie is a retraction
C(A × B, C) C(A, [B, C])
where the weak function space is given by
B → C = (W V × (1 + Y )V ×Z ) , V × Z , γ).
Theorem (Biering, 2008, 4.7, p 12)
Let C be a cartesian closed category with finite limits, and
stable, disjoint coproducts, which is locally cartesian closed.
Then the Dialectica-Kleisli category, DialL + (cod(C)), which we
denote by Dial+, has finite products and weak function spaces.
Same is true for DialL + (Sub(C)).
Main result, proof takes 6 pages, ‘souped up’ proof of dial thm
26. Discussion
Examples of fibrations that meet the conditions of Theorem 4.7:
cod(PER) → PER
cod(Set) → Set
the codomain fibration cod(C) → C for a topos C and
the subobject fibration Sub(C) → C
Biering decribes example where the fibration is
Fam(PER) → PER (a per is a partial equivalence relation), the
product as as well as the weak function space in
Dial + (Fam(PER)). But calculating there is unwieldy.
27. Discussion
Biering concludes: Two new variants of Dialectica
categories.
First, make proof theory out of Dial + . One needs to extend
¨
Godel’s T with stable, disjoint coproducts and subset types
and interpret implication by new weak function space.
Advantage: don’t need atomic formulas to be decidable.
Second, make proof theory of Dial(p). A type-theoretic
version of Dialectica. Instead of formulas over Heyting
arithmetic (original dialectica) have dependent types over
some type system and Dialectica turns the dependent type
system into a lambda-calculus without eta-rule.
Advantage?
Further work: Other comonads for Dialectica? Oliva’s
work. Do PER model better? Didn’t describe structure for
generalized dialectica categories, apart from products.
28. Conclusions
A worked out example of “proof theory in the abstract”
dialectica categories Dial(C) (1988)
dialectica pre-ordered fibration Dial(p) (2002)
Dialectica-Kleisli category Dial + (2008)
More to come Oliva, Hofstra, Triffonov, etc..?
THANK YOU!
29. References
Troelstra, A. S. (1973) Metamathematical Investigation of
Intuitionistic Arithmetic and Analysis, Springer-Verlag.
¨
W. Hodges and B. Watson: translation of K. Godel, ’Uber ¨
eine bisher noch nicht bentzte Erweiterung des finiten
Standpunktes’, JPL 9 (1980)
¨
Avigad, Feferman.Godel’s Functional (Dialectica)
Interpretation
V de Paiva. The Dialectica Categories, AMS-92, 1989
Burr, W. (1999) Concepts and aims of functional
interpretations: towards a functional interpretation of
constructive set theory.
M. Hyland, Proof Theory in the Abstract, APAL, 2002.
¨
P. Oliva. An analysis of Godel’s Dialectica interpretation via
linear logic. Dialectica,2008
Biering. Dialectica Interpretations: a categorical analysis,
PHD Thesis, 2008
¨
Collected Works of Kurt Godel, vol 2 Feferman et al, 1990