Dual Calculus Research Articles

Drinfel’d double of bialgebroids for string and M theories: dual calculus framework

We extend the notion of Lie bialgebroids for more general bracket structures used in string and M theories. We formalize the notions of calculus and dual calculi on algebroids. We achieve this by reinterpreting the main results of the matched pairs of Leibniz algebroids. By examining a rather general set of fundamental algebroid axioms, we present the compatibility conditions between two calculi on vector bundles which are not dual in the usual sense. Given two algebroids equipped with calculi satisfying the compatibility conditions, we construct its double on their direct sum. This generalizes the Drinfel’d double of Lie bialgebroids. We discuss several examples from the literature including exceptional Courant brackets. Using Nambu-Poisson structures, we construct an explicit example, which is important both from physical and mathematical point of views. This example can be considered as the extension of triangular Lie bialgebroids in the realm of higher Courant algebroids, that automatically satisfy the compatibility conditions. We extend the Poisson generalized geometry by defining Nambu-Poisson exceptional generalized geometry and prove some preliminary results in this framework. We also comment on the global picture in the framework of formal rackoids and we slightly extend the notion for vector bundle valued metrics.

Developing a Model for Understanding the Factors that Affect Individuals' Intentions to Share Personal Information

With the outbreak of COVID-19, digital contact tracing (DCT) applications have been implemented to track the transmission of the highly contagious disease. However, there has been significant resistance to using these apps due to individuals' concerns about disclosing personal information. This study aimed to examine individuals' intentions to reveal their details on DCT applications. A conceptual framework was developed incorporating theories of dual calculus, Hofstede's cultural theory, information boundary theory, and individual factors. A quantitative approach was employed, using a random sampling methodology to gather data from 533 respondents. The proposed framework was validated through partial least squares path modeling. The findings revealed that COVID-19-related stress, lack of transparency, and uncertainty avoidance negatively predicted intentions to disclose personal information. Conversely, collectivism, expected community-related outcomes of sharing information, expected personal outcomes of sharing information, and information privacy concerns positively predicted intentions to disclose personal information. This research provides insights into the factors that influence the widespread acceptance of DCT applications and can assist related authorities in ensuring their successful implementation.

Role of Algorithm Awareness in Privacy Decision-Making Process: A Dual Calculus Lens

In the context of AI, as algorithms rapidly penetrate e-commerce platforms, it is timely to investigate the role of algorithm awareness (AA) in privacy decisions because it can shape consumers’ information-disclosure behaviors. Focusing on the role of AA in the privacy decision-making process, this study investigated consumers’ personal information disclosures when using an e-commerce platform with personalized algorithms. By integrating the dual calculus model and the theory of planned behavior (TPB), we constructed a privacy decision-making model for consumers. Sample data from 581 online-shopping consumers were collected by a questionnaire survey, and SmartPLS 4.0 software was used to conduct a structural equation path analysis and a mediating effects test on the sample data. The findings suggest that AA is a potential antecedent to the privacy decision-making process through which consumers seek to evaluate privacy risks and make self-disclosure decisions. The privacy decision process goes through two interrelated trade-offs—that threat appraisals and coping appraisals weigh each other to determine the (net) perceived risk and, then, the (net) perceived risk and the perceived benefit weigh each other to decide privacy attitudes. By applying the TPB to the model, the findings further show that privacy attitudes and subjective norms jointly affect information-disclosure intention whereas perceived behavioral control has no significant impact on information-disclosure intention. The results of this study give actionable insights into how to utilize the privacy decision-making process to promote algorithm adoption and decisions regarding information disclosure, serving as a point of reference for the development of a human-centered algorithm based on AA in reference to FEAT.

A typed lambda-calculus with first-class configurations

Abstract Filinski and Griffin have independently succeeded in extending the formulae-as-types notion to deal with continuations. Whereas Griffin adopted control operators primitively, Filinski adopted the duality of functions to construct a symmetric lambda-calculus in which continuations are first-class objects. In this paper, we construct a typed lambda-calculus with first-class configurations consisting of expressions and continuations of the same types. Our calculus corresponds to a natural deduction based on Rumfitt’s bilateralism. Function types are represented as the implication and but-not connectives in intuitionistic and paraconsistent logics, respectively. Our calculus is not only logically consistent, but also computationally consistent. Our calculus with call-by-value and call-by-name strategies correspond to Wadler’s call-by-value and call-by-name dual calculi, respectively. Furthermore, we propose a notion of relaxed configurations, which loosely take expressions and continuations of different types. We confirm that the relaxedness defines control operators for delimited continuations.

Digital Health Innovation: Exploring Adoption of COVID-19 Digital Contact Tracing Apps

With the outbreak of COVID-19, contact tracing is becoming a used intervention to control the spread of this highly infectious disease. This article explores an individual's intention to adopt COVID-19 digital contact tracing (DCT) apps. A conceptual framework developed for this article combines the procedural fairness theory, dual calculus theory, protection motivation theory, theory of planned behavior, and Hofstede's cultural dimension theory. The study adopts a quantitative approach collecting data from 714 respondents using a random sampling technique. The proposed model is tested using structural equation modeling. Empirical results found that the perceived effectiveness of privacy policy negatively influenced privacy concerns, whereas perceived vulnerability had a positive influence. Expected personal and community-related outcomes of sharing information positively influenced attitudes toward DCT apps, while privacy concerns had a negative effect. The intention to adopt DCT apps were positively influenced by attitude, subjective norms, and privacy self-efficacy. This article is the first to empirically test the adoption of DCT apps of the COVID-19 pandemic and contributes both theoretically and practically toward understanding factors influencing its widespread adoption.

Classical logic with Mendler induction

AbstractWe investigate (co-) induction in classical logic under the propositions-as-types paradigm, considering propositional, second-order and (co-) inductive types. Specifically, we introduce an extension of the Dual Calculus with a Mendler-style (co-) iterator and show that it is strongly normalizing. We prove this using a reducibility argument.

Factors influencing willingness to provide personal information for personalized recommendations

By reducing the trouble of searching and selecting the right information, recommendation systems help users obtain information efficiently. However, recommendation systems also raise concerns over privacy issues. This paper thus examined consumers’ intentions to provide personal information based on the dual calculus model which incorporates two interrelated trade-offs. An online survey targeting those who have experience using YouTube was conducted. Five distinct types of personal information (usage patterns, personal identifiers, biographical information, usage context, and feedback information) were selected based on an exploratory factor analysis. The results imply that perceived benefits, convenience, and vulnerability do not have significant effects on intention to disclose information. Perceived severity, however, had a significant negative effect on intention to disclose all types of information. Coping efficacy showed positive effects on willingness to disclose personal data except for feedback information. Self-efficacy, on the other hand, had a positive effect only on disclosing feedback information. This study expands the scope of discussion on personal information by reflecting the recent expansion of the collection and use of information for recommendation systems. It also sheds light on how to identify and handle each type of information differently.

Health information privacy concerns, antecedents, and information disclosure intention in online health communities

This study explores the antecedents and consequences of health information privacy concerns in online health communities by integrating the dual calculus and protection motivation theories. On the basis of survey data from 337 users, health information privacy concerns, together with informational and emotional support, significantly influence personal health information (PHI) disclosure intention. Privacy concerns are negatively influenced by two coping appraisals (i.e., response efficacy and self-efficacy) and positively affected by two threat appraisals (i.e., perceived vulnerability and perceived severity). The perceived health status differentially moderates the effects of privacy concerns and informational support on the PHI disclosure intention.

Goodwillie calculus via adjunction and LS cocategory

In this paper, we show that for reduced homotopy endofunctors of spaces, F, and for all $n \geq 1$ there are adjoint functors $R_n, L_n$ with $T_n F \simeq R_n F L_n$, where $P_n F$ is the $n$-excisive approximation to $F$, constructed by taking the homotopy colimit over iterations of $T_n F$. This then endows $T_n$ of the identity with the structure of a monad and the $T_n F$'s are the functor version of bimodules over that monad. It follows that each $T_n F$ (and $P_nF$) takes values in spaces of symmetric Lusternik-Schnirelman cocategory $n$, as defined by Hopkins. This also recovers recent results of Chorny-Scherer. The spaces $T_n F(X)$ are in fact classically nilpotent (in the sense of Berstein-Ganea) but not nilpotent in the sense of Biedermann and Dwyer. We extend the original constructions of dual calculus to our setting, establishing the $n$-co-excisive approximation for a functor, and dualize our constructions to obtain analogous results concerning constructions $T^n$, $P^n$,and LS category.

Call-by-Value and Call-by-Name Dual Calculi with Inductive and Coinductive Types

This paper extends the dual calculus with inductive types and coinductive types. The paper first introduces a non-deterministic dual calculus with inductive and coinductive types. Besides the same duality of the original dual calculus, it has the duality of inductive and coinductive types, that is, the duality of terms and coterms for inductive and coinductive types, and the duality of their reduction rules. Its strong normalization is also proved, which is shown by translating it into a second-order dual calculus. The strong normalization of the second-order dual calculus is proved by translating it into the second-order symmetric lambda calculus. This paper then introduces a call-by-value system and a call-by-name system of the dual calculus with inductive and coinductive types, and shows the duality of call-by-value and call-by-name, their Church-Rosser properties, and their strong normalization. Their strong normalization is proved by translating them into the non-deterministic dual calculus with inductive and coinductive types.

Duality between Call-by-value Reductions and Call-by-name Reductions

Wadler proposed the dual calculus, which corresponds to classical sequent calculus LK, and studied the relationship between the λμ-calculus and the dual calculus as equational systems to explain the duality between call-by-value and call-by-name in a purely syntactical way. Wadler left an open question whether one can obtain similar results by replacing the equations with reductions. This paper gives one answer to his question. We first refine the λμ-calculus as reduction systems by reformulating sum types and omitting problematic reduction rules that are not simulated by reductions of the dual calculus. Secondly, we give translations between the call-by-name λμ-calculus and the call-by-name dual calculus, and show that they preserve the call-by-name reductions. We also show that the compositions of these translations become identity maps up to the call-by-name reductions. We also give translations for the call-by-value systems, and show that they satisfy properties similar to the call-by-name translations. Thirdly, we introduce translations between the call-by-value λμ-calculus and the call-by-name one by composing the above translations with duality on the dual calculus. We finally obtain results corresponding to Wadler's, but our results are based on reductions.

Investigations on the Dual Calculus

The Dual Calculus, proposed recently by Wadler, is the outcome of two distinct lines of research in theoretical computer science: (A) Efforts to extend the Curry–Howard isomorphism, established between the simply-typed lambda calculus and intuitionistic logic, to classical logic. (B) Efforts to establish the tacit conjecture that call-by-value (CBV) reduction in lambda calculus is dual to call-by-name (CBN) reduction. This paper initially investigates relations of the Dual Calculus to other calculi, namely the simply-typed lambda calculus and the Symmetric lambda calculus. Moreover, Church–Rosser and Strong Normalization properties are proven for the calculus’ CBV reduction relation. Finally, extensions of the calculus to second-order types are briefly introduced.

Anti-intuitionism and paraconsistency

This paper aims to help to elucidate some questions on the duality between the intuitionistic and the paraconsistent paradigms of thought, proposing some new classes of anti-intuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is shown here that anti-intuitionistic logics are paraconsistent, and in particular we develop a first anti-intuitionistic hierarchy starting with Johansson's dual calculus and ending up with Gödel's three-valued dual calculus, showing that no calculus of this hierarchy allows the introduction of an internal implication symbol. Comparing these anti-intuitionistic logics with well-known paraconsistent calculi, we prove that they do not coincide with any of these. On the other hand, by dualizing the hierarchy of the paracomplete (or maximal weakly intuitionistic) many-valued logics ( I n ) n ∈ ω we show that the anti-intuitionistic hierarchy ( I n ∗ ) n ∈ ω obtained from ( I n ) n ∈ ω does coincide with the hierarchy of the many-valued paraconsistent logics ( P n ) n ∈ ω . Fundamental properties of our method are investigated, and we also discuss some questions on the duality between the intuitionistic and the paraconsistent paradigms, including the problem of self-duality. We argue that questions of duality quite naturally require refutative systems (which we call elenctic systems) as well as the usual demonstrative systems (which we call deictic systems), and multiple-conclusion logics are used as an appropriate environment to deal with them.

Tate cohomology and periodic localization of polynomial functors

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S--module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S--modules to S--modules, there is an associated tower under F, {P_dF}, such that F --> P_dF is the universal arrow to a d--excisive functor. Our first theorem says that P_dF --> P_{d-1}F always admits a homotopy section after localization with respect to T(n) (and so also after localization with respect to Morava K--theory K(n)). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second theorem which is equivalent to the following: for any finite group G, the Tate spectrum t_G(T(n)) is weakly contractible. This strengthens and extends previous theorems of Greenlees--Sadofsky, Hovey--Sadofsky, and Mahowald--Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus.

Semantics for Dual and Symmetric Combinatory Calculi

We define dual and symmetric combinatory calculi (inequational and equational ones), and prove their consistency. Then, we introduce algebraic and set theoretical– relational and operational – semantics, and prove soundness and completeness. We analyze the relationship between these logics, and argue that inequational dual logics are the best suited to model computation.

Splitting criteria for homotopy functors of spectra

We explore the interaction between the Taylor tower and cotower, as defined in deriving calculus with cotriples and dual calculus for functors to spectra of functors of spectra. This leads to new splitting criteria which generalize the results in dual calculus for functors to spectra.

Call-by-value is dual to call-by-name

The rules of classical logic may be formulated in pairs corresponding to De Morgan duals: rules about & are dual to rules about V . A line of work, including that of Filinski (1989), Griffin (1990), Parigot (1992), Danos, Joinet, and Schellinx (1995), Selinger (1998,2001), and Curien and Herbelin (2000), has led to the startling conclusion that call-by-value is the de Morgan dual of call-by-name.This paper presents a dual calculus that corresponds to the classical sequent calculus of Gentzen (1935) in the same way that the lambda calculus of Church (1932,1940) corresponds to the intuitionistic natural deduction of Gentzen (1935). The paper includes crisp formulations of call-by-value and call-by-name that are obviously dual; no similar formulations appear in the literature. The paper gives a CPS translation and its inverse, and shows that the translation is both sound and complete, strengthening a result in Curien and Herbelin (2000).

On a line congruence which has the parameter ruled surfaces as principal ruled surfaces

In this paper, by using dual vector calculus, a line congruence which has the parameter ruled surfaces as principal ruled surfaces, is introduced. Then some well-known formulas of surfaces theory are proved. Moreover, geometric explanations of these formulas are given.

Application of screw theory to lumped-parameter modelling of elastically coupled rigid bodies

This work considers the problem of modelling elastically coupled rigid bodies, with application to modelling and computer simulation of spatial, flexural mechanisms. Two modelling methods are presented, both of which use elements of screw theory and dual number calculus. The potential utility of the methods is demonstrated by simulating the behaviour of a complex spatial, flexural mechanism.