Inductive Constructions Research Articles

An Algorithm for Synthesizing Groups of Codes in an RFID Multiple Access System

Relevance. One of the problems that must be solved when creating RFID systems is the reader's multiple access to a group of tags located in a limited space, since the reading signal causes a one-time response of many tags, which leads to collisions (conflicts) of response signals. This problem has not been solved in relation to passive tags without a chip, based on surface acoustic wave technologies, the code of which is laid down during manufacture and cannot be changed during operation. The purpose of the study is to develop algorithms that allow synthesizing such groups of codes that would provide a controlled level of pairwise correlation of the selected label signals and thereby ensure the specified accuracy of label identification. The proposed algorithms are based on the procedures of code concatenation and inductive construction of groups of codes with a given volume and correlation level. For the algorithm for forming a group of codes with the required value of the correlation coefficient and the algorithm for combining groups of codes into complete and maximum groups, properties have been proven that confirm the possibility of using them to formulate tasks for preparing groups of labels on surfactants that would correspond to the number of objects requiring identification and the accuracy of their identification and taking into account the number of labels in the group, the conditions for the propagation of radio signals in the area of operation of the reader, the number of repeated readings of the label codes, as well as algorithms for joint data processing, received with all calculations.The methods used. Methods of coding theory and correlation analysis. Result. The developed algorithm is a tool for creating modern coding systems for surfactant labels. The scientific novelty. Well-known algorithms for multiple access in RFID systems are proposed in the GEN1 and GEN2 EPC Global standards, and assume that the tag has a chip and a power supply, which makes it possible to implement protocols for influencing the tag with a reader using special commands. The proposed multiple access algorithm is applicable for passive surfactant tags, including those moving at high speed and/or located in aggressive environments, since the tags do not use silicon technology compared to active RFID tags. Practical significance. The use of the proposed set of algorithms will increase the efficiency of marking systems by reducing the identification time of objects located in a confined space.

Universal algebra in UniMath

Abstract We present our library for universal algebra in the UniMath framework dealing with multi-sorted signatures, their algebras and the basics for equation systems. We show how to implement term algebras over a signature without resorting to general inductive constructions (currently not allowed in UniMath) still retaining the computational nature of the definition. We prove that our single sorted ground term algebras are instances of homotopy W-types. From this perspective, the library enriches UniMath with a computationally well-behaved implementation of a class of W-types. Moreover, we give neat constructions of the univalent categories of algebras and equational algebras by using the formalism of displayed categories and show that the term algebra over a signature is the initial object of the category of algebras. Finally, we showcase the computational relevance of our work by sketching some basic examples from algebra and propositional logic.

On 13-crossing-critical graphs with arbitrarily large degrees

A recent result of Bokal et al. (2022) [3] proved that the exact minimum value of c such that c-crossing-critical graphs do not have bounded maximum degree is c=13. The key to that result is an inductive construction of a family of 13-crossing-critical graphs with many vertices of arbitrarily high degrees. While the inductive part of the construction is rather easy, it all relies on the fact that a certain 17-vertex base graph has the crossing number 13, which was originally verified only by a machine-readable computer proof. We provide a relatively short self-contained computer-free proof of the latter fact. Furthermore, we subsequently generalize the critical construction in order to provide a definitive answer to another long-standing question of this research area; we prove that for every c≥13 and integers d,q, there exists a c-crossing-critical graph with more than q vertices of each of the degrees 3,4,…,d.

Correct and Complete Type Checking and Certified Erasure for Coq, in Coq

Coq is built around a well-delimited kernel that performs type checking for definitions in a variant of the Calculus of Inductive Constructions ( CIC ). Although the metatheory of CIC is very stable and reliable, the correctness of its implementation in Coq is less clear. Indeed, implementing an efficient type checker for CIC is a rather complex task, and many parts of the code rely on implicit invariants which can easily be broken by further evolution of the code. Therefore, on average, one critical bug has been found every year in Coq . This paper presents the first implementation of a type checker for the kernel of Coq (without the module system, template polymorphism and η -conversion), which is proven sound and complete in Coq with respect to its formal specification. Note that because of Gödel’s second incompleteness theorem, there is no hope to prove completely the soundness of the specification of Coq inside Coq (in particular strong normalization), but it is possible to prove the correctness and completeness of the implementation assuming soundness of the specification, thus moving from a trusted code base (TCB) to a trusted theory base (TTB) paradigm. Our work is based on the MetaCoq project which provides meta-programming facilities to work with terms and declarations at the level of the kernel. We verify a relatively efficient type checker based on the specification of the typing relation of the Polymorphic, Cumulative Calculus of Inductive Constructions ( PCUIC ) at the basis of Coq . It is worth mentioning that during the verification process, we have found a source of incompleteness in Coq ’s official type checker, which has then been fixed in Coq 8.14 thanks to our work. In addition to the kernel implementation, another essential feature of Coq is the so-called extraction mechanism: the production of executable code in functional languages from Coq definitions. We present a verified version of this subtle type and proof erasure step, therefore enabling the verified extraction of a safe type checker for Coq in the future.

String diagrams for Strictification and Coherence

Whereas string diagrams for strict monoidal categories are well understood, and have found application in several fields of Computer Science, graphical formalisms for non-strict monoidal categories are far less studied. In this paper, we provide a presentation by generators and relations of string diagrams for non-strict monoidal categories, and show how this construction can handle applications in domains such as digital circuits and programming languages. We prove the correctness of our construction, which yields a novel proof of Mac Lane's strictness theorem. This in turn leads to an elementary graphical proof of Mac Lane's coherence theorem, and in particular allows for the inductive construction of the canonical isomorphisms in a monoidal category.

Comparative Historical Analysis in Political Theory

AbstractTwo important methodological debates in political theory concern (i) the place of historical research and interpretation in normative inquiry; and (ii) the importance of comparing different cultural traditions of political thought. The first question animates a long-standing controversy over the relative importance of ‘history of political thought’ versus ‘philosophy’ in political theory, while the second has been central to the recent growth of ‘comparative political theory’. Despite both debates being concerned, in fundamental ways, with history and comparison, they have remained surprisingly disconnected from broader debates in the humanities and social sciences about the use of comparative history and, in particular, of the prominent methodological approach in political science known as ‘comparative historical analysis’. This may reflect the fact that the debates over history of political thought and comparative political theory have fundamentally revolved around thought—history as the history of thought and comparison as the comparison of thought—whereas comparative historical analysis is more centrally focused on the study of political behaviour. In this paper I suggest that this neglect of comparative historical analysis in political theory represents a missed opportunity. Like other forms of social scientific inquiry, comparative historical analysis can yield important empirical knowledge for political theorists. But I wish to more ambitiously suggest that comparative historical analysis can also be adapted for conducting normative political theory itself. I summarise the key features of comparative historical analysis for political theorists, and explain how its use of historical cases can supplement both the use of imaginary thought experiments and the study of contemporary politics. I then delineate three specific ways in which comparative historical analysis might be used to support normative inquiry: deductive testing, inductive construction, and casuistic elaboration. All three, I argue, can help meet recent calls to bring the study of real political behaviour more centrally into political theory research.

Face‐simple minimal quadrangulations of surfaces

AbstractFor each surface besides the sphere, projective plane, and Klein bottle, we construct a face‐simple minimal quadrangulation, that is, a simple quadrangulation on the fewest number of vertices possible, whose dual is also a simple graph. Our result answers a question of Liu, Ellingham, and Ye while providing a simpler proof of their main result. The inductive construction is based on an earlier idea for finding near‐quadrangular embeddings of the complete graphs using the diamond sum operation.

Maude2Lean: Theorem proving for Maude specifications using Lean

Maude is a specification language based on rewriting logic whose programs can be executed, model checked, and analyzed with other automated techniques, but not easily theorem proved. On the other hand, Lean is a modern proof assistant based on the calculus of inductive constructions with a wide library of reusable proofs and definitions. This paper presents a translation from the first formalism to the second, and the maude2lean tool that predictably derives a Lean program from a Maude module. Hence, theorems can be proved in Lean about Maude specifications.

Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

AbstractLetGbe a linear semisimple algebraic group andBits Borel subgroup. Let$${\mathbb {T}}\subset B$$T⊂Bbe the maximal torus. We study the inductive construction of Bott–Samelson varieties to obtain recursive formulas for the twisted motivic Chern classes of Schubert cells inG/B. To this end we introduce two families of operators acting on the equivariant K-theory$${\text {K}}_{\mathbb {T}}(G/B)[y]$$KT(G/B)[y], the right and left Demazure–Lusztig operators depending on a parameter. The twisted motivic Chern classes coincide (up to normalization) with the K-theoretic stable envelopes. Our results imply wall-crossing formulas for a change of the weight chamber and slope parameters. The right and left operators generate a twisted double Hecke algebra. We show that in the typeAthis algebra acts on the Laurent polynomials. This action is a natural lift of the action on$${\text {K}}_{\mathbb {T}}(G/B)[y]$$KT(G/B)[y]with respect to the Kirwan map. We show that the left and right twisted Demazure–Lusztig operators provide a recursion for twisted motivic Chern classes of matrix Schubert varieties.

Lattice structures that parameterize regulatory network dynamics

We consider two types of models of regulatory network dynamics: Boolean maps and systems of switching ordinary differential equations. Our goal is to construct all models in each category that are compatible with the directed signed graph that describe the network interactions. This leads to consideration of lattice of monotone Boolean functions (MBF), poset of non-degenerate MBFs, and a lattice of chains in these sets. We describe explicit inductive construction of these posets where the induction is on the number of inputs in MBF.Our results allow enumeration of potential dynamic behavior of the network for both model types, subject to practical limitation imposed by the size of the lattice of MBFs described by the Dedekind number.

The Importance of Examples in the Philosophy of Carl Hempel

The purpose of the article is to analyze the meaning of examples in C. Hempel’s works. Hempel uses many examples referring to readings of magnetic hand, burning of white phosphorus, predictions of properties of some elements of the table of Mendeleev, to astrophysical hypotheses, terms of total solar eclipse, throwing of dice, as well as on unmarried men, on white and black swans, green mermaids, black crows and white shoes, blue roses, predictions of Jones’ recovery, the eruption of Vesuvius, the assassination of Julius Caesar, and other “special cases” and “concrete facts”. The question is raised as to whether the case studies are mere illustrative material, a form of argumentation; a way of verifying Hempel’s ideas, their “demonstrable confirmation”, and in this sense a definite empirical basis for the study. The necessity of analyzing concrete examples in the context of Hempel’s theoretical research is shown, for which the examples turn out to be an occasion for investigating the logic of confirmation and the logic of explanation, for analyzing the structures of explanans and explanandum, for understanding the differences in inductive and deductive logical constructions, and for analyzing “quasi-induction”. The paper discusses Hempel’s famous example, the crow paradox, which is named in his honor and is a kind of logical provocation.

Understanding Suicidal Behaviour and Distress in Young Muslim Canadians: A Qualitative Study

BACKGROUND: Suicide is a major public health concern worldwide, including in Canada. The onset of major mental illnesses occurs during adolescence and young adulthood. Among this group, young Muslims in Canada may be particularly vulnerable due to unique stressors and emerging evidence indicating that Muslims in the U.S. report more suicide attempts than other religious groups. To our knowledge, no prior study has undertaken an in-depth exploration of the lived experiences of Muslim young adults who have attempted suicide. This study aims to broaden our understanding of how Muslim young adults experience and understand suicidal behavior by exploring narratives of distress and help-seeking, the meanings of suicidal behavior, the impact of immigration/resettlement, discrimination, Islamophobia, and the role of negotiating multiple religious, cultural, and gender identities. METHODS: This study is currently in progress. We are recruiting approximately 25-30 Muslim youth between the ages of 15-24 who have experienced suicidal behavior with any intent to die, as reported by the participants themselves, within the Greater Toronto Area (GTA). Exclusion criteria include (1) active substance intoxication or withdrawal, (2) current admission to a psychiatric facility, and (3) low levels of intellectual functioning or a history of neurological impairment. We are conducting heterogeneous purposive sampling to ensure that our sample includes diversity in terms of gender, age, countries of origin, and first and second-generation youth. Recruitment will end when data saturation is reached. Each participant will take part in a semi-structured qualitative interview. The interview explores family and social contacts, immigration/acculturation experiences, gender and cultural roles, and views on religion. Data is being transcribed and analyzed in N-Vivo 12 software using constructivist grounded principles entailing simultaneous collection and analysis, inductive construction of abstract categories explaining social processes, sampling and categorical refinement through iterative analysis, and integration of categories into a theoretical framework. RESULTS: Preliminary data analysis is in progress based on participants (n=15) recruited thus far. Emerging themes in relation to the causes of suicidality include isolation and an inability to cope with multiple stressors, such as family conflict and physical illness. Religion has a varied effect on suicidality, ranging from minimal to strong influence and from being protective to triggering suicidality. Regarding barriers to care, emerging themes included fear of invalidation, Islamophobia, and the lack of cultural safety within the healthcare system. Notably, youth appear to have a different explanatory model for mental illness compared to their family/community. CONCLUSION: This is the first qualitative study examining the experiences of Canadian Muslim youth who have experienced suicidal behavior. Our findings will help elucidate the unique stressors and protective factors that influence suicide and suicidal behavior for Canadian Muslim youth. Understanding patterns of distress and barriers to care through qualitative analysis will provide critical context to ultimately develop appropriate and effective screening, service provision, and suicide prevention strategies.

Four-regular graphs with extremal rigidity properties

A graph G=(V,E) is called k-edge rigid (k-edge globally rigid, resp.), if it stays rigid (globally rigid, resp.) after the deletion of at most k−1 edges. We can define k-vertex rigidity and k-vertex global rigidity in a similar manner. It is known that if G is 3-edge rigid (2-edge globally rigid, 2-vertex globally rigid) with |V|≥5 then |E|≥2|V| holds. Furthermore, the graphs that satisfy the edge count with equality are all 4-regular.In this paper we show that for a 4-regular graph G the properties of 3-edge rigidity, 2-edge global rigidity, and essential 6-edge connectivity are equivalent. By sharpening a result of H. Fleischner, F. Genest, and B. Jackson we give a new inductive construction for the family of 4-regular and essentially 6-edge connected graphs (and hence also for the 4-regular graphs with these rigidity properties). We prove that G is 2-vertex globally rigid if and only if it is 4-vertex connected and essentially 6-edge connected.We also consider 2-vertex rigid graphs G=(V,E) with minimum size |E|=2|V|−1 as well as with |E|=2|V|. In the former case we use our results on essentially 6-edge connected graphs to develop a new inductive construction, complementing an earlier, different construction of B. Servatius. In the latter case we characterize the edge pairs of G whose deletion preserves rigidity, and use this result to verify the correctness of a construction of 3-vertex rigid graphs on |V|≥6 vertices and with |E|=2|V|+2 edges, proposed by S.A. Motevallian, C. Yu, and B.D.O. Anderson.

Research on the transformation of Human Resource Sharing Center under the background of Artificial Intelligence - Take Haier Group as an example

With the reform of industry 4.0, Internet of Things and other modes, sharing economy is booming as a new economic form. Through the integrated utilization of excess capacity and idle resources, enterprises build platforms for sharing technology, finance and human resources to achieve win-win scale benefits shared by users. Senior managers of enterprises re-examine the role of HR, and organize human resource management, build a human resource-sharing center, and centralize, standardize and specialize transactional human resource base services. With the background of artificial intelligence, this paper studies the intelligent transformation of the human resource-sharing center of Haier Group. Using human resources sharing center theory analysis of Haier group human resources sharing center operation status, based on zero signature, zero delay and zero distance of the best experience goal, from the guest induction, pay payment and information system construction three key business, business process analysis, found that induction process, low efficiency, business consulting information lag and talent structure. Through in-depth analysis of the causes of the problem, business process optimization, and use artificial intelligence information technology business model innovation, induction process, pay process, information system and other business modules and talent development channel and talent cultivation system transformation to save operating costs, improve efficiency and organizational efficiency. Through the study of the transformation case of the human resource-sharing center of Haier Group, it can provide successful models and reference for the construction and transformation of the human resource-sharing center of other enterprises, and enrich the theory of the human resource-sharing center.

Material dialogues for first-order logic in constructive type theory: extended version

Abstract Dialogues are turn-taking games which model debates about the satisfaction of logical formulas. A novel variant played over first-order structures gives rise to a notion of first-order satisfaction. We study the induced notion of validity for classical and intuitionistic first-order logic in the constructive setting of the calculus of inductive constructions. We prove that such material dialogue semantics for classical first-order logic admits constructive soundness and completeness proofs, setting it apart from standard model-theoretic semantics of first-order logic. Furthermore, we prove that completeness with regard to intuitionistic material dialogues fails in both constructive and classical settings. As an alternative, we propose material dialogues played over Kripke structures. These Kripke material dialogues exhibit constructive completeness when restricting to the negative fragment. The results concerning classical material dialogues have been mechanized using the Coq interactive theorem prover.

Representations on the cohomology of [formula omitted

The moduli space M‾0,n of n pointed stable curves of genus 0 admits an action of the symmetric group Sn by permuting the marked points. We provide a closed formula for the character of the Sn-action on the cohomology of M‾0,n. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of M‾0,n, equivariant with respect to the symmetric group action. Moreover we prove that H2k(M‾0,n) for k≤3 and H2k(M‾0,n)⊕H2k−2(M‾0,n) for any k are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the Sn-representation on the cohomology of the Fulton-MacPherson compactification P1[n] of the configuration space of n points on P1 and more generally on the cohomology of the moduli space M‾0,n(Pm−1,1) of stable maps.

A Monadic Second-Order Version of Tarski’s Geometry of Solids

In this paper, we are concerned with the development of a general set theory using the single axiom version of Leśniewski’s mereology. The specification of mereology, and further of Tarski’s geometry of solids will rely on the Calculus of Inductive Constructions (CIC). In the first part, we provide a specification of Leśniewski’s mereology as a model for an atomless Boolean algebra using Clay’s ideas. In the second part, we interpret Leśniewski’s mereology in monadic second-order logic using names and develop a full version of mereology referred to as CIC-based Monadic Mereology (λ-MM) allowing an expressive theory while involving only two axioms. In the third part, we propose a modeling of Tarski’s solid geometry relying on λ-MM. It is intended to serve as a basis for spatial reasoning. All parts have been proved using a translation in type theory.

Braced Triangulations and Rigidity

We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer b there is such an inductive construction of triangulations with b braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with b braces that is linear in b. In the case that b=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b=1$$\\end{document} or 2 we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^4$$\\end{document} and a class of mixed norms on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Very weak solutions of the Dirichlet problem for the two dimensional Monge-Ampère equation

In this paper, infinitely many C1,θ(θ<15) very weak solutions to the Dirichlet problem for the two dimensional Monge-Ampère equation are obtained from inductively applying the Nash-Kuiper construction of subsolutions satisfying the boundary condition. The initial subsolutions are generated from the solutions of some Possion equations. The inductive construction is then attained by successively adding compactly supported deficit matrixes.