Normal Theorem Research Articles

Natural deduction calculi for classical and intuitionistic S5

We propose an indexed natural deduction system for the modal logic , ideally following Wansing's previous work in the context of tableaux sequents. The system, given both in the classical and intuitionistic versions (called and respectively), is designed to match as much as possible the structure and properties of the standard system of natural deduction for first-order logic, exploiting the formal analogy between modalities and quantifiers. We study a (syntactical) normalization theorem for both and and its main consequences, the sub-formula principle and the consistency theorem. In particular, we propose an intuitionistic encoding of classical (via a suitable extension of the Gödel translation for first-order classical logic). Moreover, via the BHK interpretation of intuitionistic proofs, we propose a suitable Curry–Howard isomorphism for . By translation into the natural deduction system given by Galmiche and Salhi in [(2010b). Label-free proof systems for intuitionistic modal logic is5. In E. M. Clarke & A. Voronkov (Eds.), Logic for programming, artificial intelligence, and reasoning (pp. 255–271). Springer Berlin Heidelberg.], we prove the equivalence of w.r.t. an Hilbert-style axiomatization of . However, when considering the sheer provability of labelled formulas, our system is comparable to the one presented by Simpson in [(1993). The proof theory and semantics of intuitionistic modal logic [PhD thesis], University of Edinburgh, UK.]. Nevertheless, it remains uncertain whether it is feasible to establish a translation between the corresponding derivations.

A Formal Proof of the Strong Normalization Theorem for System T in Agda

We present a framework for the formal meta-theory of lambda calculi in first-order syntax, with two sorts of names, one to represent both free and bound variables, and the other for constants, and by using Stoughton's multiple substitutions. On top of the framework we formalize Girard's proof of the Strong Normalization Theorem for both the simply-typed lambda calculus and System T. As to the latter, we also present a simplification of the original proof. The whole development has been machine-checked using the Agda system.

Bounding imprecise failure probabilities in structural mechanics based on maximum standard deviation

This paper proposes a framework to calculate the bounds on failure probability of linear structural systems whose performance is affected by both random variables and interval variables. This kind of problems is known to be very challenging, as it demands coping with aleatoric and epistemic uncertainty explicitly. Inspired by the framework of the operator norm theorem, it is proposed to consider the maximum standard deviation of the structural response as a proxy for detecting the crisp values of the interval parameters, which yield the bounds of the failure probability. The scope of application of the proposed approach comprises linear structural systems, whose properties may be affected by both aleatoric and epistemic uncertainty and that are subjected to (possibly imprecise) Gaussian loading. Numerical examples indicate that the application of such proxy leads to substantial numerical advantages when compared to a traditional double-loop approach for coping with imprecise failure probabilities. In fact, the proposed framework allows to decouple the propagation of aleatoric and epistemic uncertainty.

Dispersion parameter extension of precise generalized linear mixed model asymptotics

We extend a recent asymptotic normality theorem for generalized linear mixed models to include the dispersion parameter. The maximum likelihood estimators of all model parameters have asymptotically normal distributions with asymptotic mutual independence between fixed effects, covariance and dispersion parameters.

Normalisation for Some Infectious Logics and Their Relatives

We consider certain infectious logics (Sfde, dSfde, K3w, and PWK) and several their non-infectious modifications, including two new logics, reformulate previously constructed natural deduction systems for them (or present such systems from scratch for the case of new logics) in way such that the proof of normalisation theorem becomes possible for these logics. We present such a proof and establish the negation subformula property for the logics in question.

Linear recurrent relations, power series distributions, and generalized allocation scheme

Abstract We consider distributions of the power series type determined by the generating functions of sequences satisfying linear recurrence relations with nonnegative coefficients. These functions are represented by power series with positive radius of convergence. An integral limit theorem is proved on the convergence of such distributions to the exponential distribution. For the generalized allocation scheme generated by these linear relations a local normal theorem for the total number of components is proved. As a consequence of more general results of the author, a limit theorem is stated containing sufficient conditions under which the distributions of the number of components having a given volume converge to the Poisson distribution.

Remarks on classical number-theoretic aspects of Milnor–Witt K-theory

We record a few observations on number theoretic aspects of Milnor-Witt K-theory, focusing on generalizing classical results on reciprocity laws, Hasse's norm theorem and K_2 of number fields and rings of integers.

Normalisation for Some Quite Interesting Many-Valued Logics

In this paper, we consider a set of quite interesting three- and four-valued logics and prove the normalisation theorem for their natural deduction formulations. Among the logics in question are the Logic of Paradox, First Degree Entailment, Strong Kleene logic, and some of their implicative extensions, including RM3 and RM3⊃. Also, we present a detailed version of Prawitz’s proof of Nelson’s logic N4 and its extension by intuitionist negation.

Bifurcation analysis in a diffusive phytoplankton\u2013zooplankton model with harvesting

A diffusive phytoplankton–zooplankton model with nonlinear harvesting is considered in this paper. Firstly, using the harvesting as the parameter, we get the existence and stability of the positive steady state, and also investigate the existence of spatially homogeneous and inhomogeneous periodic solutions. Then, by applying the normal form theory and center manifold theorem, we give the stability and direction of Hopf bifurcation from the positive steady state. In addition, we also prove the existence of the Bogdanov–Takens bifurcation. These results reveal that the harvesting and diffusion really affect the spatiotemporal complexity of the system. Finally, numerical simulations are also given to support our theoretical analysis.

Proof theory for heterogeneous logic combining formulas and diagrams: proof normalization

We extend natural deduction for first-order logic (FOL) by introducing diagrams as components of formal proofs. From the viewpoint of FOL, we regard a diagram as a deductively closed conjunction of certain FOL formulas. On the basis of this observation, we first investigate basic heterogeneous logic (HL) wherein heterogeneous inference rules are defined in the styles of conjunction introduction and elimination rules of FOL. By examining what is a detour in our heterogeneous proofs, we discuss that an elimination-introduction pair of rules constitutes a redex in our HL, which is opposite the usual redex in FOL. In terms of the notion of a redex, we prove the normalization theorem for HL, and we give a characterization of the structure of heterogeneous proofs. Every normal proof in our HL consists of applications of introduction rules followed by applications of elimination rules, which is also opposite the usual form of normal proofs in FOL. Thereafter, we extend the basic HL by extending the heterogeneous rule in the style of general elimination rules to include a wider range of heterogeneous systems.

Local Limit Theorems for Compound Discrete Distributions

Local limit theorems for compound discrete distributions (a normal theorem and a large deviation theorem) are put forward. Some examples of such distributions, including compound Poisson, compound ...

Hopf bifurcation in a delayed predator-prey system with asymmetric functional response and additional food

<abstract><p>In this paper, a delayed predator-prey system with additional food and asymmetric functional response is investigated. We discuss the local stability of equilibria and the existence of local Hopf bifurcation under the influence of the time delay. By using the normal form theory and center manifold theorem, the explicit formulas which determine the properties of bifurcating periodic solutions are obtained. Further, we prove that global periodic solutions exist after the second critical value of delay via Wu's theory. Finally, the correctness of the previous theoretical analysis is demonstrated by some numerical cases.</p></abstract>

Operator norm theory as an efficient tool to propagate hybrid uncertainties and calculate imprecise probabilities

This paper presents a highly efficient and effective approach to bound the responses and probability of failure of linear systems where the model parameters are subjected to combinations of epistemic and aleatory uncertainty. These combinations can take the form of imprecise probabilities or hybrid uncertainties. Typically, such computations involve solving a nested double loop problem, where the propagation of the aleatory uncertainty has to be performed for each realisation of the epistemic uncertainty. Apart from near-trivial cases, such computation is intractable without resorting to surrogate modeling schemes. In this paper, a method is presented to break this double loop by virtue of the operator norm theorem. Indeed, in case linear models are considered and under the restriction that the model definition cannot be subject to aleatory uncertainty, the paper shows that the computational efficiency, quantified by the required number of model evaluations, of propagating these parametric uncertainties can be improved by several orders of magnitude. Two case studies involving a finite element model of a clamped plate and a six-story building are included to illustrate the application of the developed technique, as well as its computational merit in comparison to existing double-loop approaches.

Test for high dimensional covariance matrices

The paper introduces a new test for testing structures of covariances for high dimensional vectors and the data dimension can be much larger than the sample size. Under proper normalization, central and noncentral limit theorems are established. The asymptotic theory is attained without imposing any explicit restriction between data dimension and sample size. To facilitate the related statistical inference, we propose the balanced Rademacher weighted differencing scheme, which is also the delete-half jackknife, to approximate the distribution of the proposed test statistics. We also develop a new testing procedure for substructures of precision matrices. The simulation results show that the tests outperform the exiting methods both in terms of size and power. Our test procedure is applied to a colorectal cancer dataset.

Term-Space Semantics of Typed Lambda Calculus

Barendregt gave a sound semantics of the simple type assignment system λ → by generalizing Tait’s proof of the strong normalization theorem. In this paper, we aim to extend the semantics so that the completeness theorem holds.

The Chevalley–Gras formula over global fields

Dans cet article, nous donnons une preuve adélique de la formule de Chevalley–Gras pour les corps de nombres qui, elle-même, est une généralisation de la formule du nombre de classes ambiges. L’idée est de réduire cette formule au théorème de la norme de Hasse et à des lois de réciprocité globaux. Nous donnons également une preuve adélique de la formule de Chevalley–Gras pour les groupes des classes des diviseurs de degré 0 dans le cas des corps de fonctions, qui étend un résultat de Rosen.

The norm theorem for semisingular quadratic forms

Abstract Let F be a field of characteristic 2. The aim of this paper is to give a complete proof of the norm theorem for singular F-quadratic forms which are not totally singular, i.e., we give necessary and sufficient conditions for which a normed irreducible polynomial of F [ x 1 , … , x n ] becomes a norm of such a quadratic form over the rational function field F ( x 1 , … , x n ) . This completes partial results proved on this question in [8] . Combining the present work with the papers [1] and [7] , we obtain the norm theorem for any type of quadratic forms in characteristic 2.

On the intersections of exceptional sets in Borel’s normal number theorem and Erdös–Rényi limit theorem

For any [Formula: see text], let [Formula: see text] be the partial summation of the first [Formula: see text] digits in the binary expansion of [Formula: see text] and [Formula: see text] be its run-length function. The classical Borel’s normal number theorem tells us that for almost all [Formula: see text], the limit of [Formula: see text] as [Formula: see text] goes to infinity is one half. On the other hand, the Erdös–Rényi limit theorem shows that [Formula: see text] increases to infinity with the logarithmic speed [Formula: see text] as [Formula: see text] for almost every [Formula: see text] in [Formula: see text]. In this paper, we are interested in the intersections of exceptional sets arising in the above two famous theorems. More precisely, for any [Formula: see text] and [Formula: see text], we completely determine the Hausdorff dimension of the following set: [Formula: see text] where [Formula: see text] and [Formula: see text] After some minor modifications, our result still holds if we replace the denominator [Formula: see text] in [Formula: see text] with any increasing function [Formula: see text] satisfying [Formula: see text] tending to [Formula: see text] and [Formula: see text]. As a result, we also obtain that the set of points for which neither the sequence [Formula: see text] nor [Formula: see text] converges has full Hausdorff dimension.

Fully decoupled reliability-based design optimization of structural systems subject to uncertain loads

Reliability-based optimization (RBO) offers the possibility of finding the best design for a system according to a prescribed criterion while explicitly taking into account the effects of uncertainty. Although the importance and usefulness of RBO is undisputed, it is rarely applied to practical problems, as the associated numerical efforts are usually extremely large due to the necessity of solving simultaneously a reliability problem nested in an optimization procedure. To alleviate this issue, this contribution proposes an approach for solving a particular class of problems in RBO: the minimization of the failure probability of a linear system subjected to an uncertain load. The main characteristic of this approach is that it is fully decoupled. That is, the solution of the RBO problem is reduced to the solution of a single deterministic optimization problem followed by a single reliability analysis. Such an approach implies a complete change of paradigm with respect to the more classical double-loop and sequential methods for RBO. The theoretical basis for the proposed approach lies in the application of the operator norm theorem. The application and capabilities of the proposed approach are illustrated by means of three examples.

Dynamics of a delayed SIR model for the transmission of PRRSV among a swine population

The objective of this paper is to propose a delayed susceptible-infectious-recovered (SIR) model for the transmission of porcine reproductive respiratory syndrome virus (PRRSV) among a swine population, including the latent period delay of the virus and the time delay due to the period the infectious swines need to recover. By taking different combinations of the two delays as the bifurcation parameter, local stability of the disease-present equilibrium and the existence of Hopf bifurcation are analyzed. Sufficient conditions for global stability of the disease-present equilibrium are derived by constructing a suitable Lyapunov function. Directly afterwards, properties of the Hopf bifurcation such as direction and stability are studied with the aid of the normal form theory and center manifold theorem. Finally, numerical simulations are presented to justify the validity of the derived theoretical results.