General Proof Research Articles

StarMalloc: Verifying a Modern, Hardened Memory Allocator

We present StarMalloc, a verified, efficient, security-oriented, and concurrent memory allocator. Using the Steel separation logic framework, we show how to specify and verify a multitude of low-level patterns and delicate security mechanisms, by relying on a combination of dependent types, SMT, and modular abstractions to enable efficient verification. We produce a verified artifact, in C, that implements the entire API surface of an allocator, and as such works as a drop-in replacement for real-world projects, notably the Firefox browser. As part of StarMalloc, we develop several generic datastructures and proof libraries directly reusable in future systems verification projects. We also extend the Steel toolchain to express several low-level idioms that were previously missing. Finally, we show that StarMalloc exhibits competitive performance by evaluating it against 10 state-of-the-art memory allocators, and against a variety of real-world projects, such as Redis, the Lean compiler, and the Z3 SMT solver.

(Almost-)Quantum Bell Inequalities and Device-Independent Applications

Investigations of the boundary of the quantum correlation set have gained increased attention in recent years. This is done through the derivation of quantum Bell inequalities, which are related to Tsirelson's problem and have significant applications in device-independent (DI) information processing. However, determining quantum Bell inequalities is a notoriously difficult task and only isolated examples are known. In this paper, we present families of (almost-)quantum Bell inequalities and highlight four foundational and DI applications. Firstly, it is known that quantum correlations on the non-signaling boundary are of crucial importance in the task of DI randomness extraction from weak sources. In the practical Bell scenario of two players with two k-outcome measurements, we derive quantum Bell inequalities that demonstrate a separation between the quantum boundary and certain portions of the boundaries of the no-signaling polytope of dimension up to 4k−8, extending previous results from nonlocality distillation and the collapse of communication complexity. Secondly as an immediate by-product, we give a general proof of Aumann’s Agreement theorem for quantum systems as well as the almost-quantum correlations, which implies Aumann’s agreement theorem is a reasonable physical principle in the context of epistemics to pick out both quantum theory and almost-quantum correlations from general no-signaling theories. Thirdly, we present a family of quantum Bell inequalities in the two players with m binary measurements scenarios, that we prove serve to self-test the two-qubit singlet and the corresponding 2m measurements. Interestingly, this claim generalizes the result for m=2 discovered by Tsirelson-Landau-Masanes and shows an improvement over the state-of-the-art Device-Independent Randomness-Amplification (DIRA). Lastly, we use our quantum Bell inequalities to derive the general form of the principle of no advantage in nonlocal computation, which is an information-theoretic principle that serves to characterize the quantum correlation set.

Mixed deterministic and stochastic disturbances in a discrete-time Nash game

This paper considers a discrete-time non-cooperative M-players linear affine quadratic game of pre-specified fixed duration, affected by stochastic noise and deterministic disturbances. The last one is seen as a pernicious fictitious player looking to maximise the expected cost functions of each player. Inspired in the discrete-time robust dynamic programming, sufficient conditions for the existence of a type of robust feedback NE are given in the solution of a set of discrete-time difference equations. A formal induction proof is provided for the closed form of the obtained robust set of strategies. Two illustrative simulation examples are included, one related to the problem of coordination of a two-echelon supply chain with uncertain seasonal demand. The goal of the agents is to reduce the expected cost of storage while satisfying a partially known demand. The second example is related to a game of government debt stabilisation. Comparing the simulation with the standard Nash strategy, the robust one achieves a better performance.

Optimal Knowledge Distillation through Non-Heuristic Control of Dark Knowledge

In this paper, a method is introduced to control the dark knowledge values also known as soft targets, with the purpose of improving the training by knowledge distillation for multi-class classification tasks. Knowledge distillation effectively transfers knowledge from a larger model to a smaller model to achieve efficient, fast, and generalizable performance while retaining much of the original accuracy. The majority of deep neural models used for classification tasks append a SoftMax layer to generate output probabilities and it is usual to take the highest score and consider it the inference of the model, while the rest of the probability values are generally ignored. The focus is on those probabilities as carriers of dark knowledge and our aim is to quantify the relevance of dark knowledge, not heuristically as provided in the literature so far, but with an inductive proof on the SoftMax operational limits. These limits are further pushed by using an incremental decision tree with information gain split. The user can set a desired precision and an accuracy level to obtain a maximal temperature setting for a continual classification process. Moreover, by fitting both the hard targets and the soft targets, one obtains an optimal knowledge distillation effect that mitigates better catastrophic forgetting. The strengths of our method come from the possibility of controlling the amount of distillation transferred non-heuristically and the agnostic application of this model-independent study.

Hurwitz enumeration problem through the perspective of Cayley graph

The Hurwitz enumeration problem studies how to determine the Hurwitz number for a branch profile, which counts the number of ways a per mutation can be factored into transpositions. In this paper, we consider the length of strictly monotone factorization from the perspective of Cayley graph theory. We represent the factorization problem using a Cayley graph, where vertices are permutations and edges are transpositions. Our focus is proving the unique monotone factorization theorem, which states that for a given permutation, there is only one monotone factorization of minimal length. To prove this, we employ inductive arguments on the structure of the Cayley graph. The key insight is using the connectivity of the graph and properties of shortest paths to characterize the uniqueness of the minimal factorization. This inductive approach allows us to rigorously connect the combinatorial Hurwitz problem to foundational graph concepts. Overall, this paper makes important theoretical advances in enumerating Hurwitz numbers by using Cayley graphs and induction to prove the novel unique monotone factorization theorem. The connections drawn between combinatorics, graphs, and inductive proofs are technically innovative. This theoretical foundation will hopefully stimulate further research into the deep links between the Hurwitz problem and other branches of mathematics.

Signature-based standard basis algorithm under the framework of GVW algorithm

The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm.

A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization

Wasserstein distributionally robust optimization has emerged as a recent topic with broader applications in operations research and machine learning. Various proofs have been presented in the literature, each differing in assumptions and levels of generality. In “A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization,” Zhang, Yang, and Gao present a novel elementary proof that not only shortens existing frameworks but also offers surprising generalizations. Leveraging classical Legendre—Fenchel duality, they demonstrate that strong duality is contingent on a certain interchangeability principle. Moreover, they extend this duality result to encompass risk-averse optimization and globalized distributionally robust counterparts.

General proof of the Tolman law

General proof of the Tolman law

Can quantum statistics help distinguish Dirac from Majorana neutrinos?

Finding out if neutrinos are Dirac or Majorana particles is known to be extremely difficult due to the smallness of neutrino mass and the fact that in the limit mν = 0 both Dirac and Majorana neutrinos become Weyl fermions, i.e. are indistinguishable. There have been suggestions in the literature that in the case of processes with production of a neutrino-antineutrino pair (if neutrinos are Dirac particles) or two neutrinos (if they are of Majorana nature) quantum statistics may be of help. This is because for Majorana neutrinos quantum indistinguishability of identical particles requires the amplitude of the process to be antisymmetrized with respect to the interchange of the final-state neutrinos, whereas no such antisymmetrization must be done for Dirac neutrinos. It has been claimed that the resulting differences between the cross sections for Dirac and Majorana neutrinos persist even for arbitrarily small but not exactly vanishing neutrino mass. We demonstrate that, at least in the framework of the Standard Model, this is not the case. We also give a general proof that within the Standard Model quantum statistics does not help tell Dirac and Majorana neutrinos apart in the limit of negligibly small mν/E.

Increasing in three units the order of convergence of iterative methods for solving nonlinear systems

In this paper, we present a technique aimed at the implementation of high accuracy iterative procedures in nonlinear systems. The main result of this research is a general proof that guarantees an increase in the order of convergence to order p+3 units for any method of order p. This methodology has as its main objective not only this increase, but also the optimization in the use of resources by requiring only one additional functional evaluation per iteration, thus keeping a tight control over the computational cost. Although there are previous proposals in the literature that approach this goal, in this work we consider such approaches as particular cases of our general proposal. This approach encompasses many methods described by other authors and can be applied to any p-order scheme to increase it by three units. Numerical evaluations, supported by a rigorous theoretical framework, validate the robustness of the proposal, highlighting its advantage over conventional techniques and demonstrating its effectiveness in the face of numerical challenges of considerable magnitude. Summing up, with this research, we offer a substantial contribution to the field of numerical methods, proposing a paradigm that consistently enhances the convergence order of iterative methods.

Inductive Diagrams for Causal Reasoning

The Lamport diagram is a pervasive and intuitive tool for informal reasoning about “happens-before” relationships in a concurrent system. However, traditional axiomatic formalizations of Lamport diagrams can be painful to work with in a mechanized setting like Agda. We propose an alternative, inductive formalization — the causal separation diagram (CSD) — that takes inspiration from string diagrams and concurrent separation logic, but enjoys a graphical syntax similar to Lamport diagrams. Critically, CSDs are based on the idea that causal relationships between events are witnessed by the paths that information follows between them. To that end, we model “happens-before” as a dependent type of paths between events. The inductive formulation of CSDs enables their interpretation into a variety of semantic domains. We demonstrate the interpretability of CSDs with a case study on properties of logical clocks , widely-used mechanisms for reifying causal relationships as data. We carry out this study by implementing a series of interpreters for CSDs, culminating in a generic proof of Lamport’s clock condition that is parametric in a choice of clock. We instantiate this proof on Lamport’s scalar clock, on Mattern’s vector clock, and on the matrix clocks of Raynal et al. and of Wuu and Bernstein, yielding verified implementations of each. The CSD formalism and our case study are mechanized in the Agda proof assistant.

New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation

In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives.

The Canonical component of the nilfibre for parabolic adjoint action in type A

This work is a continuation of [Y. Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type A]. Let P be a parabolic subgroup of an irreducible simple algebraic group G. Let P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P'$$\\end{document} be the derived group of P, and let m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {m}}$$\\end{document} be the Lie algebra of the nilradical of P. A theorem of Richardson implies that the subalgebra C[m]P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[{\\mathfrak {m}}]^{P'}$$\\end{document}, spanned by the P semi-invariants in C[m]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[{\\mathfrak {m}}]$$\\end{document}, is polynomial. A linear subvariety e+V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e+V$$\\end{document} of m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {m}}$$\\end{document} is called a Weierstrass section for the action of P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P'$$\\end{document} on m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {m}}$$\\end{document}, if the restriction map induces an isomorphism of C[m]P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[{\\mathfrak {m}}]^{P'}$$\\end{document} onto C[e+V]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[e+V]$$\\end{document}. Thus, a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. Let N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}$$\\end{document} be zero locus of the augmentation C[m]+P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}[{\\mathfrak {m}}]^{P'}_+$$\\end{document}. It is called the nilfibre relative to this action. Suppose G=SL(n,C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=\ extrm{SL}(n,{\\mathbb {C}})$$\\end{document}, and let P be a parabolic subgroup. In [Y. Fittouhi and A. Joseph, loc. cit.], the existence of a Weierstrass section e+V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e+V$$\\end{document} in m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {m}}$$\\end{document} was established by a general combinatorial construction. Notably, e∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e \\in {\\mathscr {N}}$$\\end{document} and is a sum of root vectors with linearly independent roots. The Weierstrass section e+V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e+V$$\\end{document} looks very different for different choices of parabolics but nevertheless has a uniform construction and exists in all cases. It is called the “canonical Weierstrass section”. Through [Y. Fittouhi and A. Joseph, loc. cit. Prop. 6.9.2, Cor. 6.9.8], there is always a “canonical” component Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^e$$\\end{document} of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}$$\\end{document} containing e. It was announced in [Y. Fittouhi and A. Joseph, loc. cit., Prop. 6.10.4] that one may augment e to an element eVS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e_\ extrm{VS}$$\\end{document} by adjoining root vectors. Then the linear span EVS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_\ extrm{VS}$$\\end{document} of these root vectors lies in Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {N}^e$$\\end{document} and its closure is just Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^e$$\\end{document}. Yet, this same result shows that Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^e$$\\end{document} need not admit a dense P orbit [Y. Fittouhi and A. Joseph, loc. cit., Lemma 6.10.7]. For the above [Y. Fittouhi and A. Joseph, loc. cit., Theorem 6.10.3] was needed. However, this theorem was only verified in the special case needed to obtain the example showing that Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^e$$\\end{document} may fail to admit a dense P orbit. Here a general proof is given (Theorem 4.4.5). Finally, a map from compositions to the set of distinct non-negative integers is defined. Its image is shown to determine the canonical Weierstrass section. One may anticipate that the remaining components of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}$$\\end{document} can be similarly described. However, this is a long story and will be postponed for a subsequent paper. These results should form a template for general type.

Inductive proof of Borchardt’s theorem

Inductive proof of Borchardt’s theorem

Structural causal models reveal confounder bias in linear program modelling

AbstractThe recent years have been marked by extended research on adversarial attacks, especially on deep neural networks. With this work we intend on posing and investigating the question of whether the phenomenon might be more general in nature, that is, adversarial-style attacks outside classical classification tasks. Specifically, we investigate optimization problems as they constitute a fundamental part of modern AI research. To this end, we consider the base class of optimizers namely Linear Programs (LPs). On our initial attempt of a naïve mapping between the formalism of adversarial examples and LPs, we quickly identify the key ingredients missing for making sense of a reasonable notion of adversarial examples for LPs. Intriguingly, the formalism of Pearl’s notion to causality allows for the right description of adversarial like examples for LPs. Characteristically, we show the direct influence of the Structural Causal Model (SCM) onto the subsequent LP optimization, which ultimately exposes a notion of confounding in LPs (inherited by said SCM) that allows for adversarial-style attacks. We provide both the general proof formally alongside existential proofs of such intriguing LP-parameterizations based on SCM for three combinatorial problems, namely Linear Assignment, Shortest Path and a real world problem of energy systems.

Milliken’s Tree Theorem and Its Applications: A Computability-Theoretic Perspective

Milliken’s tree theorem is a deep result in combinatorics that generalizes a vast number of other results in the subject, most notably Ramsey’s theorem and its many variants and consequences. In this sense, Milliken’s tree theorem is paradigmatic of structural Ramsey theory, which seeks to identify the common combinatorial and logical features of partition results in general. Its investigation in this area has consequently been extensive. Motivated by a question of Dobrinen, we initiate the study of Milliken’s tree theorem from the point of view of computability theory. The goal is to understand how close it is to being algorithmically solvable, and how computationally complex are the constructions needed to prove it. This kind of examination enjoys a long and rich history, and continues to be a highly active endeavor. Applied to combinatorial principles, particularly Ramsey’s theorem, it constitutes one of the most fruitful research programs in computability theory as a whole. The challenge to studying Milliken’s tree theorem using this framework is its unusually intricate proof, and more specifically, the proof of the Halpern-Laüchli theorem, which is a key ingredient. Our advance here stems from a careful analysis of the Halpern-Laüchli theorem which shows that it can be carried out effectively (i.e., that it is computably true). We use this as the basis of a new inductive proof of Milliken’s tree theorem that permits us to gauge its effectivity in turn. The key combinatorial tool we develop for the inductive step is a fast-growing computable function that can be used to obtain a finitary, or localized, version of Milliken’s tree theorem. This enables us to build solutions to the full Milliken’s tree theorem using effective forcing. The principal result of this is a full classification of the computable content of Milliken’s tree theorem in terms of the jump hierarchy, stratified by the size of instance. As usual, this also translates into the parlance of reverse mathematics, yielding a complete understanding of the fragment of second-order arithmetic required to prove Milliken’s tree theorem. We apply our analysis also to several well-known applications of Milliken’s tree theorem, namely Devlin’s theorem, a partition theorem for Rado graphs, and a generalized version of the so-called tree theorem of Chubb, Hirst, and McNicholl. These are all certain kinds of extensions of Ramsey’s theorem for different structures, namely the rational numbers, the Rado graph, and perfect binary trees, respectively. We obtain a number of new results about how these principles relate to Milliken’s tree theorem and to each other, in terms of both their computability-theoretic and combinatorial aspects. In particular, we establish new structural Ramsey-theoretic properties of the Rado graph theorem and the generalized Chubb-Hirst-McNicholl tree theorem using Zucker’s notion of big Ramsey structure.

A logical approach to validate the Goldbach conjecture: Part I

This paper is a first of the series of three papers which provide a general proof to validate the Goldbach conjecture. This conjecture states that every even number can be expressed as a summation of two prime numbers. At the onset, concept of successive-addition of‐digits‐of‐an‐integer‐number (SADN) and its properties in terms of basic algebraic functions like addition, multiplication and subtraction are discussed. SADN classifies odd numbers into 3 sequences—the S1, S3 and S5 sequences—which comprise of odd numbers having SADN 7 or 4 or 1; SADN 3 or 9 or 6 and SADN 5 or 2 or 8 respectively. The S1 and S5 sequences are of interest in the analysis. Furthermore, composites on the S1 sequence are derived as products of intra-sequence elements of the S1 and S5 sequences while composites on the S5 sequence are derived as products of inter-sequence elements of the S1 and S5 sequences. SADN also shows why such combinations for even numbers of SADN (1, 4, 7) will be found on the S5 sequence while those for even numbers of SADN (2, 5, 8) will lie on the S1 sequence and both the sequences have a role to play in identifying the prime number combinations for even numbers with SADN (3, 6, 9). Thereafter, analysis moves to calculating the total number of combinations for a given even number that would include combinations in the nature of two composites (c1 + c2); prime & composite (p + c) and two primes (p1 + p2). Identifying the total number of c1 + c2 and p + c combinations yield the number of p1 + p2 combinations. The logic employed in present discussion shows that at least one such p1 + p2 combination exists for the even numbers having SADN digit within 1 to 9. This encompasses all even numbers and hence generalizes this method for all even numbers.

Soft switching multiple model predictive control with overlapping cross-over time strategy in an industrial high speed pick and place application

In industrial machinery, position-controlled systems executing repetitive tasks are of significant importance. In such mechanisms, the load-side inertia and torque vary with the angular position of the driven axis, thus imposing a significant nonlinearity in the system dynamics. Linear controllers have a relatively low computational time, but are unable to achieve the required performance over the whole operating region. Nonlinear controllers, such as nonlinear model predictive control (NLMPC), demand an extensive computational time that is difficult to bound. The current Soft Switching Multiple MPC (SSM-MPC) techniques rely solely on the gap metric or weighting techniques to guarantee soft switching between controllers. However, in nonlinear systems with fast dynamics, it may result in poor performance and/or high computational time. In this work, we have addressed these issues by introducing an overlapping cross-over time strategy in the switching unit of the SSM-MPC to ensure a smooth transition when switching in adjacent regions. The key idea behind the overlapping cross-over time strategy is introducing a delay i.e. cross-over time during switching. The cross-over time helps to improve the completeness and non-redundancy of the model bank. The closed loop system stability is guaranteed using Lyapunov theory and a general switching stability proof is provided. The proposed SSM-MPC performance is benchmarked against the available SSM-MPC method and a single linear MPC on a high speed industrial pick and place machine through simulations and experiments. The results verify the superior performance of the presented SSM-MPC with 37.67% reduction in tracking error and 20.36% decrease in control signal oscillations at the highest operating speed of the machine compared to the available SSM-MPC technique without cross-over time. Moreover, in contrast to the nonlinear MPC, it is proven that the computational load of the SSM-MPC is 77.5% lower, which allows implementation in a standard industrial controller.

Solving problems on generalized convex graphs via mim-width

A bipartite graph G=(A,B,E) is H-convex for some family of graphs H if there exists a graph H∈H with V(H)=A such that the neighbours in A of each b∈B induce a connected subgraph of H. Many NP-complete problems are polynomial-time solvable for H-convex graphs when H is the set of paths. The underlying reason is that the class has bounded mim-width. We extend this result to families of H-convex graphs where H is the set of cycles, or H is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we strengthen many known results via one general and short proof. We also show that the mim-width of H-convex graphs is unbounded if H is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3.

General Overview and Proof of Concept of a Smart Home Energy Management System Architecture

This paper proposes and implements a smart architecture for Home Energy Management Systems (HEMS) that enables interoperability among devices from different manufacturers. This is achieved through the use of standardized elements and the design of an innovative middleware. The system comprises a control unit that communicates with smart outlets using the Wireless Smart Ubiquitous Network (WI-SUN) Home Area Network (HAN) specification, while smart metering is achieved using the WI-SUN Field Area Network (FAN) specification. To manage important data, a web platform and mobile app were created. Additionally, machine learning techniques are utilized to identify energy consumption of individual appliances when only the aggregate energy consumption of the house is available. The architecture presented here supports real-time control of energy use and generation through HEMS, and new devices can be added transparently. Finally, a comparison of the proposed system with similar systems in literature highlights its many advantages in terms of functionality.