Linear Type System Research Articles

Limit Cycles Induced by Threshold Nonlinearity in Planar Piecewise Linear Systems of Node-Focus or Node-Center Type

In this paper, we investigate limit cycles induced by threshold nonlinearity of piecewise linear (PWL) differential systems, which are node-focus type or node-center type with the focus or the center being virtual or boundary. To get the number and stability of limit cycles, we adopt a new displacement function with a better configuration than usual. For a given parameter subregion, we exhibit the exact number or the minimum number of limit cycles. In particular, sufficient conditions are established ensuring that there are exactly two limit cycles. When the focus is boundary, we not only show that the maximum number is two, but also verify that the exact number is zero, one or two by varying parameter subregions. Finally, the exact number as well as the stability are obtained in different parameter regions for the PWL differential systems of node-center type.

Boundary controllability of a one-dimensional phase-field system with one control force

In this paper, we present some controllability results for linear and nonlinear phase-field systems of Caginalp type considered in a bounded interval of R when the scalar control force acts on the temperature equation of the system by means of the Dirichlet condition on one of the endpoints of the interval. In order to prove the linear result we use the moment method providing an estimate of the cost of fast controls. Using this estimate and following the methodology developed in [19], we prove a local exact boundary controllability result to constant trajectories of the nonlinear phase-field system. To the authors' knowledge, this is the first nonlinear boundary controllability result in the framework of non-scalar parabolic systems, framework in which some “hyperbolic” behaviors could arise.

On optimal age replacement policy for a class of coherent systems

According to the well-known age replacement policy, the system is replaced preventively at time t or correctively at system failure, whichever occurs first. For a coherent system consisting of components having common failure time distribution which has increasing failure rate, we present necessary conditions for the existence of the unique optimal value which minimizes the mean cost rate. The conditions are mainly based on the signature which only depends on the system’s structure. The results are illustrated for linear and circular consecutive type systems.

Finite Observer Design for Linear Systems of Neutral Type

For the linear autonomous differential-difference systems of neutral type, a finite observer that estimates in finite time the solution to an original system with zero error is suggested. A criterion for the existence of such an observer and a corresponding design method are developed.

Duet: an expressive higher-order language and linear type system for statically enforcing differential privacy

During the past decade, differential privacy has become the gold standard for protecting the privacy of individuals. However, verifying that a particular program provides differential privacy often remains a manual task to be completed by an expert in the field. Language-based techniques have been proposed for fully automating proofs of differential privacy via type system design, however these results have lagged behind advances in differentially-private algorithms, leaving a noticeable gap in programs which can be automatically verified while also providing state-of-the-art bounds on privacy. We propose Duet, an expressive higher-order language, linear type system and tool for automatically verifying differential privacy of general-purpose higher-order programs. In addition to general purpose programming, Duet supports encoding machine learning algorithms such as stochastic gradient descent, as well as common auxiliary data analysis tasks such as clipping, normalization and hyperparameter tuning - each of which are particularly challenging to encode in a statically verified differential privacy framework. We present a core design of the Duet language and linear type system, and complete key proofs about privacy for well-typed programs. We then show how to extend Duet to support realistic machine learning applications and recent variants of differential privacy which result in improved accuracy for many practical differentially private algorithms. Finally, we implement several differentially private machine learning algorithms in Duet which have never before been automatically verified by a language-based tool, and we present experimental results which demonstrate the benefits of Duet's language design in terms of accuracy of trained machine learning models.

Estimates of the exponential decay of solutions to linear systems of neutral type with periodic coefficients

Estimates of the exponential decay of solutions to linear systems of neutral type with periodic coefficients

Implicit Computational Complexity of Subrecursive Definitions and Applications to Cryptographic Proofs

We define a call-by-value variant of Godel’s system $$\textsf {T} $$ with references, and equip it with a linear dependent type and effect system, called $$\textsf {d}\ell \textsf {T} $$ , that can estimate the time complexity of programs, as a function of the size of their inputs. We prove that the type system is intentionally sound, in the sense that it over-approximates the complexity of executing the programs on a variant of the CEK abstract machine. Moreover, we define a sound and complete type inference algorithm which critically exploits the subrecursive nature of $$\textsf {d}\ell \textsf {T} $$ . Finally, we demonstrate the usefulness of $$\textsf {d}\ell \textsf {T} $$ for analyzing the complexity of cryptographic reductions by providing an upper bound for the constructed adversary of the Goldreich–Levin theorem.

Synthesizing differentially private programs

Inspired by the proliferation of data-analysis tasks, recent research in program synthesis has had a strong focus on enabling users to specify data-analysis programs through intuitive specifications, like examples and natural language. However, with the ever-increasing threat to privacy through data analysis, we believe it is imperative to reimagine program synthesis technology in the presence of formal privacy constraints. In this paper, we study the problem of automatically synthesizing randomized, differentially private programs, where the user can provide the synthesizer with a constraint on the privacy of the desired algorithm. We base our technique on a linear dependent type system that can track the resources consumed by a program, and hence its privacy cost. We develop a novel type-directed synthesis algorithm that constructs randomized differentially private programs. We apply our technique to the problems of synthesizing database-like queries as well as recursive differential privacy mechanisms from the literature.

Estimates of the Exponential Decay of Solutions to Linear Systems of Neutral Type with Periodic Coefficients

We consider the class of linear systems of delay differential equations with periodic coefficients. Using a special class of Lyapunov–Krasovskiĭ functionals, we establish conditions for the exponential stability of the zero solution and obtain estimates characterizing the exponential decay rate of solutions at infinity.

Formalization of Metatheory of the Quipper Quantum Programming Language in a Linear Logic

We develop a linear logical framework within the Hybrid system and use it to reason about the type system of a quantum lambda calculus. In particular, we consider a practical version of the calculus called Proto-Quipper, which contains the core of Quipper. Quipper is a quantum programming language under active development and recently has gained much popularity among the quantum computing communities. Hybrid is a system that is designed to support the use of higher-order abstract syntax for representing and reasoning about formal systems implemented in the Coq Proof Assistant. In this work, we extend the system with a linear specification logic (SL) in order to reason about the linear type system of Quipper. To this end, we formalize the semantics of Proto-Quipper by encoding the typing and evaluation rules in the SL, and prove type soundness.

Stability of neutral type delay systems: A joint Lyapunov–Krasovskii and Razumikhin approach

In this paper, a new stability criterion for linear time-delay systems of neutral type is presented. By virtue of the well-known Krasovskii theorem, the system is exponentially stable, if and only if there exists a functional which (i) has a negative definite derivative along the solutions of the system, and (ii) admits a quadratic lower bound, of course under an assumption that the difference operator corresponding to the system is stable. We modify the second condition in the following manner: The functional is required to admit a quadratic lower bound only on the special set of initial functions in our stability result instead of the set of all appropriate functions in the Krasovskii theorem. This special set consists of the functions satisfying a Razumikhin-type inequality and a similar inequality on the derivative. Basing on such modification and staying within the framework of the functionals with a given derivative, we suggest a methodology for the stability analysis, which is described and tested on the examples in the case of a scalar equation. Unlike most of recent results, we use the functional with a derivative prescribed just as a negative definite quadratic form of the “current” state of a system, what does not impose restrictions on the approach.

Synthesis of Observers for Linear Systems of Neutral Type

Two types of observers are proposed for linear autonomous differential-difference systems of neutral type. The first observer can be applied to completely identifiable systems (generalization of the spectral observability property to systems of neutral type), and it asymptotically exactly reconstructs the current state of the system. The second observer can be applied to systems without the above-cited property. The estimation error of the second observer is described, and conditions under which it asymptotically exactly reconstructs the current state of the system are proposed, as well as conditions under which the estimation error is bounded. The results are illustrated with examples.

Approximate symmetries, conservation laws and numerical solutions for a class of perturbed linear wave type system

The present work considers the Lie group analysis of a system of linear wave type perturbed systems. The methodology is based on finding approximate symmetry operators of a given system. Approximate conservation laws are found via an approximate version of Noether’s theorem. This is based on the modified Noether’s method provided by Ibragimov. Finally a numerical method is applied to solve the considered system.

On the Robust Stability of Solutions to Periodic Systems of Neutral Type

Under consideration is some class of linear systems of neutral type with periodic coefficients. We obtain the conditions on perturbations of the coefficients which preserve the exponential stability of the zero solution. Using a special Lyapunov–Krasovskii functional, we establish some estimates that characterize the rate of exponential decay at infinity of the solutions of the perturbed systems.

Linear Haskell: practical linearity in a higher-order polymorphic language

Linear type systems have a long and storied history, but not a clear path forward to integrate with existing languages such as OCaml or Haskell. In this paper, we study a linear type system designed with two crucial properties in mind: backwards-compatibility and code reuse across linear and non-linear users of a library. Only then can the benefits of linear types permeate conventional functional programming. Rather than bifurcate types into linear and non-linear counterparts, we instead attach linearity to function arrows . Linear functions can receive inputs from linearly-bound values, but can also operate over unrestricted, regular values. To demonstrate the efficacy of our linear type system — both how easy it can be integrated in an existing language implementation and how streamlined it makes it to write programs with linear types — we implemented our type system in ghc, the leading Haskell compiler, and demonstrate two kinds of applications of linear types: mutable data with pure interfaces; and enforcing protocols in I/O-performing functions.

Necessary conditions for exponential stability of linear neutral type systems with multiple time delays

In this paper, we will give necessary conditions for the exponential stability of linear neutral type systems with multiple time delays by employing the Lyapunov–Krasovskii functional approach. These conditions not only extend the existing results of the neutral-delay-free systems, but also provide a new tool for stability analysis of linear neutral type systems with multiple time delays by characterizing instability domains. As a medium step, we will investigate several crucial properties which are involved with both the fundamental matrix and Lyapunov matrix. Numerical examples illustrate the validity of the theoretical results.

Profust reliability of linear and circular type F and G systems having two failure criteria under Markov dependency

ABSTRACTIn this study, the profust reliabilities of (n, f, k): F(G) and < n, f, k > : F(G) systems for Markov dependent components are investigated. Having two failure criteria are the common features of these four systems. The usage of both fuzzy approach and two failure criteria in the same system provides us more realistic approach to evaluate the reliability of more complex systems. The component configurations are examined for both linear and circular sequences and the working principle of systems are studied for both F and G systems. Under all these assumptions, the profust reliabilities of (n, f, k): F(G) and < n, f, k > : F(G) systems are obtained using the distribution of run statistics. Also a new membership function different from the linear membership function which is generally used in the literature is proposed. Some numerical results which allow the comparison of the systems from various perspectives and various figures for both linear and circular type systems are presented. Some special cases (between Markov – iid assumption, conventional – profust reliability) are also considered.

Modal controllability for systems of neutral type in classes of differential-difference controllers

We propose a solution for the modal controllability problem for linear autonomous differential systems of neutral type with commensurate delays in two classes of differentialdifference controllers.

System Programming in Rust

Rust is a new system programming language that offers a practical and safe alternative to C. Rust is unique in that it enforces safety without runtime overhead, most importantly, without the overhead of garbage collection. While zero-cost safety is remarkable on its own, we argue that the superpowers of Rust go beyond safety. In particular, Rust's linear type system enables capabilities that cannot be implemented efficiently in traditional languages, both safe and unsafe, and that dramatically improve security and reliability of system software. We show three examples of such capabilities: zero-copy software fault isolation, efficient static information flow analysis, and automatic checkpointing. While these capabilities have been in the spotlight of systems research for a long time, their practical use is hindered by high cost and complexity. We argue that with the adoption of Rust these mechanisms will become commoditized.

Cooperative decoupled processes

Event-driven programming has become a major paradigm in developing concurrent, distributed systems. Its benefits are often informally captured by the key tenet of “decoupling,” a notion which roughly captures the ability of processes to join and leave (or fail) applications dynamically, and to be developed by independent parties. Programming models for event-driven programming either make it hard to globally reason about control flow, thus hampering sound execution, or sacrifice decoupling to aid in reasoning about control flow. This work fills the gap by introducing a programming model—dubbed cooperative decoupled processes—that achieves both decoupling and global reasoning about control flow. We introduce this programming model through an event calculus, loosely inspired by the Join calculus, that enables reasoning about cooperative decoupled processes through the concepts of pre- and postconditions. A linear type system controls aliasing of events to avoid a break of control flow and thus safe exchange of shared events. Fundamental properties of the type system such as subject reduction, migration safety, and progress are established.