5,255 research outputs found

    ON THE APPLICATIONS OF INTERACTIVE THEOREM PROVING IN COMPUTATIONAL SCIENCES AND ENGINEERING

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    Interactive Theorem Proving (ITP) is one of the most rigorous methods used in formal verification of computing systems. While ITP provides a high level of confidence in the correctness of the system under verification, it suffers from a steep learning curve and the laborious nature of interaction with a theorem prover. As such, it is desirable to investigate whether ITP can be used in unexplored (but high-impact) domains where other verification methods fail to deliver. To this end, the focus of this dissertation is on two important domains, namely design of parameterized self-stabilizing systems, and mechanical verification of numerical approximations for Riemann integration. Self-stabilization is an important property of distributed systems that enables recovery from any system configuration/state. There are important applications for self-stabilization in network protocols, game theory, socioeconomic systems, multi-agent systems and robust data structures. Most existing techniques for the design of self-stabilization rely on a ‘manual design and after-the-fact verification’ method. In a paradigm shift, we present a novel hybrid method of ‘synthesize in small scale and generalize’ where we combine the power of a finite-state synthesizer with theorem proving. We have used our method for the design of network protocols that are self-stabilizing irrespective of the number of network nodes (i.e., parameterized protocols). The second domain of application of ITP that we are investigating concentrates on formal verification of the numerical propositions of Riemann integral in formal proofs. This is a high-impact problem as Riemann Integral is considered one of the most indispensable tools of modern calculus. That has significant applications in the development of mission-critical systems in many Engineering fields that require rigorous computations such as aeronautics, space mechanics, and electrodynamics. Our contribution to this problem is three fold: first, we formally specify and verify the fundamental Riemann Integral inclusion theorem in interval arithmetic; second, we propose a general method to verify numerical propositions on Riemann Integral for a large class of integrable functions; third, we develop a set of practical automatic proof strategies based on formally verified theorems. The contributions of Part II have become part of the ultra-reliable NASA PVS standard library

    Separation of Circulating Tokens

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    Self-stabilizing distributed control is often modeled by token abstractions. A system with a single token may implement mutual exclusion; a system with multiple tokens may ensure that immediate neighbors do not simultaneously enjoy a privilege. For a cyber-physical system, tokens may represent physical objects whose movement is controlled. The problem studied in this paper is to ensure that a synchronous system with m circulating tokens has at least d distance between tokens. This problem is first considered in a ring where d is given whilst m and the ring size n are unknown. The protocol solving this problem can be uniform, with all processes running the same program, or it can be non-uniform, with some processes acting only as token relays. The protocol for this first problem is simple, and can be expressed with Petri net formalism. A second problem is to maximize d when m is given, and n is unknown. For the second problem, the paper presents a non-uniform protocol with a single corrective process.Comment: 22 pages, 7 figures, epsf and pstricks in LaTe

    Stable Optimization of a Tensor Product Variational State

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    We consider a variational problem for three-dimensional (3D) classical lattice models. We construct the trial state as a two-dimensional product of local variational weights that contain auxiliary variables. We propose a stable numerical algorithm for the maximization of the variational partition function per layer. The numerical stability and efficiency of the new method are examined through its application to the 3D Ising model.Comment: 9 pages, 5 figures, in LaTex2e style. accepted for publication in Prog. Theor. Phys. 11

    Competitive Gradient Descent

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    We introduce a new algorithm for the numerical computation of Nash equilibria of competitive two-player games. Our method is a natural generalization of gradient descent to the two-player setting where the update is given by the Nash equilibrium of a regularized bilinear local approximation of the underlying game. It avoids oscillatory and divergent behaviors seen in alternating gradient descent. Using numerical experiments and rigorous analysis, we provide a detailed comparison to methods based on \emph{optimism} and \emph{consensus} and show that our method avoids making any unnecessary changes to the gradient dynamics while achieving exponential (local) convergence for (locally) convex-concave zero sum games. Convergence and stability properties of our method are robust to strong interactions between the players, without adapting the stepsize, which is not the case with previous methods. In our numerical experiments on non-convex-concave problems, existing methods are prone to divergence and instability due to their sensitivity to interactions among the players, whereas we never observe divergence of our algorithm. The ability to choose larger stepsizes furthermore allows our algorithm to achieve faster convergence, as measured by the number of model evaluations.Comment: Appeared in NeurIPS 2019. This version corrects an error in theorem 2.2. Source code used for the numerical experiments can be found under http://github.com/f-t-s/CGD. A high-level overview of this work can be found under http://f-t-s.github.io/projects/cgd
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