arXiv subjects

Mohit Tekriwal

Publications and source records attributed to Mohit Tekriwal.

6 recordsLinked to original sources

arXiv2025

Towards Richer Challenge Problems for Scientific Computing Correctness

Correctness in scientific computing (SC) is gaining increasing attention in the formal methods (FM) and programming languages (PL) community. Existing PL/FM verification techniques struggle with the complexities of realistic SC applications. Part of the problem is a lack of a common understanding between the SC and PL/FM communities of machine-verifiable correctness challenges and dimensions of correctness in SC applications. To address this gap, we call for specialized challenge problems to inform the development and evaluation of FM/PL verification techniques for correctness in SC. These specialized challenges are intended to augment existing problems studied by FM/PL researchers for general programs to ensure the needs of SC applications can be met. We propose several dimensions of correctness relevant to scientific computing, and discuss some guidelines and criteria for designing challenge problems to evaluate correctness in scientific computing.

cs.SE↗

arXiv2025

Towards Verified Compilation of Floating-point Optimization in Scientific Computing Programs

Scientific computing programs often undergo aggressive compiler optimization to achieve high performance and efficient resource utilization. While performance is critical, we also need to ensure that these optimizations are correct. In this paper, we focus on a specific class of optimizations, floating-point optimizations, notably due to fast math, at the LLVM IR level. We present a preliminary work, which leverages the Verified LLVM framework in the Rocq theorem prover, to prove the correctness of Fused-Multiply-Add (FMA) optimization for a basic block implementing the arithmetic expression $a * b + c$ . We then propose ways to extend this preliminary results by adding more program features and fast math floating-point optimizations.

cs.PL↗

arXiv2024

Formalization of Asymptotic Convergence for Stationary Iterative Methods

Solutions to differential equations, which are used to model physical systems, are computed numerically by solving a set of discretized equations. This set of discretized equations is reduced to a large linear system, whose solution is typically found using an iterative solver. We start with an initial guess, $x_0$, and iterate the algorithm to obtain a sequence of solution vectors, $x_k$, which are approximations to the exact solution of the linear system, $x$. The iterative algorithm is said to converge to $x$, in the field of reals, if and only if $x_k$ converges to $x$ in the limit of $k \to \infty$. In this paper, we formally prove the asymptotic convergence of a particular class of iterative methods called the stationary iterative methods, in the Coq theorem prover. We formalize the necessary and sufficient conditions required for the iterative convergence, and extend this result to two classical iterative methods: the Gauss--Seidel method and the Jacobi method. For the Gauss--Seidel method, we also formalize a set of easily testable conditions for iterative convergence, called the Reich theorem, for a particular matrix structure, and apply this on a model problem of the one-dimensional heat equation. We also apply the main theorem of iterative convergence to prove convergence of the Jacobi method on the model problem.

math.NA↗

arXiv2024

Formally verified asymptotic consensus in robust networks

Distributed architectures are used to improve performance and reliability of various systems. Examples include drone swarms and load-balancing servers. An important capability of a distributed architecture is the ability to reach consensus among all its nodes. Several consensus algorithms have been proposed, and many of these algorithms come with intricate proofs of correctness, that are not mechanically checked. In the controls community, algorithms often achieve consensus asymptotically, e.g., for problems such as the design of human control systems, or the analysis of natural systems like bird flocking. This is in contrast to exact consensus algorithm such as Paxos, which have received much more recent attention in the formal methods community. This paper presents the first formal proof of an asymptotic consensus algorithm, and addresses various challenges in its formalization. Using the Coq proof assistant, we verify the correctness of a widely used consensus algorithm in the distributed controls community, the Weighted-Mean Subsequence Reduced (W-MSR) algorithm. We formalize the necessary and sufficient conditions required to achieve resilient asymptotic consensus under the assumed attacker model. During the formalization, we clarify several imprecisions in the paper proof, including an imprecision on quantifiers in the main theorem.

cs.PL↗

arXiv2022

Dandelion: Certified Approximations of Elementary Functions

Elementary function operations such as sin and exp cannot in general be computed exactly on today's digital computers, and thus have to be approximated. The standard approximations in library functions typically provide only a limited set of precisions, and are too inefficient for many applications. Polynomial approximations that are customized to a limited input domain and output accuracy can provide superior performance. In fact, the Remez algorithm computes the best possible approximation for a given polynomial degree, but has so far not been formally verified. This paper presents Dandelion, an automated certificate checker for polynomial approximations of elementary functions computed with Remez-like algorithms that is fully verified in the HOL4 theorem prover. Dandelion checks whether the difference between a polynomial approximation and its target reference elementary function remains below a given error bound for all inputs in a given constraint. By extracting a verified binary with the CakeML compiler, Dandelion can validate certificates within a reasonable time, fully automating previous manually verified approximations.

cs.PL↗

arXiv2021

A formal proof of the Lax equivalence theorem for finite difference schemes

The behavior of physical systems is typically modeled using differential equations which are too complex to solve analytically. In practical problems, these equations are discretized on a computational domain, and numerical solutions are computed. A numerical scheme is called convergent, if in the limit of infinitesimal discretization, the bounds on the discretization error is also infinitesimally small. The approximate solution converges to the "true solution" in this limit. The Lax equivalence theorem enables a proof of convergence given consistency and stability of the method. In this work, we formally prove the Lax equivalence theorem using the Coq Proof Assistant. We assume a continuous linear differential operator between complete normed spaces, and define an equivalent mapping in the discretized space. Given that the numerical method is consistent (i.e., the discretization error tends to zero as the discretization step tends to zero), and the method is stable (i.e., the error is uniformly bounded), we formally prove that the approximate solution converges to the true solution. We then demonstrate convergence of the difference scheme on an example problem by proving both its consistency and stability, and then applying the Lax equivalence theorem. In order to prove consistency, we use the Taylor-Lagrange theorem by formally showing that the discretization error is bounded above by the nth power of the discretization step, where n is the order of the truncated Taylor polynomial.

math.NA↗