Related papers: Canonical for Automated Theorem Proving in Lean
The canonical duality theory has provided with a unified analytic solution to a range of discrete and continuous problems in global optimization, which can transform a nonconvex primal problem to a concave maximization dual problem over a…
Several approaches exist to data-mining big corpora of formal proofs. Some of these approaches are based on statistical machine learning, and some -- on theory exploration. However, most are developed for either untyped or simply-typed…
Time-dependent linear differential equations are a common type of problem that needs to be solved in classical physics. Here we provide a quantum algorithm for solving time-dependent linear differential equations with logarithmic dependence…
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness…
Verifying mathematical proofs is difficult, but can be automated with the assistance of a computer. Autoformalization is the task of automatically translating natural language mathematics into a formal language that can be verified by a…
Canonical matrices are given for (a) bilinear forms over an algebraically closed or real closed field; (b) sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and (c) sesquilinear…
Massive data analysis calls for distributed algorithms and theories. We design a multi-round distributed algorithm for canonical correlation analysis. We construct principal directions through the convex formulation of canonical correlation…
We describe arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects specified as a Haskell type class. Our algorithms work on a {\em generic} representation that we illustrate on…
Automated theorem proving, or more broadly automated reasoning, aims at using computer programs to automatically prove or disprove mathematical theorems and logical statements. It takes on an essential role across a vast array of…
The structure of the Euler-Lagrange equations for a general Lagrangian theory is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to…
Integrable systems have provided various insights into physical phenomena and mathematics. The way of constructing many-body integrable systems is limited to few ansatzes for the Lax pair, except for highly inventive findings of conserved…
We develop the theory of canonical-dissipative systems, based on the assumption that both the conservative and the dissipative elements of the dynamics are determined by invariants of motion. In this case, known solutions for conservative…
Logics with team semantics provide alternative means for logical characterization of complexity classes. Both dependence and independence logic are known to capture non-deterministic polynomial time, and the frontiers of tractability in…
This paper considers the development of an AI-based provably-correct mathematical proof tutor. While Large Language Models (LLMs) allow seamless communication in natural language, they are error prone. Theorem provers such as Lean allow for…
This paper considers the development of an AI-based provably-correct mathematical proof tutor. While Large Language Models (LLMs) allow seamless communication in natural language, they are error prone. Theorem provers such as Lean allow for…
We propose a new methodology, called numerical canonical quantization, to solve quantum Maxwell's equations useful for mathematical modeling of quantum optics physics, and numerical experiments on arbitrary passive and lossless…
We consider the problem of constrained motion along a conic path under a given external potential function. The model is described as a second-class system capturing the behavior of a certain class of specific quantum field theories. By…
We show that time complexity analysis of higher-order functional programs can be effectively reduced to an arguably simpler (although computationally equivalent) verification problem, namely checking first-order inequalities for validity.…
Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time…
We report on the development of an optimized and verified decision procedure for orthologic equalities and inequalities. This decision procedure is quadratic-time and is used as a sound, efficient and predictable approximation to classical…