Parametric Computation in Axiom Towards Indefinite Symbolic Computing

目前的计算研究集中在如何对计算机进行编程，以执行涉及具体数学对象的代数计算。在具体情况下，可以使用显式数值数据来影响和控制运行时的计算。当对象是不特定的(不确定的)时，没有这样的信息可用，然而数学家经常用它们进行手工计算，作为获得有洞察力的定理的初步步骤。在很大程度上，涉及不确定的计算还没有得到深入的研究。这项研究探索了不确定对象的代数操作可以自动化的范围和方法，并为符号计算中一个全新的抽象水平奠定了基础。不定计算的计算机实现极大地扩展了计算机在代数中的应用范围，是一项最高级别的智力挑战。在理解和开发进行此类计算的方法方面取得的每一项微小进展都将有非常广泛的应用。计划的研究包括几个阶段：(1)实例调查和分析。(2)具体情况下的实验样机。(3)为不确定计算定义了一个实用的分类框架。(4)最终在开源系统Axiom中全面实现。(5)申请开题。研究人员将应用代数和计算机科学中的最新算法来定义框架。如有必要，将整合用于求解参数方程、符号求和、递归方程、Grobner基、柱面代数分解、动态评估和懒惰评估的前沿方法。为了更好地理解一些基本的困难，我们将研究在群论和微分代数中非常重要的特定开放问题。学生将通过课程、研讨会和动手实施参与。

Current research in computation focuses on how to program a computer to perform algebraic calculations that involve concrete mathematical objects. In a concrete situation, explicit numerical data is available to effect and control the calculations at run time. No such information is available when the objects are unspecified (indefinite), and yet mathematicians routinely carry out manual calculations with them as preliminary steps to obtain insightful theorems. For the most part, computations involving indefinites have not been studied in depth. This research explores both the scope and the methods by which algebraic manipulations of indefinite objects can be automated and lays the foundation for an entirely new level of abstraction in symbolic computation.Computer implementation of indefinite computation considerably expands the scope of computer use in algebra and is an intellectual challenge of the highest order. Every tiny inroad into understanding and developing methods to do such computations will have very wide applications. The planned research consists of several phases: (1) Investigation and analysis of examples. (2) Experimental prototypes in specific cases. (3) Defining a practical and categorical framework for indefinite computation. (4) Eventually full implementation in the open source system Axiom. (5) Applications to open problems. The investigators will apply state of the art algorithms in algebra and in computer science to define the framework. Cutting edge methods for solving parametric equations, symbolic summation, recurrence equations, Grobner basis, cylindrical algebraic decomposition, dynamical evaluation, and lazy evaluation will be integrated as necessary. Specific open problems of great importance in group theory and differential algebra will be studied in order to better understand some of the basic difficulties. Students will participate through courses, seminars, and hands-on implementation.