使用Pacheck和Pastecque和Nuss-Checker检查器进行实用的代数计算和Nullstellensatz
Daniela Kaufmann1, Mathias Fleury1, Armin Biere2
1Institute for Formal Models and Verification, Johannes Kepler University, Linz, Austria.
概括
本研究介绍了实用的代数微积分和LPAC,用于自动推理中的高效,可验证的证明,增强对算术电路的正式验证的信任.
科学领域:
- 自动推理自动推理
- 正式验证是正式的验证.
- 计算机代数计算机代数.
背景情况:
- 使用计算机代数的自动推理对于正式验证至关重要,但验证过程中的错误需要可信的证明证书.
- 现有的证明系统如Nullstellensatz和多项式微积分在实际应用和证明混合方面存在局限性.
研究的目的:
- 将实用代数微积分 (PAC) 作为多项式微积分的有效可检查实例.
- 引入LPAC (PAC + 线性组合),统一Nullstellensatz和多项式微积分证明.
- 增强实用的重写技术的扩展和删除规则的证明系统.
主要方法:
- 开发了实用代数计算 (PAC) 以进行高效的证明检查.
- 通过将线性组合纳入PAC,引入了LPAC.
- 实施的扩展和删除规则用于实际重写.
- 关于算术电路验证的证明形式.
- 呈现的校对器:帕切克 (Pacheck),帕斯特克 (Pasteque) 和努斯-检查器.
主要成果:
- PAC提供了对多项式微积分证明的高效检查.
- LPAC成功地将Nullstellensatz和多项式微积分证明格式结合在一起.
- 开发的检验器 (Pacheck,Pasteque,Nuss-Checker) 证明了新检验系统的实际应用.
- 帕斯蒂克是正式使用伊莎贝尔/HOL进行验证,确保其可靠性.
结论:
- 拟议的实用代数计算和LPAC增强了对正式验证的自动推理的信任和效率.
- 这些进步使得能够生成可验证的证明证书,作为正式验证过程的副产品.
- 开发的工具为检查各种代数证明格式提供了实用解决方案.
相关概念视频
Theorems of Pappus and Guldinus: Problem Solving
659
Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
659
SFG Algebra
87
In Signal Flow Graph (SFG) algebra, the value a node represents is determined by the sum of all signals entering that node. This summed value is then transmitted through every branch leaving the node, making the SFG a powerful tool for visualizing and analyzing control systems.
Each node in an SFG corresponds to a variable, and the interactions between nodes are represented by branches with associated gains. When multiple branches lead into a node, the value at that node is the sum of the...
Each node in an SFG corresponds to a variable, and the interactions between nodes are represented by branches with associated gains. When multiple branches lead into a node, the value at that node is the sum of the...
87
Castigliano's Theorem: Problem Solving
508
The deflection of a simply supported beam that carries a central point load can be analyzed using structural mechanics principles, particularly by applying Castigliano's theorem. This theorem relates the displacement at the load application point to the partial derivatives of the strain energy in the structure. The simply supported beam with a point load at its center has symmetric reaction forces at the supports, each bearing half of the load. The bending moment at any point along the beam...
508
Euler's Formula to Columns: Problem Solving
124
Euler's formula is used in structural engineering to determine the buckling load of columns under various conditions. However, when dealing with systems that incorporate both rigid elements and elastic components, such as springs, the analysis requires a finer approach to determine the critical load. The problem described involves two rigid bars connected at a pivot point with a spring attached and a vertical load applied at one end.
The system comprises two vertical rigid bars, AB and BC,...
The system comprises two vertical rigid bars, AB and BC,...
124
Euler's Formula to Columns with Other End Conditions
403
Euler's formula is very important in the field of structural engineering, providing a foundation for understanding the critical loading conditions of pin-ended columns. This formula links the modulus of elasticity, the moment of inertia of the cross-section, and the column's length, offering a precise calculation of the critical load at which a column is prone to buckling.
403
Statically Indeterminate Problem Solving
334
Statically indeterminate problems are those where statics alone can not determine the internal forces or reactions. Consider a structure comprising two cylindrical rods made of steel and brass. These rods are joined at point B and restrained by rigid supports at points A and C. Now, the reactions at points A and C and the deflection at point B are to be determined. This rod structure is classified as statically indeterminate as the structure has more supports than are necessary for maintaining...
334


