轨道有限线性规划
Orbit-finite linear programming
AI总结：
研究约束为轨道有限线性不等式组的线性规划推广，提出判定实数解存在性及计算目标函数极值的决策过程，并证明整数解情形不可判定。
摘要：
若一个无限集在原子的底层结构置换下仅有有限多个元素，则称其为轨道有限的。我们研究线性规划的一种推广，其约束由轨道有限的线性不等式组表示。作为主要贡献，我们提供了一种判定过程，用于检查此类系统是否存在实数解，并计算线性目标函数在解集上的最小/最大值。我们还证明了当仅考虑整数解时这些问题是不可判定的。因此，轨道有限线性规划是可判定的，而轨道有限整数线性规划则不可判定。
英文摘要：
An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.
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