In this paper, we study the complexity of the recurrence evaluation problem. We are interested in finitely valued recurrent functions. We present two results in this direction. First, we study the recurrence problem for sequences, assuming that a recurrence relation is defined by a fixed function, while the offsets are part of the input. Depending on the form of presentation (whether the offsets are given in unary or in binary), the problem is PSPACE-complete or EXP-complete. Second, we study recurrences defined by the NAND function. They are related to impartial games. We prove PP-hardness of the recurrence evaluation problem for a very simple 3-dimensional game, in which the offset vectors are coordinate vectors (1,0,0), (0,1,0) and (0,0,1) but the boundary conditions are arbitrary. In other words, we consider generalized winning conditions for the game extending the normal and the misère winning conditions.

翻译：本文研究了递归求值问题的复杂性。我们关注有限值递归函数，并在此方向上提出了两个结果。首先，我们研究了序列的递归问题，假设递归关系由固定函数定义，而偏移量作为输入的一部分。根据表示形式（偏移量以一元或二进制形式给出），该问题为PSPACE完全或EXP完全。其次，我们研究了由NAND函数定义的递归，它们与无偏博弈相关。我们证明了一个极简三维博弈（其偏移向量为坐标向量(1,0,0)、(0,1,0)和(0,0,1)，但边界条件任意）的递归求值问题具有PP难度。换言之，我们考虑了该博弈在常规获胜条件和反常获胜条件之外的广义获胜条件。