Artem Parfenov | Searcharxiv

arXiv2026

On the Complexity of Recurrence Evaluation

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.

cs.CC↗

arXiv2023

Experimental Study of the Game Exact Nim(5, 2)

We compare to different extensions of the ancient game of nim: Moore's nim$(n, \leq k)$ and exact nim$(n, = k)$. Given integers $n$ and $k$ such that $0 < k \leq n$, we consider $n$ piles of stones. Two players alternate turns. By one move it is allowed to choose and reduce any (i) at most $k$ or (ii) exactly $k$ piles of stones in games nim$(n, \leq k)$ and nim$(n, = k)$, respectively. The player who has to move but cannot is the loser. Both games coincide with nim when $k=1$. Game nim$(n, \leq k)$ was introduced by Moore (1910) who characterized its Sprague-Grundy (SG) values 0 (that is, P-positions) and 1. The first open case is SG values 2 for nim$(4, \leq 2)$. Game nim$(n, = k)$, was introduced in 2018. An explicit formula for its SG function was computed for $2k \geq n$. In contrast, case $2k < n$ seems difficult: even the P-positions are not known already for nim$(5,=2)$. Yet, it seems that the P-position of games nim$(n+1,=2)$ and nim$(n+1,\leq 2)$ are closely related. (Note that P-positions of the latter are known.) Here we provide some theoretical and computational evidence of such a relation for $n=5$.

math.CO↗