Catalytic computing concerns space bounded computation which starts with memory full of data that have to be restored by the end of the computation. Lossy catalytic computing, defined by Gupta et al. (2024) and fully characterized by Folkertsma et al. (ITCS 2025), is the study of allowing a small number of errors when resetting the catalytic tape at the end of a computation. Such a notion is useful when considering the robust use of catalytic techniques in the study of ordinary space-bounded algorithms. To that end however, defining and characterizing less strict notions of error was left open by Folkertsma et al. (ITCS 2025) and other works such as Mertz (B. EATCS, 2023). We expand the definition of possible resetting error in three natural ways: 1. randomized catalytic computation which can completely destroy the catalytic tape with some probability over the randomness 2. randomized catalytic computation which makes a bounded number of errors in expectation over the randomness 3. deterministic catalytic computation which makes a bounded number of errors in expectation over the initial catalytic tape itself We show a near complete characterization of the above models, both in the general case and in the logspace polynomial-time regime, by showing equivalences either between one another, to errorless catalytic space models, or to standard time or space complexity classes. Under a derandomization assumption, we show a near full collapse of all existing catalytic classes in the logspace regime.

翻译：催化计算关注的是空间有界计算，其特点是计算开始时内存中充满数据，且计算结束时这些数据必须还原。由Gupta等人(2024)定义、Folkertsma等人(ITCS 2025)完全刻画的有损催化计算，研究的是允许在计算结束重置催化带时出现少量错误的情形。在研究普通空间有界算法中鲁棒使用催化技术时，这一概念非常有用。然而，Folkertsma等人(ITCS 2025)以及Mertz(B. EATCS, 2023)等其他工作，将定义和刻画更宽松的错误概念留作未解决问题。我们以三种自然方式扩展了可能重置错误的定义：1. 随机化催化计算，以一定概率在随机性上完全破坏催化带；2. 随机化催化计算，在随机性上期望错误数有界；3. 确定性催化计算，在初始催化带本身的期望上错误数有界。我们通过展示这些模型之间、与无错催化空间模型、或与标准时间或空间复杂性类之间的等价关系，给出了上述模型在一般情形和对数空间多项式时间情形下的近乎完整的刻画。在去随机化假设下，我们证明了对数空间情形下所有现有催化类近乎完全坍缩。