Fully Characterizing Lossy Catalytic Computation

A catalytic machine is a model of computation where a traditional space-bounded machine is augmented with an additional, significantly larger, "catalytic" tape, which, while being available as a work tape, has the caveat of being initialized with an arbitrary string, which must be preserved at the end of the computation. Despite this restriction, catalytic machines have been shown to have surprising additional power; a logspace machine with a polynomial length catalytic tape, known as catalytic logspace ($CL$), can compute problems which are believed to be impossible for $L$. A fundamental question of the model is whether the catalytic condition, of leaving the catalytic tape in its exact original configuration, is robust to minor deviations. This study was initialized by Gupta et al. (2024), who defined lossy catalytic logspace ($LCL[e]$) as a variant of $CL$ where we allow up to $e$ errors when resetting the catalytic tape. They showed that $LCL[e] = CL$ for any $e = O(1)$, which remains the frontier of our understanding. In this work we completely characterize lossy catalytic space ($LCSPACE[s,c,e]$) in terms of ordinary catalytic space ($CSPACE[s,c]$). We show that $$LCSPACE[s,c,e] = CSPACE[\Theta(s + e \log c), \Theta(c)]$$ In other words, allowing $e$ errors on a catalytic tape of length $c$ is equivalent, up to a constant stretch, to an equivalent errorless catalytic machine with an additional $e \log c$ bits of ordinary working memory. As a consequence, we show that for any $e$, $LCL[e] = CL$ implies $SPACE[e \log n] \subseteq ZPP$, thus giving a barrier to any improvement beyond $LCL[O(1)] = CL$. We also show equivalent results for non-deterministic and randomized catalytic space.