Lossy Catalytic Computation
Chetan Gupta, Rahul Jain, Vimal Raj Sharma, Raghunath Tewari
TL;DR
This work addresses whether allowing a constant loss on the auxiliary tape of catalytic Turing machines increases power in the logspace regime. It defines the constant-$k$ lossy variant, and proves that $k$-$\LCL = \CL$ by a space-efficient reduction that simulates any $k$-lossy machine with a standard catalytic machine using the Fredman–Komlós–Szemerédi hashing scheme. The construction builds an $\mathcal{M'}$ that (i) finds a good prime $p$ with injective hashing for a neighborhood $W$ of auxiliary contents, (ii) stores the hash, (iii) simulates the original $\mathcal{M}$ on $(x,w)$, and (iv) recovers a valid auxiliary word within a set $Z$ of strings with $\hamdist \le k$, using the good prime to resolve ambiguity, then halts according to the original result. The result reinforces that bounded loss does not broaden the power of logspace catalytic computation and highlights hash-based encoding as a key technique in space-bounded computation, connecting to classical classes such as $\textsf{L}$, $\textsf{ZPP}$, and uniform $\textsf{TC}_1$.
Abstract
A catalytic Turing machine is a variant of a Turing machine in which there exists an auxiliary tape in addition to the input tape and the work tape. This auxiliary tape is initially filled with arbitrary content. The machine can read and write on the auxiliary tape, but it is constrained to restore its initial content when it halts. Studying such a model and finding its powers and limitations has practical applications. In this paper, we study catalytic Turing machines with O(log n)-sized work tape and polynomial-sized auxiliary tape that are allowed to lose at most constant many bits of the auxiliary tape when they halt. We show that such catalytic Turing machines can only decide the same set of languages as standard catalytic Turing machines with the same size work and auxiliary tape.