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Fully Characterizing Lossy Catalytic Computation
Marten Folkertsma1,2, Ian Mertz3, Florian Speelman4,2
1CWI, Amsterdam, The Netherlands.
Algorithmica
|May 25, 2026
Summary
This study characterizes lossy catalytic space, showing that a small number of errors on a catalytic tape is equivalent to increased working memory in errorless catalytic machines. This finding establishes a barrier for improving the power of lossy catalytic logspace beyond constant errors.
Area of Science:
- Theoretical Computer Science
- Computational Complexity Theory
- Automata Theory
Background:
- Catalytic machines offer enhanced computational power beyond traditional space-bounded models.
- Catalytic logspace (CL) utilizes a special tape that must remain unchanged.
- Lossy catalytic logspace (LCL[e]) allows a limited number of errors (e) on the catalytic tape.
Purpose of the Study:
- To fully characterize the computational power of lossy catalytic space (LCSPACE[s,c,e]).
- To relate lossy catalytic space to standard catalytic space (CSPACE[s,c]).
- To investigate the implications of LCL[e] = CL for complexity classes.
Main Methods:
- Formal analysis of space and tape requirements for lossy catalytic machines.
- Derivation of equivalences between LCSPACE[s,c,e] and CSPACE[s,c].
- Establishing complexity-theoretic consequences of the characterized equivalences.
Main Results:
- LCSPACE[s,c,e] is equivalent to CSPACE[Θ(s + e log c), Θ(c)], meaning 'e' errors add e log c working memory.
- This characterization implies that LCL[e] = CL for any 'e' is equivalent to SPACE[e log n] ⊆ ZPP.
- A fundamental barrier is identified for improving lossy catalytic logspace beyond constant errors.
Conclusions:
- The catalytic condition's robustness to minor deviations is precisely quantified.
- The study provides a complete characterization of lossy catalytic space complexity.
- Results offer insights into the power of catalytic computation and its relation to standard complexity classes.
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