Reversibility is a fundamental property of microscopic physical systems, implied by the laws of quantum mechanics, which seems to be at odds with the Second Law of Thermodynamics (Schiff 2008; Toffoli and Margolus 1990). Nonreversibility always implies energy dissipation, in practice, in the form of heat. Using reversible cellular automata (CAs) to simulate such systems has caused wide attention since the early days of the investigation of CAs (Toffoli and Margolus 1990; Kari 2005). On the other hand, if a CA is not reversible but reversible over an invariant closed subset, e.g., the limit set (Taaki 2007), it can also be used to describe physical systems locally. In this chapter, (Theorems 11.1, 11.2, and 11.4 were reproduced from Zhang and Zhang (2015) with permission @ 2015 Old City Publishing Inc. Theorems 11.3 and 11.5 were reproduced from Taaki (2007) with permission @ 2007 Old City Publishing Inc.) we present a formal definition to represent this class of generalized reversible CAs, and investigate some of their topological properties. We refer the reader to Zhang and Zhang (2015), Taaki (2007) for further reading. Other variants of generalized reversibility can be found in Castillo-Ramirez and Gadouleau (2017).
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