Reversible AND universal elementary cellular automata

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The discussion centers on the computational universality of reversible extensions of one-dimensional cellular automata, specifically referencing Wolfram's Rule 37R and its potential for universal computation like Rule 110. Participants explore whether these reversible automata can construct universal Turing machines (UTMs) through the combination of NOR gates, which are known for their logical completeness. However, it is clarified that while NOR gates are universal in a logical sense, they do not guarantee universal computation without specific conditions regarding input and output topology. The conversation highlights the need for unbounded memory and coherent access methods for true computational universality. Additionally, the limitations of reversible computing are discussed, referencing historical work by Charles Bennett, emphasizing that reversible machines cannot perform certain tasks, such as polynomial-time factorization. Overall, the thread raises intriguing questions about the nature of computational universality in reversible cellular automata, but definitive answers remain elusive.
Giulio Prisco
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Has any of the reversible extensions of the elementary one-dimensional cellular automata described by Wolfram in NKS (e.g. Rule 37R) been shown to be computationally universal (like Rule 110)? If so, please give me links. Otherwise, could this be the case? Or is there a proof that no nR can be universal?
Has any of the reversible extensions of the elementary one-dimensional cellular automata described by Wolfram in NKS (e.g. Rule 37R) been shown to be computationally universal (like Rule 110)? If so, please give me links. Otherwise, could this be the case? Or is there a proof that no nR can be universal?
 
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pbuk said:
Interesting question, answers don't seem easy to find. Given that we can easily construct e.g. a NOR gate with a second order (reversible) cellular automaton I would have thought we should be able to construct a UTM. This paywalled paper could be useful: https://link.springer.com/referenceworkentry/10.1007/978-3-540-92910-9_7

Thanks, I have that paper.

Wolfram mentions Rule 37R as an example of “cellular automata with class 4 overall behavior” (NKS, Chapter 11). “I strongly suspect that all class 4 rules, like rule 110, will turn out to be universal,” he says.
 
The NOR gate is universal, so I would think that if a second-order reversible 1D cellular automata supports NOR gates that can be arbitrarily combined, then it is universal. Or not?
 
Giulio Prisco said:
The NOR gate is universal, so I would think that if a second-order reversible 1D cellular automata supports NOR gates that can be arbitrarily combined, then it is universal. Or not?
NOR gates are only universal when we fix the topology of the inputs and outputs in particular ways. So what do you mean by arbitrarily combined?
 
Nor gates are logically complete or logically universal, but it's not the same as universal computation (Turing complete). For universal computation you need unbounded memory and a way to traverse/access it in a coherent way.
 
Giulio Prisco said:
The NOR gate is universal, so I would think that if a second-order reversible 1D cellular automata supports NOR gates that can be arbitrarily combined, then it is universal.
As I indicated in #2 that is my instinct too (and more valuably it also seems to be Stephen Wolfram's), but instinct is not enough as @Jarvis323 points out.

Just in case anyone is thinking that we could do some impossible things with a reversible computing machine (like "unmultiply" a number to find its factors in polynomial time), note that (i) we can't (which hints at some of the characteristics of a reversible computer) and (ii) reversible computing is widely studied and has origins at least as far back as Charles Bennett's definitive paper Logical Reversibility of Computation from 1973.
 

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