I was reading about Cellular Automata and I read in this question that Matthew Cook proved that rule 110 is universal, and that his proof relied upon showing how rule 110 can simulate a tag system. Is there another method or information that would still prove the universality of rule 110? In other words, is there another way to to do that without considering tag systems, and if so, can you explain the proof or give a reference to it?
More a bunch of references than an answer .... :-)
As said by @vzn in its comment Cook's proof is very complex and required a lot of work; nevertheless the universality of rule 110 renewed the interest on small model of computations.
Among the most notable results was the proof that rule 110 can simulate a deterministic Turing machine efficiently, i.e. with only a polynomial-time slowdown:
- Turlough Neary, Damien Woods: P-completeness of Cellular Automaton Rule 110. ICALP (1) 2006: 132-143
- Damien Woods, Turlough Neary: On the time complexity of 2-tag systems and small universal Turing machines. FOCS 2006: 439-448
Along the direction of a simplified/different proof of the universality of rule 110, you can find some work, e.g.:
Juan Carlos Seck Tuoh Mora, Genaro Juárez Martínez, Norberto Hernandez-Romero, Joselito Medina Marín: Elementary cellular automaton Rule 110 explained as a block substitution system. Computing 88(3-4): 193-205 (2010)
For a good survey on universality in cellular automata and other ideas for proving universality:
Finally a *personal* idea on an alternative approach that could be investigated:
- Proving universality of larger CAs (with larger neighbourhood or more states) is much easier and you can even get rid of the intermediate tag-system simulation, so you could investigate if using rule 110 gliders it is possible to simulate an arbitrary larger CA; a slightly similar approach has been used by Nicolas Ollinger and Gaétan Richard in A Particular Universal Cellular Automaton to prove the universality of colliding particles systems (they prove that a particular system can simulate an arbitrary CA)
And don't forget that instead of working on a different proof for rule 110 universality, you can work on the open problem regarding rule 54 ... is it (weakly) universal ? :-)