Do you happen to have definition of "energy" for cellular automata? I guess you could group states reachable via the reversible law (thus being on one loop) into equivalence classes, but that does not say anything about cells in any local area.
Physics is continuous and has Noether's theorem; for it, time shift symmetry (not even reversing time direction) implies conservation of energy.
For the physics case, I'm asking essentially whether there can be a physical rule (at least in a hypothetical universe different than our real one) that is time-reversible, but not having time shift symmetry, and thus not implying conservation of energy, or if time-reversible physical rules always imply time shift symmetry/time symmetry.
Another way to say it is I'm asking if there are hypothetical time-reversible rules that don't have the first law of conservation of energy due to not implying time symmetry.
Scott Aaronson claims that the analogue of energy...
I'll just answer the physics question, since I don't know anything about cellular automata.
When you say time-reversal symmetry, do you mean that t -> T-t is a symmetry for any T?
If so, the composition of two such transformations is a time-translation, so we automatically get time-translation symmetry, which implies the 1st law.
If not, then the 1st law needn't hold. E.g. take any time-dependent hamiltonian satisfying H(t) = H(-t). This has time-reversal symmetry about t=0, but H is not conserved.
“Time-Symmetric” and “reversible” mean the same thing to me: if you look at the system with reversed time, it obeys the same law. But apparently they don’t mean the same to OP, and I notice I am confused. In any event, as Mr Drori points out, symmetry/reversibility implies symmetry under time translation. If, further, the system can be described by a Hamiltonian (like all physical systems) then Noether’s Theorem applies, and energy is conserved.
Hm, I'm talking about time reversible physical laws, not necessarily time symmetric physical laws, so my question is do you always get time-symmetric physical laws that are symmetric for any T, out of time-reversible physical laws?
See also this question in another comment:
...For the physics case, I'm asking essentially whether there can be a physical rule (at least in a hypothetical universe different than our real one) that is time-reversible, but not having time shift symmetry, and thus not implying conservation of energy, or if time-reversible physical r
This question is kind of self-explanatory, but for people who are physicists, if a time reversible rule of physics/cellular automaton exists in a world, does this automatically imply the first law of thermodynamics, that is energy may not be created or destroyed?
Note I'm not talking about time-symmetry or the 2nd law of thermodynamics, which states that you can't have a 100% efficient machine, just time-reversible physical laws/cellular automatons and the first law of thermodynamics.
Edit: @jacob_drori has clarified what exactly I'm supposed to be asking, so the edited question is this:
Do you always get time-symmetric physical laws that are symmetric for any T, out of time-reversible physical laws?
The question of do you always get time-symmetric physical laws from time reversible laws is also a valid question to answer here, but the important part for the first law of thermodynamics to hold is that it's symmetric for all times T, and in principle, the question of time reversible laws of physics always implying time symmetry could have a positive answer while having a negative answer to the original question, because it doesn't imply time symmetric laws of physics for all T.