Relative to any cellular automata capable of universal computation, initial states can be classified according to a nested hierarchy’of complexity classes. The first three levels of the hierarchy were informally known since the beginnings of cellular automata theory in the 20th century, and the next two levels were also speculated to exist, motivated by the idea of formalizing an abstract notion of “organism” and an abstract notion of “sentience”, respectively. EM-brain 202623897, a descendant of the Musk-Tao-Mirzahkani mergemind, formalized a definition of another two levels of the hierarchy, and argued in their 2107 paper that the formalization provides a basis for the aforementioned theories of “organism” and “sentience”. The EM138716-EM198274 theorem established that states of the fifth hierarchy exist for any universal CA; on the other hand, no examples of states of the fourth hierarchy have been found for any cellular automata, other than somewhat unsatisfying examples such as “an encoding of a program designed to systematically search for states of the fourth hierarchy”.
CA theorists generally agree that the EM20263897 hierarchy represents a conceptual leap in CA theory, but there is still debate as to whether the fourth and fifth hierarchy actually constitute a concept of “organism” and “sentience” as claimed. In particular, according to the transcoding theorem, any software program; and in particular, a simulated universe (of potentially unbounded size) containing a simulated genetic self-replicator or emulated brain, can be encoded as a finite state for a universal CA; but it is not known whether such encodings are included in the fourth level of the hierarchy. Amusingly, the main issue is the uncertainty as to whether or not all such self-replicators are destined to self-destruct [citation needed].
1. All states. The highest level attainable for any state sequence with a repeated state.
2. Aperiodic. No state ever repeats. A basic example of this is the glider gun) in the game of life.
3. Compuational Irreducibility. The state sequence cannot be matched by a non-universal cellular automata; for example, the state consisting of a single 1 for rule 30.
4. Level IV. The state sequence attains states of unbounded fractal entropy dimension. EM202623997 showed that the proportion states of size N in the third hierarchy with a supremum fractal entropy dimension above any constant times exp(N) goers to zero as N goes to infinity, i.e. “most” states in the third hierarchy are not contained in the fourth hierarchy. EM202623997 further argued that fractal entropy dimension captures the “macrostructures” exhibited by ecosystems. Additional work has confirmed that simulations of interacting and evolving self-replicators seem to increase in fractal entropy dimension.
5. Level V. The state sequence reaches an unbounded level of simuiation dimension. The formal definition is extremely technical, but the idea is that the state sequence contains space-localized subsequences which can be interpreted as simulations of the CA on a coarser scale, which may in turn contain space-localized subsequences with the same property, etc. The simulation dimension is a measure of such nesting: however, due to difficulties in formalizing what it means for a space-localized subsequence to contain a “simulation of the CA on a coarser scale”, the simulation dimension is a real-valued quantity (and in practice, uncomputable) rather than a whole number, EM202623997 proved that this property implied unbounded fractal dimension, and argued that requiring unboundedness agrees with the work of philosophers studying “universal characteristics of sentient, rational agents”.
Levels IV and V were originally named “life-like” and “sentience-llike”, respectively, in the original paper by EM202623997, but the term was never widely adopted.
Philosophical reaction
Philosopher EM19387 criticized the definition of the fifth hierarchy as conflating sentience with self-preservation. EM202623997 responded by speculating that “any universe capable of producing sentience can also produce ambitious sentience”, and hence the fifth hierarchy will, in practice, capture “most” state sequences which can be considered sentient.
Quantum analogues
Developing an analogous hierarchy for quantum cellular automata is an active area of research. In particular, a quantum CA could easily encode a universe based on string theory; hence, conditional on the accuracy of string theory in describing our universe, one can very directly ask whether universes similar to our (but modified to have unbounded informational capacity) own falls into the fourth hierarchy. Our own universe is probably not, according to most theories of cosmology, which indicate a bound on the informational capacity if our universe.
EM202623997 state complexity hierarchy
Relative to any cellular automata capable of universal computation, initial states can be classified according to a nested hierarchy’of complexity classes. The first three levels of the hierarchy were informally known since the beginnings of cellular automata theory in the 20th century, and the next two levels were also speculated to exist, motivated by the idea of formalizing an abstract notion of “organism” and an abstract notion of “sentience”, respectively. EM-brain 202623897, a descendant of the Musk-Tao-Mirzahkani mergemind, formalized a definition of another two levels of the hierarchy, and argued in their 2107 paper that the formalization provides a basis for the aforementioned theories of “organism” and “sentience”. The EM138716-EM198274 theorem established that states of the fifth hierarchy exist for any universal CA; on the other hand, no examples of states of the fourth hierarchy have been found for any cellular automata, other than somewhat unsatisfying examples such as “an encoding of a program designed to systematically search for states of the fourth hierarchy”.
CA theorists generally agree that the EM20263897 hierarchy represents a conceptual leap in CA theory, but there is still debate as to whether the fourth and fifth hierarchy actually constitute a concept of “organism” and “sentience” as claimed. In particular, according to the transcoding theorem, any software program; and in particular, a simulated universe (of potentially unbounded size) containing a simulated genetic self-replicator or emulated brain, can be encoded as a finite state for a universal CA; but it is not known whether such encodings are included in the fourth level of the hierarchy. Amusingly, the main issue is the uncertainty as to whether or not all such self-replicators are destined to self-destruct [citation needed].
1. All states. The highest level attainable for any state sequence with a repeated state.
2. Aperiodic. No state ever repeats. A basic example of this is the glider gun) in the game of life.
3. Compuational Irreducibility. The state sequence cannot be matched by a non-universal cellular automata; for example, the state consisting of a single 1 for rule 30.
4. Level IV. The state sequence attains states of unbounded fractal entropy dimension. EM202623997 showed that the proportion states of size N in the third hierarchy with a supremum fractal entropy dimension above any constant times exp(N) goers to zero as N goes to infinity, i.e. “most” states in the third hierarchy are not contained in the fourth hierarchy. EM202623997 further argued that fractal entropy dimension captures the “macrostructures” exhibited by ecosystems. Additional work has confirmed that simulations of interacting and evolving self-replicators seem to increase in fractal entropy dimension.
5. Level V. The state sequence reaches an unbounded level of simuiation dimension. The formal definition is extremely technical, but the idea is that the state sequence contains space-localized subsequences which can be interpreted as simulations of the CA on a coarser scale, which may in turn contain space-localized subsequences with the same property, etc. The simulation dimension is a measure of such nesting: however, due to difficulties in formalizing what it means for a space-localized subsequence to contain a “simulation of the CA on a coarser scale”, the simulation dimension is a real-valued quantity (and in practice, uncomputable) rather than a whole number, EM202623997 proved that this property implied unbounded fractal dimension, and argued that requiring unboundedness agrees with the work of philosophers studying “universal characteristics of sentient, rational agents”.
Levels IV and V were originally named “life-like” and “sentience-llike”, respectively, in the original paper by EM202623997, but the term was never widely adopted.
Philosophical reaction
Philosopher EM19387 criticized the definition of the fifth hierarchy as conflating sentience with self-preservation. EM202623997 responded by speculating that “any universe capable of producing sentience can also produce ambitious sentience”, and hence the fifth hierarchy will, in practice, capture “most” state sequences which can be considered sentient.
Quantum analogues
Developing an analogous hierarchy for quantum cellular automata is an active area of research. In particular, a quantum CA could easily encode a universe based on string theory; hence, conditional on the accuracy of string theory in describing our universe, one can very directly ask whether universes similar to our (but modified to have unbounded informational capacity) own falls into the fourth hierarchy. Our own universe is probably not, according to most theories of cosmology, which indicate a bound on the informational capacity if our universe.