Similarly, when q = 1 the transition probability of the edge with symbol-label 0 from state C to state D goes to zero and we remove that edge from the epsilon machine representation. Now, states D, A, and B become transient and are removed from the epsilon machine representation. This leaves us with a single state epsilon machine that emits a single symbol, explaining why hµ = Cµ = E = 0 along the top boundary of the parameter space. Fig.5: Fig.6: Results Using these concepts, we prove the following theorem and state the following conjecture. The potential implications of these to the study of the structure of the space of probabilistic epsilon machines are the objective of current research, and promising ideas will be introduced below. Theorem Theorem: Topological epsilon machines do not “collapse” in a region of the transition probability distribution parameter space that is not also a “killing” region. Proof: In order for states to “collapse”, states “collapsing” must have the same probability distribution over the future. Suppose we have a topological epsilon machine. Now suppose that this topological epsilon machine has a symbol-labeling scheme that supports a transition probability distribution that allows for some states to have the same probability distribution over the future. We wish to show this is impossible. We will do this by means of reaching a contradiction. Suppose the epsilon machine has a symbol-labeling scheme that allows for some states to also have the same probability distribution over the future. Then there must be the same number of edges exiting these states, and they must have the same symbol-labeling. By definition, the transition probabilities of topological epsilon machines are uniformly distributed. Since the number of edges exiting these states is the same, the exiting probabilities must also match up. As we already assumed that the symbol-labeling scheme of the machine was such that it supported a probability distribution that allows for these states to “collapse”, we know the symbol-labels of edges exiting the states into which the original exiting edges entered match and that the
number of edges exiting these states is the same, as well. As the transition probabilities of the edges leaving these states must be uniformly distributed as well, we can conclude that the exiting probabilities from these states match up as well. Continuing with this logic, we see that an epsilon machine with a symbol-labeling scheme that would allow for states to have the same probability distribution over the future would, at the time of construction, have the transition probability distribution that did so. This means that these potentially “collapsible” states have equivalent probability distributions for the future; thus, our machine is not minimal. We must conclude that this machine is not an epsilon machine. But, we supposed from the beginning that our machine was a topological epsilon machine. Having reached a contradiction, we can therefore say that we cannot have a topological epsilon machine that is “collapsible” without first “killing” off an edge. Conjecture Conjecture: (The Converse) An epsilon machine with a symbol-labeling scheme that does not “collapse” in a region of the transition probability distribution parameter space being transition parameter space that is not also a “killing” region is isomorphic to a topological epsilon machine. One way to approach this conjecture would be to consider the restrictions imposed on an epsilon machine by the condition of the “collapsing” region being contained within “killing” region of the transition probability distribution parameter space. The next step would be to demonstrate the equivalence between this set of restrictions and the enumeration scheme of topological epsilon machines found in “Enumerating Finitary Processes” as a pruning of a larger class of objects. Implications A potential implication of the above theorem and conjecture relates back to the work done by Crutchfield, Johnson, Ellison, and McTague in “Enumerating Finitary Processes” in which they enumerate all topological epsilon machines with n states and alphabet size k. Starting with an arbitrary probabilistic epsilon machine, we can look at the transition probability distribution parameter space and identify the region where “collapsing” occurs. Properties of this space as a submanifold of the transition probability distribution parameter space can alone be explored as a means of identifying structure to the space of all epsilon machines. Yet were the conjecture above proven true, the epsilon machine representations in the parameter space where “collapsing” occurs are all topological epsilon machines, and thus can be identified