Similarly, when q = 1 the transition probability of the edge with symbol-label 0 from state C to state goes to zero and we remove that edge from the epsilon machine representation. Now, D 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 in a region of the transition “collapse” probability distribution parameter space that is not also a region. “killing” Proof: In order for states to states must have the same probability distribution “collapse”, “collapsing” 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 we know the symbol-labels of “collapse”, edges exiting the states into which the original exiting edges entered match and that the
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
Research%20Report%20on%20Killing%20and%20Collapsing.docx