# Novel Results on Random Walk-Jump Chains That Possess *Tree*-Based Transitions

## Abstract

The most difficult task in analyzing and appraising algorithms in Artificial Intelligence (AI) involves their formal mathematical analysis. In general, such an analysis is intractable because of the size of the search space and the fact that the transitions between the states within this space can be very intricate. That is why AI algorithms are, for the most part, evaluated *empirically* and *experimentally*, i.e., by simulations. However, whenever such an analysis is undertaken, it usually involves an analysis of the underlying stochastic process. In this connection, the most common tools used involve Random Walks (RWs), which is a field that has been extensively studied for more than a century [6]. These walks have traditionally been on a *line*, and the generalizations for two and three dimensions, have been by extending the random steps to the corresponding neighboring positions in one or many of the dimensions. The analysis of RWs on a *tree* have received little attention, even though it is an important topic since a tree is a counter-part space representation of a line whenever there is some ordering on the nodes on the line.

Nevertheless, RWs on a tree entail moving to *non-neighbor* states in the space, which makes the analysis involved, and in many cases, impossible. This is precisely what we achieve in this rather pioneering paper. The applications of this paper are numerous. Indeed, the RW on the tree that this paper models, is a type of generalization of dichotomous search with faulty feedback about the direction of the search, rendering the real-life application of the model to be pertinent. To resolve this, we advocate the concept of “backtracking” transitions in order to efficiently explore the search space. Interestingly, it is precisely these “backtracking” transitions that naturally render the chain to be “time reversible”. By doing this, we are able to bridge the gap between deterministic dichotomous search and its faulty version, explained, in detail, in [21].

## Keywords

Time reversibility Controlled random walk Random walk with jumps Dichotomous search Learning systems## References

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