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
J. Oommen—Chancellor’s Professor; Fellow: IEEE and Fellow: IAPR. This author is also an Adjunct Professor with the University of Agder in Grimstad, Norway. The work of this author was partially supported by NSERC, the Natural Sciences and Engineering Research Council of Canada.
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- 1.
These algorithms can be designed to yield a high probabilistic guarantee of reaching the target. To do this, they formulate a limiting assumption: In order to choose a number of tests at each level that is large enough, one needs to know the exact probability by which the environment suggests the right direction of the search. In this paper, we assume that the latter probability is unknown and that the only known clue is that this quantity has a lower bound of \(\frac{1}{2}\). The case when the probability of correct transition is less than \(\frac{1}{2}\) is referred to as the symmetric environment [22].
- 2.
This is also in line with the theoretical results where we did not impose any condition on distributing \(1-p\). Thus, for the sake of brevity, we will not report those experiments with “uneven” distributions of \(1-p\) over the incorrect alternatives, and limit the reported results to the case of “even” distributions, i.e., \(\frac{1-p}{2}\).
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Yazidi, A., John Oommen, B. (2018). Novel Results on Random Walk-Jump Chains That Possess Tree-Based Transitions. In: Kurzynski, M., Wozniak, M., Burduk, R. (eds) Proceedings of the 10th International Conference on Computer Recognition Systems CORES 2017. CORES 2017. Advances in Intelligent Systems and Computing, vol 578. Springer, Cham. https://doi.org/10.1007/978-3-319-59162-9_5
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