Fuzzy Decision Trees

The first (crisp) decision tree techniques were introduced in the 1960s (Hunt, Marin, & Stone, 1966), their appeal to decision makers is due in no part to their comprehensibility in classifying objects based on their attribute values (Janikow, 1998). With early techniques such as the ID3 algorithm (Quinlan, 1979), the general approach involves the repetitive partitioning of the objects in a data set through the augmentation of attributes down a tree structure from the root node, until each subset of objects is associated with the same decision class or no attribute is available for further decomposition, ending in a number of leaf nodes. This article considers the notion of decision trees in a fuzzy environment (Zadeh, 1965). The first fuzzy decision tree (FDT) reference is attributed to Chang and Pavlidis (1977), which defined a binary tree using a branch-bound-backtrack algorithm, but limited instruction on FDT construction. Later developments included fuzzy versions of crisp decision techniques, such as fuzzy ID3, and so forth (see Ichihashi, Shirai, Nagasaka, & Miyoshi, 1996; Pal & Chakraborty, 2001) and other versions (Olaru & Wehenkel, 2003).

AN EMERGENCE OF GAME STRATEGY IN MULTIAGENT SYSTEMS

Emergence of game strategy in multiagent systems is studied. Symbolic and subsymbolic (neural network) approaches are compared. Symbolic approach is represented by a backtrack algorithm with specified search depth, whereas the subsymbolic approach is represented by feedforward neural networks that are adapted by reinforcement temporal difference TD(λ) technique. As a test game, we use simplified checkers. The problem is studied in the framework of multiagent system, where each agent is endowed with a neural network used for a classification of checkers positions. Three different strategies are used. The first strategy corresponds to a single agent that repeatedly plays games against MinMax version of a backtrack search method. The second strategy corresponds to single agents that are repeatedly playing a megatournament, where each agent plays two different games with all other agents, one game with white pieces and the other game with black pieces. After finishing each game, both agents modify their neural networks by reinforcement learning. The third strategy is an evolutionary modification of the second one. When a megatournament is finished, each agent is evaluated by a fitness, which reflects its success in the given megatournament (more successful agents have greater fitness). It is demonstrated that all these approaches lead to a population of agents very successfully playing checkers against a backtrack algorithm with the search depth 3.

Construction of Occurrence Graphs with Permutation Symmetries Aided by the Backtrack Method

<p>This paper recalls the concept of occurrence graphs with permuta- tion symmetries (OS-graphs) for Coloured Petri Nets. It is explained how so-called self-symmetries can help to speed up construction of OS- graphs. The contribution of the paper is to suggest a new method for calculation of self-symmetries, the Backtrack Method. The method is based on the so-called Backtrack Algorithm, which originates in com- putational group theory. The suggestion of the method is justified, both by identifying an important general complexity property and by obtaining encouraging experimental performance measures.</p><p><strong>Topics.</strong> Coloured Petri Nets, reduced state spaces, occurrence graphs with permutation symmetries, self-symmetries, computational group theory, backtrack searches.</p>

The Gn,m Phase Transition is Not Hard for the Hamiltonian Cycle Problem

Using an improved backtrack algorithm with sophisticated pruning techniques, we revise previous observations correlating a high frequency of hard to solve Hamiltonian Cycle instances with the Gn,m phase transition between Hamiltonicity and non-Hamiltonicity. Instead all tested graphs of 100 to 1500 vertices are easily solved. When we artificially restrict the degree sequence with a bounded maximum degree, although there is some increase in difficulty, the frequency of hard graphs is still low. When we consider more regular graphs based on a generalization of knight's tours, we observe frequent instances of really hard graphs, but on these the average degree is bounded by a constant. We design a set of graphs with a feature our algorithm is unable to detect and so are very hard for our algorithm, but in these we can vary the average degree from O(1) to O(n). We have so far found no class of graphs correlated with the Gn,m phase transition which asymptotically produces a high frequency of hard instances.