Problems in which some elementary entities interact with each other are common in computational intelligence. This scenario, typical for coevolving artificial life agents, learning strategies for games, and machine learning from examples, can be formalized as a test-based problem and conveniently embedded in the common conceptual framework of coevolution. In test-based problems, candidate solutions are evaluated on a number of test cases (agents, opponents, examples). It has been recently shown that every test of such problem can be regarded as a separate objective, and the whole problem as multi-objective optimization. Research on reducing the number of such objectives while preserving the relations between candidate solutions and tests led to the notions of underlying objectives and internal problem structure, which can be formalized as a coordinate system that spatially arranges candidate solutions and tests. The coordinate system that spans the minimal number of axes determines the so-called dimension of a problem and, being an inherent property of every problem, is of particular interest. In this study, we investigate in-depth the formalism of a coordinate system and its properties, relate them to properties of partially ordered sets, and design an exact algorithm for finding a minimal coordinate system. We also prove that this problem is NP-hard and come up with a heuristic which is superior to the best algorithm proposed so far. Finally, we apply the algorithms to three abstract problems and demonstrate that the dimension of the problem is typically much lower than the number of tests, and for some problems converges to the intrinsic parameter of the problem–its a priori dimension.

We propose a multitask learning method of visual concepts within the genetic programming (GP) framework. Each GP individual is composed of several trees that process visual primitives derived from input images. Two trees solve two different visual tasks and are allowed to share knowledge with each other by commonly calling the remaining GP trees (subfunctions) included in the same individual. The performance of a particular tree is measured by its ability to reproduce the shapes contained in the training images. We apply this method to visual learning tasks of recognizing simple shapes and compare it to a reference method. The experimental verification demonstrates that such multitask learning often leads to performance improvements in one or both solved tasks, without extra computational effort.