Spatial complexity measure for characterising cellular automata generated 2D patterns

Cellular automata (CA) are known for their capacity to generate complex patterns through the local interaction of rules. Often the generated patterns, especially with multi-state two-dimensional CA, can exhibit interesting emergent behaviour. This paper addresses quantitative evaluation of spatial characteristics of CA generated patterns. It is suggested that the structural characteristics of two-dimensional (2D) CA patterns can be measured using mean information gain. This information-theoretic quantity, also known as conditional entropy, takes into account conditional and joint probabilities of cell states in a 2D plane. The effectiveness of the measure is shown in a series of experiments for multi-state 2D patterns generated by CA. The results of the experiments show that the measure is capable of distinguishing the structural characteristics including symmetry and randomness of 2D CA patterns

A quantitative approach for detecting symmetries and complexity in 2D plane

Aesthetic evaluation of computer generated patterns is a growing filed with several challenges. This paper focuses on the quantitative evaluation of order and complexity in multi-state two-dimensional (2D) cellular automata (CA). CA are known for their ability to generate highly complex patterns through simple and well defined local interaction of rules. It is suggested that the order and complexity of 2D patterns can be quantified by using mean information gain. This measure, also known as conditional entropy, takes into account conditional and joint probabilities of the elements of a configuration in a 2D plane. A series of experiments is designed to demonstrate the effectiveness of the mean information gain in quantifying the structural order and complexity, including the orientation of symmetries of multi-state 2D CA configurations

A Comparative Analysis of Detecting Symmetries in Toroidal Topology

In late 1940s and with the introduction of cellular automata, various types of problems in computer science and other multidisciplinary fields have started utilising this new technique. The generative capabilities of cellular automata have been used for simulating various natural, physical and chemical phenomena. Aside from these applications, the lattice grid of cellular automata has been providing a by-product interface to generate graphical patterns for digital art creation. One notable aspect of cellular automata is symmetry, detecting of which is often a difficult task and computationally expensive. This paper uses a swarm intelligence algorithm—Stochastic Diffusion Search—to extend and generalise previous works and detect partial symmetries in cellular automata generated patterns. The newly proposed technique tailored to address the spatially-independent symmetry problem is also capable of identifying the absolute point of symmetry (where symmetry holds from all perspectives) in a given pattern. Therefore, along with partially symmetric areas, the centre of symmetry is highlighted through the convergence of the agents of the swarm intelligence algorithm. Additionally this paper proposes the use of entropy and information gain measure as a complementary tool in order to offer insight into the structure of the input cellular automata generated images. It is shown that using these technique provides a comprehensive picture about both the structure of the images as well as the presence of any complete or spatially-independent symmetries. These technique are potentially applicable in the domain of aesthetic evaluation where symmetry is one of the measures

Aesthetic Automata: Synthesis and Simulation of Aesthetic Behaviour in Cellular Automata

This thesis addresses the computational notion of aesthetics in the framework of multistate two-dimensional cellular automata (2D CA). The measure of complexity is a core concept in computational approaches to aesthetics. Shannon's information theory provided an objective measure of complexity, which led to the emergence of various informational theories of aesthetics. However, entropy fails to take into account the spatial characteristics of 2D patterns; these characteristics are fundamental in addressing the aesthetic problem, in general, and of CA-generated patterns, in particular. This thesis proposes two empirically evaluated alternative measures of complexity, taking into account the spatial characteristics of 2D patterns and experimental studies on human aesthetic perception in the visual domain. The measures are extended to robustly quantify the complexity of multi-state 2D CA-generated patterns. The first model, spatial complexity, is based on the probabilistic spatial distribution of homogeneous/heterogeneous neighbouring cells over the lattice of a multi-state 2D cellular automaton. The second model is based on algorithmic information theory (Kolmogorov complexity) which is extended to estimate the complexity of 2D patterns. The spatial complexity measure presents performance advantage over information-theoretic models, specifically in discriminating symmetries and the orientation in CA-generated patterns, enabling more accurate measurement of complexity in relation to aesthetic evaluations of 2D patterns. A series of experimental stimuli with various structural characteristics and levels of complexity were generated by seeding 3-state 2D CA with different initial configurations for psychological experiments. The results of experimentation demonstrate the presence of correlation between spatial complexity measures and aesthetic judgements of experimental stimuli. The same results were obtained for the estimations of Kolmogorov complexity of experimental stimuli

Information gain measure for structural discrimination of cellular automata configurations

Cellular automata (CA) are known for their capability in exhibiting interesting emergent behaviour and capacity to generate complex and often aesthetically appealing patterns through the local interaction of rules. Mean information gain has been suggested as a measure of discriminating structurally different two-dimensional (2D) patterns. This paper addresses quantitative evaluation of the complexity of CA generated configurations. In particular, we examine information gain as a spatial complexity measure for discriminating multi-state 2D CA generated configurations. This information-theoretic quantity, also known as conditional entropy, takes into account conditional and joint probabilities of cell states in a 2D plane. The effectiveness of the measure is shown in a series of experiments for multi-state 2D patterns generated by CA. The results of the experiments show that the measure is capable of distinguishing the structural characteristics including symmetries and randomness of 2D CA patterns

Analysis of information gain and Kolmogorov complexity for structural evaluation of cellular automata configurations

Shannon entropy fails to discriminate structurally different patterns in two-dimensional images. We have adapted information gain measure and Kolmogorov complexity to overcome the shortcomings of entropy as a measure of image structure. The measures are customised to robustly quantify the complexity of images resulting from multi-state cellular automata (CA). Experiments with a two-dimensional multi-state cellular automaton demonstrate that these measures are able to predict some of the structural characteristics, symmetry and orientation of CA generated patterns