Formal concept analysis and category theory in modeling interactions of living systems and their environments

Abstract

All functional systems are able to receive inputs, perform some transformations on them, and produce proper outputs. When dealing with mechanical systems, identifying the input/output relation is usually unambiguous because it is purposely designed. However, in biological systems, the situation is more complicated. First, because of their complexity, many specific purposes can be identified, such as feeding, moving, or reproducing. Therefore, decisions regarding what signals or changes should be identified as inputs or outputs are governed by the purpose of investigation. In this case, the difference between mechanical and biological systems is only a matter of scale. Second, and fundamentally different, is the ability of biological systems to treat the same signal in different manners, depending on their current state or context. As a consequence, it is very difficult to straightforwardly identify functional inputs or outputs for any process within a biological system. Usually, this problem is avoided by assuming that the common form of processing within the defined context is the only one. In this way, a modeling approach is reduced to the standard form of sequential machines and can be performed efficiently and satisfy the power of prediction. However, we believe that this is accompanied by the cost of losing deeper insight into some of the specific and important features of biological system functions, such as flexibility and evolvability. In this chapter, after some theoretical considerations, we develop a formal representation of the manner in which organisms "observe" the environment using formal concept analysis (FCA) and category theory. FCA is a branch of applied lattice theory that defines a concept as a unit of two parts: extension and intension. The extension covers all objects belonging to the particular concept, and the intension comprises all attributes that are valid for all of those objects. Both attributes and objects are united by a triple (G, M, I), which is called a formal context, while the ordered set of all formal concepts of (G, M, I) forms a concept lattice that is always complete. By using FCA alone, we can clearly represent how the available set of attributes governs the process of concept formation. However, it is not powerful enough to examine the variation of "observations" because it deals only with isolated lattices. To analyze the algebraic properties of such changeable systems, we further consider concept lattices as objects in the category of lattices and morphisms that preserve lattice completeness. Based on this analysis, we elaborate on some of the functional consequences of the ability of biological systems to perform at different scales with the variable identification of environmental signals. © 2012 by Nova Science Publishers, Inc. All rights reserved.