Computable formal contexts

Abstract

This paper studies effective aspects of mathematical structures in formal concept analysis from the standpoint of computability theory. Firstly, we consider the notion of formal concepts of formal contexts and prove that the first and second derivation of a subset of objects of a computable context are ${\Pi }_1^0$- and ${\Pi }_2^0$-complete, respectively. Secondly, we examine the complexity of two representative processes in finding reductions of a context. To study the complexity of the process of merging objects with the same object intents of a context, we define a natural equivalence relation on objects of the context and show that the object equivalence relation of a computable context is ${\Pi }_1^0$-complete. We directly formalize the other process of removing reducible objects of a context via reducible objects themselves and prove that the set of reducible objects of a computable context is ${\Pi }_3^0$-complete. The down-left arrow relation is a useful tool to find reducible objects of a context. Lastly, we show that the arrow relation of a computable context is ${\Pi }_2^0$-complete. By dealing with dual contexts, we obtain the same complexity results on corresponding structures based on attributes of contexts.