A cell is a minimal self-sustaining system that can move and compute. Previous work has shown that a unicellular slime mold, Physarum, can be utilized as a biological computer based on cytoplasmic flow encapsulated by a membrane. Although the interplay between the modification of the boundary of a cell and the cytoplasmic flow surrounded by the boundary plays a key role in Physarum computing, no model of a cell has been developed to describe this interplay. Here we propose a toy model of a cell that shows amoebic motion and can solve a maze, Steiner minimum tree problem and a spanning tree problem. Only by assuming that cytoplasm is hardened after passing external matter (or softened part) through a cell, the shape of the cell and the cytoplasmic flow can be changed. Without cytoplasm hardening, a cell is easily destroyed. This suggests that cytoplasmic hardening and/or sol-gel transformation caused by external perturbation can keep a cell in a critical state leading to a wide variety of shapes and motion.

We study the permutation complexity of finite-state stationary stochastic processes based on a duality between values and orderings between values. First, we establish a duality between the set of all words of a fixed length and the set of all permutations of the same length. Second, on this basis, we give an elementary alternative proof of the equality between the permutation entropy rate and the entropy rate for a finite-state stationary stochastic processes first proved in [Amigó, J.M., Kennel, M. B., Kocarev, L., 2005. Physica D 210, 77-95]. Third, we show that further information on the relationship between the structure of values and the structure of orderings for finite-state stationary stochastic processes beyond the entropy rate can be obtained from the established duality. In particular, we prove that the permutation excess entropy is equal to the excess entropy, which is a measure of global correlation present in a stationary stochastic process, for finite-state stationary ergodic Markov processes.

The central notion of a rough set is indiscernibility based on equivalence relation. Since equivalence relation shows strong bondage in an equivalence class, it forms a Galois connection and the difference between upper and lower approximations is lost. We here introduce two different equivalence relations, the one for upper approximation, and the other for lower approximation, and construct composite approximation operator consisting of different equivalence relations. We show that a collection of fixed points with respect to the operator is a lattice, and that there exists a representation theorem for that construction. 1. Introduction This paper is written to make a difference between topological space and rough set theory [1, 2] clear in a term of lattice theory [3, 4]. Rough set provides a method for data analysis, based on the notion of indiscernibility which is defined by equivalence relation [5, 6]. Since equivalence classes can be analogously used as open sets, similar notions used in topological space can be defined. Upper and lower approximations in a rough set theory correspond to closure and internal set, respectively [7, 8]. On one hand, closure and internal set are defined under the constraint of a topological space (i.e., closed with respect to finite intersection and to any union). On the other hand, upper and lower approximation can be defined independent of such a kind of constraint. The essential difference between operations in a topological space and approximations in a rough set is the relationship between an element and a set (open set or equivalence class) containing the element. Any elements in an equivalence class have the same equivalence class, different from the case of topological space.

To overcome the dualism between mind and matter and to implement consciousness in science, a physical entity has to be embedded with a measurement process. Although quantum mechanics have been regarded as a candidate for implementing consciousness, nature at its macroscopic level is inconsistent with quantum mechanics. We propose a measurement-oriented inference system comprising Bayesian and inverse Bayesian inferences. While Bayesian inference contracts probability space, the newly defined inverse one relaxes the space. These two inferences allow an agent to make a decision corresponding to an immediate change in their environment. They generate a particular pattern of joint probability for data and hypotheses, comprising multiple diagonal and noisy matrices. This is expressed as a nondistributive orthomodular lattice equivalent to quantum logic. We also show that an orthomodular lattice can reveal information generated by inverse syllogism as well as the solutions to the frame and symbol-grounding problems. Our model is the first to connect macroscopic cognitive processes with the mathematical structure of quantum mechanics with no additional assumptions.