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One important problem in current cognitive development research is the lack of theory. In this article, therefore, we propose a cognitive development theory based on mathematical structures. Specifically, we first focus on the concept of structure, which is the concept Piaget introduced to cognitive developmental research. Piaget's theory was mainly inspired by mathematical group and lattice, but many concepts Piaget himself invented (e.g., grouping) were difficult to deal with in a mathematically rigorous manner. Therefore, here, the authors recapture some of the concepts proposed by Piaget in a mathematically understandable form based on the concept of mathematical category, a generalization of the group. Furthermore, we would like to introduce cognitive developmental research on structural concepts since Piaget and suggest directions for future research.

Motivated by Aoki et al.'s recent research on conserved charges and entropy current, we investigated the conservation of relativistic Ertel's current, which has received little attention outside the field of geophysical fluid dynamics. Ertel's charge is an important indicator of the correlation between vortex vectors and entropy gradient fields in Earth's meridional heat transport. We first show that in the generalized Hamiltonian structure of baroclinic fluids, the duality between the total energy and the Casimir as a function of Ertel's charge plays an important role in the nonrelativistic case. Then, by extending the result to relativistic cases, we show that this finding has far-reaching implications not only for spacetime issues in cosmology but also for the foundation of quantum field theory. An especially important finding is that, as an unreported dual form of the Einstein field equation, we identify a special equation satisfied not only by the vortex tensor field generated by the conserved charge but also by the Weyl tensor in interpreting the physical nature of the metric tensor $g^{\mu\nu}$, which appears in the cosmological term $\Lambda g^{\mu\nu}$., Comment: 16 pages, Submitted to Int. J. Mod. Phys. A

Generation of irregular time series based on physical processes is indispensable in computing and artificial intelligence. In this report, we propose and demonstrate the generation of Schubert polynomials, which are the foundation of versatile permutations in mathematics, via optical near-field processes introduced in a photochromic crystal of diarylethene combined with a simple photon detection protocol. Optical near-field excitation on the surface of a photochromic single crystal yields a chain of local photoisomerization, forming a complex pattern on the opposite side of the crystal. The incoming photon travels through the nanostructured photochromic crystal, and the exit position of the photon exhibits a versatile pattern. We emulated trains of photons based on the optical pattern experimentally observed through double-probe optical near-field microscopy, where the detection position was determined based on a simple protocol, leading to Schubert matrices corresponding to Schubert polynomials. The versatility and correlations of the generated Schubert matrices could be reconfigured in either a soft or hard manner by adjusting the photon detection sensitivity. This is the first study of Schubert polynomial generation via physical processes or nanophotonics, paving the way for future nano-scale intelligence devices and systems.

Characterizing consciousness in and of itself is notoriously difficult. Here, we propose an alternative approach to characterize, and eventually define, consciousness through exhaustive descriptions of consciousness’ relationships to all other consciousness. This approach is founded in category theory. Indeed, category theory can prove that two objects A and B in a category can be equivalent if and only if all the relationships that A holds with others in the category are the same as those of B; this proof is called the Yoneda lemma. To introduce the Yoneda lemma, we gradually introduce key concepts of category theory to consciousness researchers. Along the way, we propose several possible definitions of categories of consciousness, both in terms of level and contents, through the usage of simple examples. We propose to use the categorical structure of consciousness as a gold standard to formalize empirical research (e.g. color qualia structure at fovea and periphery) and, especially, the empirical testing of theories of consciousness.

Qualitative relationships between two instances of conscious experiences can be quantified through the perceived similarity. Previously, we proposed that by defining similarity relationships as arrows and conscious experiences as objects, we can define a category of qualia in the context of category theory. However, the example qualia categories we proposed were highly idealized and limited to cases where perceived similarity is binary: either present or absent without any gradation. When similarity is graded, a situation can arise where A0 is similar to A1, A1 is similar to A2, and so on, yet A0 is not similar to An, which is called the Sorites paradox. Here, we introduce enriched category theory to address this situation. Enriched categories generalize the concept of a relation between objects as a directed arrow (or morphism) in ordinary category theory to a more flexible notion, such as a measure of distance. As an alternative relation, here we propose a graded measure of perceived dissimilarity between the two objects. These measures combine in a way that addresses the Sorites paradox; even if the dissimilarity between Ai and Ai+1 is small for i = 0 … n, hence perceived as similar, the dissimilarity between A0 and An can be large, hence perceived as different. In this way, we show how dissimilarity-enriched categories of qualia resolve the Sorites paradox. We claim that enriched categories accommodate various types of conscious experiences. An important extension of this claim is the application of the Yoneda lemma in enriched category; we can characterize a quale through a collection of relationships between the quale and the other qualia up to an (enriched) isomorphism.

We have previously observed an irregular spatial distribution of photon transmission through a photochromic crystal photoisomerized by a local optical near-field excitation, manifesting complex branching processes via the interplay of deformation of the material and near-field photon transfer therein. Furthermore, by combining such naturally constructed complex photon transmission with a simple photon detection protocol, Schubert polynomials, the foundation of versatile permutation operations in mathematics, have been generated. In this study, we demonstrate an order recognition algorithm inspired by Schubert calculus using optical near-field statistics via nanometre-scale photochromism. More specifically, by utilizing Schubert polynomials generated via optical near-field patterns, we show that the order of slot machines with initially unknown reward probability is successfully recognized. We emphasize that, unlike conventional algorithms in the literature, the proposed principle does not estimate the reward probabilities. Instead, it exploits the inversion relations contained in the Schubert polynomials. To quantitatively evaluate the impact of the Schubert polynomials generated from an optical near-field pattern, order recognition performances are compared with uniformly distributed and spatially strongly skewed probability distributions, where the optical near-field pattern outperforms the others. We found that the number of singularities contained in Schubert polynomials and that of the given problem or considered environment exhibits a clear correspondence, indicating that superior order recognition performances may be attained if the singularity of the given problem is presupposed. This study paves a new way toward nanophotonic intelligent devices and systems by the interplay of complex natural processes and mathematical insights gained by Schubert calculus.

Machine learning techniques have realized some principal cognitive functionalities such as nonlinear generalization and causal model construction, as far as huge amount of data are available. A next frontier for cognitive modelling would be the ability of humans to transfer past knowledge to novel, ongoing experience, making analogies from the known to the unknown. Novel metaphor comprehension may be considered as an example of such transfer learning and analogical reasoning that can be empirically tested in a relatively straightforward way. Based on some concepts inherent in category theory, we implement a model of metaphor comprehension called the theory of indeterminate natural transformation (TINT), and test its descriptive validity of humans' metaphor comprehension. We simulate metaphor comprehension with two models: one being structure-ignoring, and the other being structure-respecting. The former is a sub-TINT model, while the latter is the minimal-TINT model. As the required input to the TINT models, we gathered the association data from human participants to construct the ``latent category'' for TINT, which is a complete weighted directed graph. To test the validity of metaphor comprehension by the TINT models, we conducted an experiment that examines how humans comprehend a metaphor. While the sub-TINT does not show any significant correlation, the minimal-TINT shows significant correlations with the human data. It suggests that we can capture metaphor comprehension processes in a quite bottom-up manner realized by TINT.

Characterizing consciousness in and of itself is notoriously difficult. Any effort to define consciousness seems to evade what it tries to achieve. In particular, definitions often involve comparisons of different kinds of “consciousness” in a self-referential manner. The tautological nature of characterizing consciousness has led some scholars to propose that establishing a science of consciousness is infeasible. Here, we propose an alternative approach to characterize, and to eventually define, consciousness through exhaustive descriptions of consciousness’ relationships to all other consciousness. This approach is mathematically founded in category theory. Indeed, category theory can prove two objects A and B in a category can be equivalent if and only if all the relationships that A holds with others in the category are the same as those of B; this proof is called the Yoneda lemma. To introduce the Yoneda lemma, we gradually introduce key concepts of category theory to consciousness researchers in this paper. Along the way, we propose several possible definitions of categories of consciousness, both in terms of level and contents, through the usage of simple examples. We also propose empirical research programs that can test the validity of our proposed categories of consciousness and to improve them. We propose to use the categorical structure of consciousness as a gold standard if one tries to empirically test some structural theories of consciousness, such as Integrated Information Theory of consciousness.

In our recent essay on Cognitive Science [Naotsugu Tsuchiya & Hayato Saigo (2019) “Understanding Consciousness Through Category Theory” vol 26, pp 462 - 477], we provided a general introduction of category theory to consciousness researchers. Further, we also provided our tentative theoretical sketches on our latest ideas on how to apply tools in category theory into consciousness research. In particular, we discussed how we can propose categories of level of consciousness and categories of contents of consciousness. We also speculated what (if any) these efforts will bring into consciousness research. In this short piece, we will address several comments we received on our essay on the same volume from six experts, providing some clarification on three issues: 1) significance of our proposal of a novel viewpoint to enrich what it means to define consciousness, 2) possibility of category theoretical interpretation of consciousness, and 3) understanding of consciousness through the enriched category theoretical framework.

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor $\mathcal{U}$ on a topos of sets $S$ together with a natural transformation $\upsilon$, instead of the terms as "standard", "internal" or "external". Moreover, we propose a general notion of a space called $\mathcal{U}$-space, and the category $\mathcal{U}space$ whose objects are $\mathcal{U}$-spaces and morphisms are functions called $\mathcal{U}$-spatial morphisms. The category $\mathcal{U}Space$, which is shown to be cartesian closed, will give a unified viewpoint toward topological and coarse geometric structure. It will also useful to study symmetries/asymmetries of the systems with infinite degrees of freedom such as quantum fields., Comment: 11 pages

我々が主観的に感じる意識と、脳の中で生じている物理・化学現象の間をつなぐ科学法則とは一体どのようなものだろうか？ 意識をめぐる謎は、現代科学が直面している謎の中でも最も巨大でかつ魅惑的なものの一つである。この謎を避け、人間や動物を外部から観測可能な行動のみから規定しようとする世界観では、幻覚・妄想などを訴える患者の意識の理解、さらには、言語報告のできない患者・乳児・動物の主観感覚の理解といった問題に、科学的なメスを入れることができない。近年、そのような世界観を乗り越え、意識の問題に真っ向から取り込もうとする科学者が中心となり、意識と脳の関係性を巡る実証的研究が大幅に進んできた。本稿では、現状見込みがあると考えられている意識研究のいくつかの研究の枠組みを概観する。さらに、従来の枠組みでは乗り越えるのが難しい意識の問題を解決するために、著者らが近年考えている方策を紹介する。著者らは意識経験の構造、さらに脳から生じる情報の構造、そして構造の間の関係性を明らかにする必要性があると考えている。本稿ではそれを可能にする数学的ツール、圏論も簡単に紹介し、今後の展望を述べる。

Towards the realization of a sustainable, fair and inclusive society, we proposed a novel decision-making model that incorporates social norms in a rational choice model from the standpoints of deontology and utilitarianism. We proposed a hypothesis that interprets choice of action as the X-point for individual utility function that increases with actions and social norm function that decreases with actions. This hypothesis is based on humans psychologically balancing the value of utility and norms in selecting actions. Using the hypothesis and approximation, we were able to isolate and infer utility function and norm function from real-world measurement data of actions on environmental conditions and elucidate the interaction between the both functions that led from current status to target actions. As examples of collective data that aggregate decision-making of individuals, we looked at the changes in power usage before and after the Great East Japan Earthquake and the correlation between national GDP and CO2 emission in different countries. The first example showed that the perceived benefits of power (i.e., utility of power usage) was stronger than the power usage restrictions imposed by norms after the earthquake, contrary to our expectation. The second example showed that a reduction of CO2 emission in each country was not related to utility derived from GDP but to norms related to CO2 emission. Going forward, we will apply this new X-point model to actual social practices involving normative problems, and design the approaches for the diagnosis, prognosis and intervention of social systems by IT systems., Comment: 15 pages, 13 figures. Published online at http://redfame.com/journal/index.php/aef/article/view/4890 . Related to arXiv:2002.09037

One of the most mysterious phenomena in science is the nature of conscious experience. Due to its subjective nature, a reductionist approach is having a hard time in addressing some fundamental questions about consciousness. These questions are squarely and quantitatively tackled by a recently developed theoretical framework, called integrated information theory (IIT) of consciousness. In particular, IIT proposes that a maximally irreducible conceptual structure (MICS) is identical to conscious experience. However, there has been no principled way to assess the claimed identity. Here, we propose to apply a mathematical formalism, category theory, to assess the proposed identity and suggest that it is important to consider if there exists a proper translation between the domain of conscious experience and that of the MICS. If such translation exists, we postulate that questions in one domain can be answered in the other domain; very difficult questions in the domain of consciousness can be resolved in the domain of mathematics. We claim that it is possible to empirically test if such a functor exists, by using a combination of neuroscientific and computational approaches. Our general, principled and empirical framework allows us to assess the relationship between the domain of consciousness and the domain of mathematical structures, including those suggested by IIT.

We propose the theory of indeterminate natural transformation (TINT) to investigate the dynamical creation of meaning as an association relationship between images, focusing on metaphor comprehension as an example. TINT models meaning creation as a type of stochastic process based on mathematical structure and defined by association relationships, such as morphisms in category theory, to represent the indeterminate nature of structure-structure interactions between the systems of image meanings. Such interactions are formulated in terms of the so-called coslice categories and functors as structure-preserving correspondences between them. The relationship between such functors is “indeterminate natural transformation”, the central notion in TINT, which models the creation of meanings in a precise manner. For instance, metaphor comprehension is modeled by the construction of indeterminate natural transformations from a canonically defined functor, which we call the base-of-metaphor functor.

Soft robotics is an emerging field of research where the robot body is composed of flexible and soft materials. It allows the body to bend, twist, and deform to move or to adapt its shape to the environment for grasping, all of which are difficult for traditional hard robots with rigid bodies. However, the theoretical basis and design principles for soft robotics are not well-founded despite their recognized importance. For example, the control of soft robots is outsourced to morphological attributes and natural processes; thus, the coupled relations between a robot and its environment are particularly crucial. In this paper, we propose a mathematical foundation for soft robotics based on category theory, a branch of abstract mathematics where any notions can be described by objects and arrows. It allows for a rigorous description of the inherent characteristics of soft robots and their relation to the environment as well as the differences compared to conventional hard robots. We present a notion called the category of mobility to well describe the subject matter. The theory has been applied to a model system and analysis to highlight the adaptation behavior observed in universal grippers, which are a typical example of soft robotics. The aim of the present study is not to offer concrete engineering solutions to existing robotics but to provide clear mathematical description of soft robots by category theory and to imply its potential abilities by a simple soft gripper demonstration. This paper paves the way to developing a theoretical background and design principles for soft robotics.

We introduced a decision-making model based on value functions that included individualistic utility function and socio-constructivistic norm function and proposed a norm-fostering process that recursively updates norm function through mutual recognition between the self and others. As an example, we looked at the resource-sharing problem typical of economic activities and assumed the distribution of individual actions to define the (1) norm function fostered through mutual comparison of value/action ratio based on the equity theory (progressive tax-like), (2) norm function proportional to resource utilization (proportional tax-like) and (3) fixed norm function independent of resource utilization (fixed tax-like). By carrying out numerical simulation, we showed that the progressive tax-like norm function (i) does not increase disparity for the distribution of the actions, unlike the other norm functions, and (ii) has high resource productivity and low Gini coefficient. Therefore the progressive tax-like norm function has the highest sustainability and fairness., Comment: 19 pages, 12 figures. Published online at http://redfame.com/journal/index.php/aef/article/view/4825 . Related to arXiv:2002.09036

Decision making based on behavioral and neural observations of living systems has been extensively studied in brain science, psychology, neuroeconomics, and other disciplines. Decision-making mechanisms have also been experimentally implemented in physical processes, such as single photons and chaotic lasers. The findings of these experiments suggest that there is a certain common basis in describing decision making, regardless of its physical realizations. In this study, we propose a local reservoir model to account for choice-based learning (CBL). CBL describes decision consistency as a phenomenon where making a certain decision increases the possibility of making that same decision again later. This phenomenon has been intensively investigated in neuroscience, psychology, and other related fields. Our proposed model is inspired by the viewpoint that a decision is affected by its local environment, which is referred to as a local reservoir. If the size of the local reservoir is large enough, consecutive decision making will not be affected by previous decisions, thus showing lower degrees of decision consistency in CBL. In contrast, if the size of the local reservoir decreases, a biased distribution occurs within it, which leads to higher degrees of decision consistency in CBL. In this study, an analytical approach for characterizing local reservoirs is presented, as well as several numerical demonstrations. Furthermore, a physical architecture for CBL based on single photons is discussed, and the effects of local reservoirs are numerically demonstrated. Decision consistency in human decision-making tasks and in recruiting empirical data is evaluated based on the local reservoir. This foundation based on a local reservoir offers further insights into the understanding and design of decision making.

Reinforcement learning involves decision-making in dynamic and uncertain environments and constitutes a crucial element of artificial intelligence. In our previous work, we experimentally demonstrated that the ultrafast chaotic oscillatory dynamics of lasers can be used to efficiently solve the two-armed bandit problem, which requires decision-making concerning a class of difficult trade-offs called the exploration–exploitation dilemma. However, only two selections were employed in that research; hence, the scalability of the laser-chaos-based reinforcement learning should be clarified. In this study, we demonstrated a scalable, pipelined principle of resolving the multi-armed bandit problem by introducing time-division multiplexing of chaotically oscillated ultrafast time series. The experimental demonstrations in which bandit problems with up to 64 arms were successfully solved are presented where laser chaos time series significantly outperforms quasiperiodic signals, computer-generated pseudorandom numbers, and coloured noise. Detailed analyses are also provided that include performance comparisons among laser chaos signals generated in different physical conditions, which coincide with the diffusivity inherent in the time series. This study paves the way for ultrafast reinforcement learning by taking advantage of the ultrahigh bandwidths of light wave and practical enabling technologies.

In the present paper we propose a new framework for the investigation of dressed photon from the viewpoint of quantum probability. As a result, we show that a probability measure called the Arcsine law plays crucial roles in nanophotonics. For instance, a concrete formula for the probability density of dressed photon in a fiber probe and a new prediction in photon breeding phenomena are given in terms of the Arcsine law.

In the present paper, a new aspect of the interplay is examined between mathematical–physical arguments and light–matter fusion technologies in terms of the concept of “effective mass”, starting from a question: Who has seen a free photon? Owing to the general results due to Newton–Wigner and to Wightman, a position operator is absent for massless free particles with non-zero finite spins, and hence, we cannot observe free photons in any local space regions. To solve this paradox of “photon localization”, the effective mass of a photon needs to be generated through the couplings of photons with matter. Here “polaritons” picture as a basic notion in optical and solid physics is shown to verify this viewpoint, which is seen to apply also to more general settings . Focusing on the role played by nanoparticles, we reach a new look at the notion of “dressed photons” as off-shell particles. The perspective above shows that essential mathematical structure of quantum field theory for the so-called elementary particles in subatomic scale can also be applied to certain phenomena in the nano-scale.

The subject of the present paper is an application of quantum probability to [Formula: see text]-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for [Formula: see text], where [Formula: see text] is a [Formula: see text]-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for [Formula: see text].

In the case of monotone independence, the transparent understanding of the mechanism to validate the central limit theorem (CLT) has been lacking, in sharp contrast to commutative, free and Boolean cases. We have succeeded in clarifying it by making use of simple combinatorial structure of peakless pair partitions., Comment: 5 pages