Plasmodium of Physarum polycephalum is a large single cell capable for distributed sensing, information processing, decentralized decision-making and collective action. In the paper, we interpret basic features of the plasmodium foraging behavior in terms of process calculus and spatial logic and show that this behavior could be regarded as one of the natural implementations of spatial logic without modal operators.

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We define a kind of simple actions of labelled transition systems. These actions cannot be atomic; consequently, their compositions cannot be inductive. Their informal meaning is that in one simple action we can suppose the maximum of its modifications. Such actions are called hybrid. Then we propose two formal theories on hybrid actions (the hybrid actions are defined there as non-well-founded terms and non-well-founded formulas): group theory and Boolean algebra. Both theories possess many unusual properties such as the following one: the same member of this group theory behaves as multiplicative zero in respect to one members and as multiplicative unit in respect to other members.

In the paper we build up the ontology of Leśniewski's type for formalizing synthetic propositions. We claim that for these propositions an unconventional square of opposition holds, where a, i are contrary, a, o (resp. e, i) are contradictory, e, o are subcontrary, a, e (resp. i, o) are said to stand in the subalternation. Further, we construct a non-Archimedean extension of Boolean algebra and show that in this algebra just two squares of opposition are formalized: conventional and the square that we invented. As a result, we can claim that there are only two basic squares of opposition. All basic constructions of the paper (the new square of opposition, the formalization of synthetic propositions within ontology of Leśniewski's type, the non-Archimedean explanation of square of opposition) are introduced for the first time.

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PurposeThe purpose of this paper is to fill a gap between experimental and abstract‐theoretic models of reaction‐diffusion computing. Chemical reaction‐diffusion computers are amongst leading experimental prototypes in the field of unconventional and nature‐inspired computing. In the reaction‐diffusion computers, the data are represented by concentration profiles of reagents, information is transferred by propagating diffusive and phase waves, computation is implemented in interaction of the traveling patterns, and results of the computation are recorded as a final concentration profile.Design/methodology/approachThe paper analyzes a possibility of co‐algebraic representation of the computation in reaction‐diffusion systems using reaction‐diffusion cellular‐automata models.FindingsUsing notions of space‐time trajectories of local domains of a reaction‐diffusion medium the logic of trajectories is built, where well‐formed formulas and their truth‐values are defined by co‐induction. These formulas are non‐well‐founded set‐theoretic objects. It is demonstrated that the logic of trajectories is a co‐algebra.Research limitations/implicationsThe paper uses the logic defined to establish a semantical model of the computation in reaction‐diffusion media.Originality/valueThe work presents the first ever attempt toward mathematical formalization of reaction‐diffusion processes and is built building up semantics of reaction‐diffusion computing. It is envisaged that the formalism produced will be used in developing programming techniques of reaction‐diffusion chemical media.