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14
First Steps Towards a Geometry of Computation
"... Summary. We introduce a geometrical 3 setting which seems promising for the study of computation in multiset rewriting systems, but could also be applied to register machines and other models of computation. This approach will be applied here to membrane systems (also known as P systems) without dyn ..."
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Summary. We introduce a geometrical 3 setting which seems promising for the study of computation in multiset rewriting systems, but could also be applied to register machines and other models of computation. This approach will be applied here to membrane systems (also known as P systems) without dynamical membrane creation. We discuss the rôle of maximum parallelism and further simplify our model by considering only one membrane and sequential application of rules, thereby arriving at asynchronous multiset rewriting systems (AMR systems). Considering only one membrane is no restriction, as each static membrane system has an equivalent AMR system. It is further shown that AMR systems without a priority relation on the rules are equivalent to Petri Nets. For these systems we introduce the notion of asymptotically exact computation, which allows for stochastic appearance checking in a priori bounded (for some complexity measure) computations. The geometrical analogy in the lattice N d 0, d ∈ N, is developed, in which a computation corresponds to a trajectory of a random walk on the directed graph induced by the possible rule applications. Eventually this leads to symbolic dynamics on the partition generated by shifted positive cones C + p, p ∈ N d 0, which are associated with the rewriting rules, and their intersections. Complexity measures are introduced and we consider non–halting, loop–free computations and the conditions imposed on the rewriting rules. Eventually, two models of information processing, control by demand and control by availability are discussed and we end with a discussion of possible future developments. 1
A New Biology: A Modern Perspective on the Challenge of Closing the Gap between the Islands of Knowledge
"... Abstract. This paper discusses the rebirth of the old quest for the principles of biology along the discourse line of machineorganism disanalogy and within the context of biocomputation from a modern perspective. It reviews some new attempts to revise the existing body of research and enhance it wi ..."
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Abstract. This paper discusses the rebirth of the old quest for the principles of biology along the discourse line of machineorganism disanalogy and within the context of biocomputation from a modern perspective. It reviews some new attempts to revise the existing body of research and enhance it with new developments in some promising fields of mathematics and computation. The major challenge is that the latter are expected to also answer the need for a new framework, a new language and a new methodology capable of closing the existing gap between the different levels of complex system organization.
Adaptive Behavior Journal Defining Agency
"... individuality, normativity, asymmetry and spatiotemporality in action ..."
ONTOLOGIES AND WORLDS IN CATEGORY THEORY: IMPLICATIONS FOR NEURAL SYSTEMS
"... ABSTRACT. We propose category theory, the mathematical theory of structure, as a vehicle for defining ontologies in an unambiguous language with analytical and constructive features. Specifically, we apply categorical logic and model theory, based upon viewing an ontology as a subcategory of a cate ..."
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ABSTRACT. We propose category theory, the mathematical theory of structure, as a vehicle for defining ontologies in an unambiguous language with analytical and constructive features. Specifically, we apply categorical logic and model theory, based upon viewing an ontology as a subcategory of a category of theories expressed in a formal logic. In addition to providing mathematical rigor, this approach has several advantages. It allows the incremental analysis of ontologies by basing them in an interconnected hierarchy of theories, with an operation on the hierarchy that expresses the formation of complex theories from simple theories that express first principles. Another operation forms abstractions expressing the shared concepts in an array of theories. The use of categorical model theory makes possible the incremental analysis of possible worlds, or instances, for the theories, and the mapping of instances of a theory to instances of its more abstract parts. We describe the theoretical approach by applying it to the semantics of neural networks.
Adap. Beh. J. Special Issue “Agency” Defining Agency
"... individuality, normativity, asymmetry and spatiotemporality in action ..."
Interactome Nonlinear Dynamic Models: Functors and Natural Transformations of Łukasiewicz Logic Algebras as Representations of Neural Network Development and Neoplastic Transformations of Tissues
"... A categorical and ŁukasiewiczTopos framework for Łukasiewicz Algebraic Logic models of nonlinear dynamics in complex functional systems such as neural networks, genomes and cell interactomes is proposed. Łukasiewicz Algebraic Logic models of genetic networks and signaling pathways in cells are form ..."
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A categorical and ŁukasiewiczTopos framework for Łukasiewicz Algebraic Logic models of nonlinear dynamics in complex functional systems such as neural networks, genomes and cell interactomes is proposed. Łukasiewicz Algebraic Logic models of genetic networks and signaling pathways in cells are formulated in terms of nonlinear dynamic systems with nstate components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable 'nextstate functions ' is extended to a Łukasiewicz Topos with an nvalued Łukasiewicz Algebraic Logic subobject classifier description that represents nonrandom and nonlinear network activities as well as their transformations in developmental processes and carcinogenesis. 1. Introduction. Previously, the assumption was made (Baianu,1977) that certain genetic activities have n levels of intensity, and this assumption is justified both by the existence of epigenetic controls, as well as by the coupling of the genome to the rest of the cell through specific signaling pathways that are involved in the modulation of both translation and transcription control
SUMMARY Contents
"... D2.2.1 Application of models of cooperation to network operation, design of P2P application, and social research through ..."
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D2.2.1 Application of models of cooperation to network operation, design of P2P application, and social research through
Summary of Changes
, 2009
"... 3 3,4,5 Split Ch 3 into Ch 3 and 4; began Logic chapter 4 1,4,5,7 Finished intro, algebra chapter, logic chapter, and conclusion ..."
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3 3,4,5 Split Ch 3 into Ch 3 and 4; began Logic chapter 4 1,4,5,7 Finished intro, algebra chapter, logic chapter, and conclusion
FROM SIMPLE TO HIGHLYCOMPLEX SYSTEMS: A PARADIGM SHIFT TOWARDS NONABELIAN EMERGENT SYSTEM DYNAMICS AND METALEVELS
"... Abstract. The evolution of nonlinear dynamical system theory and supercomplex systems–that are defined by classes of variable topologies and their associated transformations–is presented from a categorial and generalised, or extended topos viewpoint. A generalisation of dynamical systems, general ..."
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Abstract. The evolution of nonlinear dynamical system theory and supercomplex systems–that are defined by classes of variable topologies and their associated transformations–is presented from a categorial and generalised, or extended topos viewpoint. A generalisation of dynamical systems, general systems theory is then considered for the metalevel dynamical systems with variable topology and variable phase space, within the framework of an “universal”, or generalised Topos–a logicomathematical construction that covers both the commutative and noncommutative structures based on logic classifiers that are multivalued (MV) logic algebras. The extended topos concept was previously developed in conjunction with dynamic networks that were shown to be relevant to Complex Systems Biology. In so doing, we shall distinguish three major phases in the development of systems theory (two already completed and one currently unfolding). The three phases will be respectively called The Age of Equilibrium, The Age of Complexity and The Age of SuperComplexity. The first two may be taken as lasting from approximately 1850 to 1960, and the third which is now rapidly developing in applications to various types of systems after it began in the 1970s after the works of Rosen, Maturana and others. The mathematical theory of categories–which began in the 1940s [44],[45] with a seminal paper by Eilenberg and Mac Lane in 1945 [45] – is an unifying trend in modern mathematics [40], and has proved especially suitable for modeling the novelties raised by the third phase of systems ’ theory, which became associated with applications to system supercomplexity problems in the late 1950s and 70s [84][85],[2],[6],[8], [88][89]; it was continued by applications to logical programming involving categorical logic in computer science [58] , as well as the categorical foundations of mathematics [59][60].
Athel CornishBowden c, Marı´a Luz Ca´rdenas c
, 2005
"... www.elsevier.com/locate/yjtbi Organizational invariance and metabolic closure: Analysis in terms of ðM; RÞ systems ..."
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www.elsevier.com/locate/yjtbi Organizational invariance and metabolic closure: Analysis in terms of ðM; RÞ systems