Search

What is space computing,simulation, or understanding? Converging from several sources, this seems to be something more primitive than what is meant nowadays by computation, something that was along with us since antiquity (the word "choros", "chora", denotes "space" or "place" and is seemingly the most mysterious notion from Plato, described in Timaeus 48e - 53c) which has to do with cybernetics and with the understanding of the front end visual system. It may have some unexpected applications, also. Here, inspired by Bateson (see Supplementary Material), I explore from the mathematical side the point of view that there is no difference between the map and the territory, but instead the transformation of one into another can be understood by using a formalism of tangle diagrams.

chemSKI with tokens is a confluent graph rewrite system where all rewrites are local, which moreover can be used to do SKI calculus reductions. The graph rewrites of chemSKI are made conservative by the use of tokens. We thus achieve several goals: conservative rewrites in a chemical style, a solution to the problem of new edge names in a distributed, decentralized graphical reduction and a new estimation of the cost of a combinatory calculus computation. This formalism can be used either as an artificial chemistry or as a model of a virtual decentralized machine which performs only local reductions. A programs repository and the same article with simulations are available at github at https://mbuliga.github.io/chemski/chemski-with-tokens.html

We explain a dissipative version of hamiltonian mechanics, based on the information content of the deviation from hamiltonian dynamics. From this formulation we deduce minimal dissipation principles, dynamical inclusions, or constrained evolution with hamiltonian drift reformulations. Among applications we recover a dynamics generalization of Mielke et al quasistatic rate-independent processes. This article gives a clear and unitary presentation of the theory of hamiltonian inclusions with convex dissipation or symplectic Brezis-Ekeland-Nayroles principle, presented under various conventions first in arXiv:0810.1419, then in arXiv:1408.3102 and, for the appearance of bipotentials in relation to the symplectic duality, in arXiv:1902.04598v1., Comment: typos corrected, otherwise same as v1

Emergent algebras, first time introduced in arXiv:0907.1520 , are families of quasigroup operations indexed by a commutative group, which satisfy some algebraic relations and also topological (convergence and continuity) relations. Besides sub-riemannian geometry arXiv:math/0608536, they appear as a semantics of a family of graph-rewrite systems related to interaction combinators arXiv:2007.10288, or lambda calculus arXiv:1305.5786 . In arXiv:1807.02058 there is a lambda calculus version of emergent algebras. In this article we prove that for emergent algebras the condition (COLIN), or right-distributivity for emergent algebras, implies (LIN), or left-distributivity for emergent algebras. It means that any emergent algebra which is right-distributive has to come from a commutative group endowed with a family of dilations. This is surprising, because there are many examples of emergent algebras which satisfy (LIN), but not (COLIN), namely those who are associated to non-commutative conical groups, in particular to non-commutative Carnot groups.

Here I report about the modifications of and relations between graphic lambda calculus, various formalisms which appeared under the name chemlambda and a version of directed interaction combinators. This is part of the study and experiments with the artificial chemistry chemlambda and the relations with lambda calculus or interaction combinators, as described in arXiv:2003.14332 and available from the entry page at https://chemlambda.github.io/index.html, Comment: arXiv admin note: text overlap with arXiv:2003.14332

Given a graph rewrite system, a graph G is a quine graph if it has a non-void maximal collection of non-conflicting matches of left patterns of graphs rewrites, such that after the parallel application of the rewrites we obtain a graph isomorphic with G. Such graphs exhibit a metabolism, they can multiply or they can die, when reduced by a random rewriting algorithm. These are introductory notes to the pages of artificial chemistry experiments with chemlambda, lambda calculus or interaction combinators, available from the entry page https://chemlambda.github.io/index.html . The experiments are bundled into pages, all of them based on a library of programs, on a database which contains hundreds of graphs and on a database of about 150 pages of text comments and a collection of more than 200 animations, most of them which can be re-done live, via the programs. There are links to public repositories of other contributors to these experiments, with versions of these programs in python, haskell, awk or javascript.

A dissipative version of hamiltonian mechanics is proposed via a principle of minimal information content of the deviation from hamiltonian evolution. We show that we can cover viscosity, plasticity, damage and unilateral contact. This article continues arXiv:1807.10480

We propose the chemlambda artificial chemistry, whose behavior strongly suggests that real molecules which embed Interaction Nets patterns and real chemical reactions which resemble Interaction Nets graph rewrites could be a realistic path towards molecular computers, in the sense explained in the article., Comment: Github repository of the project at https://github.com/chorasimilarity/chemlambda-gui/blob/gh-pages/dynamic/README.md

The statistical counterpart of the formalism of hamiltonian systems with convex dissipation arXiv:0810.1419 , arXiv:1408.3102 is a completely open subject. Here are described a stochastic version of the SBEN principle and a Liouville type theorem which uses a minimal dissipation cost functional.

We introduce and study em (or "emergent"), a lambda calculus style rewrite system inspired from dilations structures in metric geometry. Then we add a new axiom (convex) and explore its consequences. Although (convex) forces commutativity of the infinitesimal operations, Theorems 6.2, 8.9 and Proposition 8.7 appear as a lambda calculus style version of Gleason and Montgomery-Zippin solution to the Hilbert 5th problem.

We propose a modification of the hamiltonian formalism which can be used for dissipative systems. This work continues arXiv:0810.1419 and advances by the introduction of a symplectic version of the Brezis-Ekeland-Nayroles principle [Brezis Ekeland 1976] [Nayroles 1976]. As an application we show how standard plasticity can be treated in our formalism.

Zipper logic is a graph rewrite system, consisting in only local rewrites on a class of zipper graphs. Connections with the chemlambda artificial chemistry and with knot diagrammatics based computation are explored in the article., Comment: 16 pages, 24 colour figures

We present chemlambda (or the chemical concrete machine), an artificial chemistry with the following properties: (a) is Turing complete, (b) has a model of decentralized, distributed computing associated to it, (c) works at the level of individual (artificial) molecules, subject of reversible, but otherwise deterministic interactions with a small number of enzymes, (d) encodes information in the geometrical structure of the molecules and not in their numbers, (e) all interactions are purely local in space and time. This is part of a larger project to create computing, artificial chemistry and artificial life in a distributed context, using topological and graphical languages., Comment: 8 pages, 21 colour figures, conference paper submitted to ALIFE 14

Based on graphic lambda calculus, we propose a program for a new model of asynchronous distributed computing, inspired from Hewitt Actor Model, as well as several investigation paths, concerning how one may graft lambda calculus and knot diagrammatics.

The chemical concrete machine is a graph rewriting system which uses only local moves (rewrites), seen as chemical reactions involving molecules which are graphs made up by 4 trivalent nodes. It is Turing complete, therefore it might be used as a model of computation in algorithmic chemistry.

We introduce and study graphic lambda calculus, a visual language which can be used for representing untyped lambda calculus, but it can also be used for computations in emergent algebras or for representing Reidemeister moves of locally planar tangle diagrams., Comment: v2: Minor typos and figure corrections in section 3. v1: Massive revision of all previous descriptions of graphic lambda calculus, based on arXiv:1207.0332 and arXiv:1302.0778, with a lot of material added and harmonized exposition

This is an expository article concerning the last section "Why is the tangent space a group?" (section 8) of the article by A. Bellaiche, The tangent space in sub-riemannian geometry, from the viewpoint of emergent algebras.

The Appendix of the article arXiv:1212.5056 [math.CO] "On growth in an abstract plane" by Nick Gill, H. A. Helfgott, Misha Rudnev, contains a general "geometric Ruzsa triangle inequality" in a Desarguesian projective plane. The purpose of this note is to give a similar inequality for metric spaces with dilations, that is in the absence of an algebraic or incidence structure., Comment: extended version, detailed proofs, graph rewrites on binary trees used for a short proof of lemma 2

This is a short description of graphic lambda calculus, with special emphasis on a duality suggested by the two different appearances of knot diagrams, in lambda calculus and emergent algebra sectors of the graphic lambda calculus respectively. This duality leads to the introduction of the dual of the graphic beta move. While the graphic beta move corresponds to beta reduction in untyped lambda calculus, the dual graphic beta move appears in relation to emergent algebras.

In arXiv:1207.0332 [cs.LO] was proposed a graphic lambda calculus formalism, which has sectors corresponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus.

We give a description of local and global moves on a class of locally planar trivalent graphs and we show that it contains $\lambda$-Scale calculus, therefore in particular untyped lambda calculus. Surprisingly, the beta reduction rule comes from a local "sewing" transformation of trivalent locally planar graphs.

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carath\'eodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homogeneous Lie group. Siebert characterizes homogeneous Lie groups as locally compact groups admitting a contracting and continuous one-parameter group of automorphisms. Siebert result has not a metric character. In these notes I show that sub-riemannian geometry may be described by about 12 axioms, without using any a priori given differential structure, but using dilation structures instead. Dilation structures bring forth the other intrinsic ingredient, namely the dilations, thus blending Gromov metric point of view with Siebert algebraic one., Comment: These are the notes prepared for the course "Metric spaces with dilations and sub-riemannian geometry from intrinsic point of view", CIMPA research school on sub-riemannian geometry (2012). Unitary exposition based on several previous papers of mine, especially arXiv:0810.5042, hopefully with many details, explanations and some proofs which I previously skipped

$\lambda$-Scale is an enrichment of lambda calculus which is adapted to emergent algebras. It can be used therefore in metric spaces with dilations., Comment: massively re-written version, condensed, relative calculus introduced

Inspired by the algorithm of Berga and de Saxce for solving the discretisation in time of the evolution problem for an implicit standard material, we propose a general variational formulation in terms of bipotentials.

We study normed groupoids with dilations and their induced deformations., Comment: arXiv admin note: this contains the content of arXiv:0911.1300

This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its "physical" meaning, and dilation structures as a convenient simplification of an exhaustive database of maps of a metric space into another. See arXiv:1103.6007 for the context., Comment: arXiv admin note: This is the appendix of arXiv:1103.6007, separated out as a separate submission

I try to give mathematical evidence to the following equivalence, which is based on ideas from Plato (Timaeus): reality emerges from a more primitive, non-geometrical, reality in the same way as the brain construct (understands, simulates, transforms, encodes or decodes) the image of reality, starting from intensive properties (like a bunch of spiking signals sent by receptors in the retina), without any use of extensive (i.e. spatial or geometric) properties.

The 19th century impressionist movement, as a reaction to "art pompier", is proposed as a comparison with modern evolutions in mathematics., Comment: with respect to the previous version, only fixed the hyperref

With the help of link diagrams with decorated crossings, I explain computations in emergent algebras, introduced in arXiv:0907.1520, as the kind of computations done in the front end visual system., Comment: comments welcomed

This paper gives a short introduction into the metric theory of spaces with dilations., Comment: comments welcome

Braided sets which are also spaces with dilations are presented and explored in this paper, in the general frame of emergent algebras arxiv:0907.1520. Examples of such spaces are the sub-riemannian symmetric spaces. Keywords: braided sets, quandles; emergent algebras; dilatation structures (spaces with dilations); sub-riemannian symmetric spaces., Comment: Final version, appeared in: Geometry. Exploratory Workshop on Differential Geometry and its Applications, eds. D. Andrica, S. Moroianu, Cluj-Napoca 2011, 21-35

For the importance of differentiation theorems in metric spaces (starting with Pansu Rademacher type theorem in Carnot groups) and relations with rigidity of embeddings see the section 1.2 in Cheeger and Kleiner paper arXiv:math/0611954 and its bibliographic references. Here we propose another type of differentiation theorem, which does not involve measures. It is therefore different from Rademacher type theorems. Instead, this differentiation theorem (and the concept of uniformly topological derivable function) is formulated in terms of filters in topological spaces.

Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $\delta$-structures. Algebraic study of normed groups endowed with dilatation structures is contained in the differential calculus on metric spaces. Thus all algebraic properties of the small world of normed groups with dilatation structures have equivalent formulations (of comparable complexity) in the big world of metric spaces admitting a differential calculus. Moreover these results non trivially extend beyond metric spaces, by using the language of groupoids.

Inspired from research subjects in sub-riemannian geometry and metric geometry, we propose uniform idempotent right quasigroups and emergent algebras as an alternative to differentiable algebras. Idempotent right quasigroups (irqs) are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). To any uniform idempotent right quasigroup can be associated an approximate differential calculus, with Pansu differential calculus in sub-riemannian geometry as an example. An emergent algebra A over a uniform idempotent right quasigroup X is a collection of operations such that each operation emerges from X, meaning that it can be realized as a combination of the operations of the uniform irq X, possibly by taking limits which are uniform with respect to a set of parameters. Two applications are considered: we prove a bijection between contractible groups and distributive uniform irqs (uniform quandles) and that some symmetric spaces in the sense of Loos may be seen as uniform quasigroups with a distributivity property., Comment: Modified title, some notations and improvements of exposition

Let X be a reflexive Banach space and Y its dual. In this paper we find necessary and sufficient conditions for the existence of a bipotential for a blurred maximal cyclically monotone graph. Equivalently, we find a necessary and sufficient condition on $\phi \in \Gamma_{0}(X)$ for that the differential inclusion $y \in \bar{B}(\epsilon) + \partial \phi(x)$ can be put in the form $y \in \partial b(\cdot, y)(x)$, with $b$ a bipotential., Comment: Revised version, corrections in theorem 6.2

In many practical situations, incertitudes affect the mechanical behaviour that is given by a family of graphs instead of a single one. In this paper, we show how the bipotential method is able to capture such blurred constitutive laws, using bipotential convex covers.

We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained from pairs of length dilatation structures, the first being a tempered one and the second obtained via a coherent projection. Thus we get an intrinsic, synthetic, axiomatic description of sub-riemannian geometry, which transforms the classical construction of a Carnot-Caratheodory distance on a regular sub-riemannian manifold into a model for this abstract sub-riemannian geometry., Comment: extended introduction and final discussion

We propose a generalization of hamiltonian mechanics, as a hamiltonian inclusion with convex dissipation function. We obtain a dynamical version of the approach of Mielke to quasistatic rate-independent processes. Then we show that a class of models of dynamical brittle damage can be formulated in this setting.

This is a survey of recent results about bipotentials representing multivalued operators. The notion of bipotential is based on an extension of Fenchel's inequality, with several interesting applications related to non associated constitutive laws in non smooth mechanics, such as Coulomb frictional contact or non-associated Drucker-Prager model in plasticity. Relations betweeen bipotentials and Fitzpatrick functions are described. Selfdual lagrangians, introduced and studied by Ghoussoub, can be seen as bipotentials representing maximal monotone operators. We show that bipotentials can represent some monotone but not maximal operators, as well as non monotone operators. Further we describe results concerning the construction of a bipotential which represents a given non monotone operator, by using convex lagrangian covers or bipotential convex covers., Comment: to appear in Acta Applicandae Mathematicae

We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry, endowed with a noncommutative vector addition operation and with a modified version of ratio of three collinear points. This is the geometry of normed affine group spaces, a category which contains the ones of homogeneous groups, Carnot groups or contractible groups. In this category group operations are not fundamental, but derived objects, and the generalization of affine geometry is not based on incidence relations., Comment: to appear in the Houston Journal of Mathematics

We show a surprising connexion between a property of the inf convolution of a family of convex lower semicontinuous functions and the fact that the intersection of maximal cyclically monotone graphs is the critical set of a bipotential. We then extend the results from arXiv:math/0608424v4 to bipotentials convex covers, generalizing the notion of a bi-implicitly convex lagrangian cover. As an application we prove that the bipotential related to Coulomb's friction law is related to a specific bipotential convex cover with the property that any graph of the cover is non maximal cyclically monotone., Comment: accepted by the Journal of Convex Analysis

In this short note we give an interpretation of the Kirchheim-Magnani counterexample to metric differentiability in terms of dilatation structures.

We show that on the boundary of the dyadic tree, any self-similar dilatation structure induces a web of interacting automata. This is a short version, for publication, of the paper arXiv:math/0612509v2

Based on the notion of dilatation structure arXiv:math/0608536, we give an intrinsic treatment to sub-riemannian geometry, started in the paper arXiv:0706.3644 . Here we prove that regular sub-riemannian manifolds admit dilatation structures. From the existence of normal frames proved by Bellaiche we deduce the rest of the properties of regular sub-riemannian manifolds by using the formalism of dilatation structures., Comment: to appear in "Contemporary Geometry and Topology and Related Topics. Cluj-Napoca, August 19-25, 2007"

In this paper I explain what is a pair of dilatation structures, one looking down to another. Such a pair of dilatation structures leads to the intrinsic definition of a distribution as a field of topological filters. To any pair of dilatation structures there is an associated notion of differentiability which generalizes the Pansu differentiability. This allows the introduction of the Radon-Nikodym property for dilatation structures, which is the straightforward generalization of the Radon-Nikodym property for Banach spaces. After an introducting section about length metric spaces and metric derivatives, is proved that for a dilatation structure with the Radon-Nikodym property the length of absolutely continuous curves expresses as an integral of the norms of the tangents to the curve, as in Riemannian geometry. Further it is shown that Radon-Nikodym property transfers from any "upper" dilatation structure looking down to a "lower" dilatation structure, theorem \ref{ttransfer}. Im my opinion this result explains intrinsically the fact that absolutely continuous curves in regular sub-Riemannian manifolds are derivable almost everywhere, as proved by Margulis-Mostow, Pansu (for Carnot groups) or Vodopyanov.

Here we prove that for dilatation structures linearity (see arXiv:0705.1440v1) is equivalent to a statement about the inverse semigroup generated by the family of dilatations of the space. The result is new for Carnot groups and the proof seems to be new even for vector spaces., Comment: larger version

A dilatation structure on a metric space, arXiv:math/0608536v4, is a notion in between a group and a differential structure, accounting for the approximate self-similarity of the metric space. The basic objects of a dilatation structure are dilatations (or contractions). The axioms of a dilatation structure set the rules of interaction between different dilatations. Linearity is also a property which can be explained with the help of a dilatation structure. In this paper we show that we can speak about two kinds of linearity: the linearity of a function between two dilatation structures and the linearity of the dilatation structure itself. Our main result here is a characterization of contractible groups in terms of dilatation structures. To a normed conical group (normed contractible group) we can naturally associate a linear dilatation structure. Conversely, any linear and strong dilatation structure comes from the dilatation structure of a normed contractible group., Comment: larger, updated version

We want to understand he concentration of damage in microfractured elastic media. Due to the different scallings of the volume and area (or area and length in two dimensions) the traditional method of homogenization using periodic arrays of cells seems to fail when applied to the Mumford-Shah functional and to periodically fractured domains. In the present paper we are departing from traditional homogenization. The main result implies the use of Mumford-Shah energies and leads to an explanation of the observed concentration of damage in microfractured elastic bodies.

By a combination of geometrical and configurational analysis we study the properties of absolute minimal and equilibrium states of general Mumford-Shah functionals, with applications to models of quasistatic brittle fracture propagation. The main results concern the mathematical relations between physical quantities as energy release rate and energy concentration for 3D cracks with complex shapes, seen as outer measures living on the crack edge., Comment: larger version

In this paper we continue the study of dilatation structures, introduced in math.MG/0608536 . A dilatation structure on a metric space is a kind of enhanced self-similarity. By way of examples this is explained here with the help of the middle-thirds Cantor set. Linear and self-similar dilatation structures are introduced and studied on ultrametric spaces, especially on the boundary of the dyadic tree (same as the middle-thirds Cantor set). Some other examples of dilatation structures, which share some common features, are given. Another class of examples, coming from sub-Riemannian geometry, will make the subject of an article in preparation. In the particular case of ultrametric spaces the axioms of dilatation structures take a simplified form, leading to a description of all possible weak dilatation structures on the Cantor set. As an application we prove that there is more than one linear and self-similar dilatation structure on the Cantor set, compatible with the iterated functions system which defines the Cantor set. Applications to self-similar groups are reserved for a further paper.