arXiv subjects

Florent Becker

Publications and source records attributed to Florent Becker.

10 recordsLinked to original sources

arXiv2026

Powers and Limitations of Synchronous Self-Assembly

In abstract models of algorithmic self-assembly, synchronization between attachments has emerged as a crucial distinction between the classical asynchronous model (aTAM) and a new synchronous model, the syncTAM. This paper presents recent advances in gauging the additional power afforded by the syncTAM. While it is known that the syncTAM and the aTAM are each unable to fully simulate the other, this paper offers evidence that the syncTAM is computationally significantly more powerful than the aTAM, especially in the non-cooperative setting. The additional power of the non-cooperative syncTAM is witnessed by the following constructions, all impossible in the non-cooperative aTAM: a flagpole, a strict self-assembly of a variant of the discrete Sierpinski triangle, and the ability to build the same assemblies (modulo scale factor) as directed aTAM systems. The second topic is that of limited synchronization, wherein, when the number of attachments is smaller than some threshold $l$, they happen synchronously, but attachments in excess of that number must wait. In that context, the precise value of $l$ is crucial, and changes to that value prevent simulation and can change which shapes can be obtained.

cs.CG↗

arXiv2024

Strict Self-Assembly of Discrete Self-Similar Fractals in the abstract Tile-Assembly Model

This paper answers a long-standing open question in tile-assembly theory, namely that it is possible to strictly assemble discrete self-similar fractals (DSSFs) in the abstract Tile-Assembly Model (aTAM). We prove this in 2 separate ways, each taking advantage of a novel set of tools. One of our constructions shows that specializing the notion of a quine, a program which prints its own output, to the language of tile-assembly naturally induces a fractal structure. The other construction introduces self-describing circuits as a means to abstractly represent the information flow through a tile-assembly construction and shows that such circuits may be constructed for a relative of the Sierpinski carpet, and indeed many other DSSFs, through a process of fixed-point iteration. This later result, or more specifically the machinery used in its construction, further enable us to provide a polynomial time procedure for deciding whether any given subset of $\mathbb{Z}^2$ will generate an aTAM producible DSSF. To this end, we also introduce the Tree Pump Theorem, a result analogous to the important Window Movie Lemma, but with requirements on the set of productions rather than on the self-assembling system itself.

cs.CG↗

arXiv2024

Strict Self-Assembly of Discrete Self-Similar Fractal Shapes

This paper gives a (polynomial time) algorithm to decide whether a given Discrete Self-Similar Fractal Shape can be assembled in the aTAM model.In the positive case, the construction relies on a Self-Assembling System in the aTAM which strictly assembles a particular self-similar fractal shape, namely a variant $K^\infty$ of the Sierpinski Carpet. We prove that the aTAM we propose is correct through a novel device, \emph{self-describing circuits} which are generally useful for rigorous yet readable proofs of the behaviour of aTAMs.We then discuss which self-similar fractals can or cannot be strictly self-assembled in the aTAM. It turns out that the ability of iterates of the generator to pass information is crucial: either this \emph{bandwidth} is eventually sufficient in both cardinal directions and $K^\infty$ appears within the fractal pattern after some finite number of iterations, or that bandwidth remains ever insufficient in one direction and any aTAM trying to self-assemble the shape will end up either bounded with an ultimately periodic pattern covering arbitrarily large squares. This is established thanks to a new characterization of the productions of systems whose productions have a uniformly bounded treewidth.

cs.DM↗

arXiv2022

DNA tile self-assembly for 3D-surfaces: Towards genus identification

We introduce a new DNA tile self-assembly model: the Surface Flexible Tile Assembly Model (SFTAM), where 2D tiles are placed on host 3D surfaces made of axis-parallel unit cubes glued together by their faces, called polycubes. The bonds are flexible, so that the assembly can bind on the edges of the polycube. We are interested in the study of SFTAM self-assemblies on 3D surfaces which are not always embeddable in the Euclidean plane, in order to compare their different behaviors and to compute the topological properties of the host surfaces. We focus on a family of polycubes called cuboids. Order-0 cuboids are polycubes that have six rectangular faces, and order-1 cuboids are made from two order-0 cuboids by substracting one from the other. Thus, order-1 cuboids can be of genus 0 or of genus 1 (then they contain a tunnel). We are interested in the genus of these structures, and we present a SFTAM tile assembly system that determines the genus of a given order-1 cuboid. The SFTAM tile assembly system which we design, contains a specific set $Y$ of tile types with the following properties. If the assembly is made on a host order-1 cuboid $C$ of genus 0, no tile of $Y$ appears in any producible assembly, but if $C$ has genus 1, every terminal assembly contains at least one tile of $Y$. Thus, we are able to distinguish the host surfaces according to their genus, by the tiles used in the assembly. This system is specific to order-1 cuboids but the techniques we use should be generalizable to other families of shapes.

cs.DM↗

arXiv2018

Universality in Freezing Cellular Automata

Cellular Automata have been used since their introduction as a discrete tool of modelization. In many of the physical processes one may modelize thus (such as bootstrap percolation, forest fire or epidemic propagation models, life without death, etc), each local change is irreversible. The class of freezing Cellular Automata (FCA) captures this feature. In a freezing cellular automaton the states are ordered and the cells can only decrease their state according to this "freezing-order". We investigate the dynamics of such systems through the questions of simulation and universality in this class: is there a Freezing Cellular Automaton (FCA) that is able to simulate any Freezing Cellular Automata, i.e. an intrinsically universal FCA? We show that the answer to that question is sensitive to both the number of changes cells are allowed to make, and geometric features of the space. In dimension 1, there is no universal FCA. In dimension 2, if either the number of changes is at least 2, or the neighborhood is Moore, then there are universal FCA. On the other hand, there is no universal FCA with one change and Von Neumann neighborhood. We also show that monotonicity of the local rule with respect to the freezing-order (a common feature of bootstrap percolation) is also an obstacle to universality.

cs.CC↗

arXiv2018

Abstract Geometrical Computation 10: An Intrinsically Universal Family of Signal Machines

Signal machines form an abstract and idealised model of collision computing. Based on dimensionless signals moving on the real line, they model particle/signal dynamics in Cellular Automata. Each particle, or signal, moves at constant speed in continuous time and space. When signals meet, they get replaced by other signals. A signal machine defines the types of available signals, their speeds and the rules for replacement in collision. A signal machine A simulates another one B if all the space-time diagrams of B can be generated from space-time diagrams of A by removing some signals and renaming other signals according to local information. Given any finite set of speeds S, we construct a signal machine that is able to simulate any signal machine whose speeds belong to S. Each signal is simulated by a macro-signal, a ray of parallel signals. Each macro-signal has a main signal located exactly where the simulated signal would be, as well as auxiliary signals which encode its id and the collision rules of the simulated machine. The simulation of a collision, a macro-collision, consists of two phases. In the first phase, macro-signals are shrunk, then the macro-signals involved in the collision are identified and it is ensured that no other macro-signal comes too close. If some do, the process is aborted and the macro-signals are shrunk, so that the correct macro-collision will eventually be restarted and successfully initiated. Otherwise, the second phase starts: the appropriate collision rule is found and new macro-signals are generated accordingly. Considering all finite set of speeds S and their corresponding simulators provides an intrinsically universal family of signal machines.

cs.FL↗

arXiv2015

It's a Tough Nanoworld: in Tile Assembly, Cooperation is not (strictly) more Powerful than Competition

We present a strict separation between the class of "mismatch free" self-assembly systems and general aTAM systems. Mismatch free systems are those systems in which concurrently grown parts must always agree with each other. Tile self-assembly is a model of the formation of crystal growth, in which a large number of particles concurrently and selectively stick to each other, forming complex shapes and structures. It is useful in nanotechnologies, and more generally in the understanding of these processes, ubiquitous in natural systems. The other property of the local assembly process known to change the power of the model is cooperation between two tiles to attach another. We show that disagreement (mismatches) and cooperation are incomparable: neither can be used to simulate the other one. The fact that mismatches are a hard property is especially surprising, since no known, explicit construction of a computational device in tile assembly uses mismatches, except for the recent construction of an intrinsically universal tileset, i.e. a tileset capable of simulating any other tileset up to rescaling. This work shows how to use intrinsic universality in a systematic way to highlight the essence of different features of tile assembly. Moreover, even the most recent experimental realizations do not use competition, which, in view of our results, suggests that a large part of the natural phenomena behind DNA self-assembly remains to be understood experimentally.

cs.CG↗

arXiv2013

Abstract Geometrical Computation 8: Small Machines, Accumulations and Rationality

In the context of abstract geometrical computation, computing with colored line segments, we study the possibility of having an accumulation with small signal machines, ie, signal machines having only a very limited number of distinct speeds. The cases of 2 and 4 speeds are trivial: we provide a proof that no machine can produce an accumulation in the case of 2 speeds and exhibit an accumulation with 4 speeds. The main result is the twofold case of 3 speeds. On the one hand, we prove that accumulations cannot happen when all ratios between speeds and all ratios between initial distances are rational. On the other hand, we provide examples of an accumulation in the case of an irrational ratio between 2 speeds and in the case of an irrational ratio between two distances in the initial configuration. This dichotomy is explained by the presence of a phenomenon computing Euclid's algorithm (gcd): it stops if and only if its input is commensurate (ie, of rational ratio).

cs.OH↗

arXiv2011

Interconnection network with a shared whiteboard: Impact of (a)synchronicity on computing power

In this work we study the computational power of graph-based models of distributed computing in which each node additionally has access to a global whiteboard. A node can read the contents of the whiteboard and, when activated, can write one message of O(log n) bits on it. When the protocol terminates, each node computes the output based on the final contents of the whiteboard. We consider several scheduling schemes for nodes, providing a strict ordering of their power in terms of the problems which can be solved with exactly one activation per node. The problems used to separate the models are related to Maximal Independent Set, detection of cycles of length 4, and BFS spanning tree constructions.

cs.DC↗

arXiv2010

Adding a referee to an interconnection network: What can(not) be computed in one round

In this paper we ask which properties of a distributed network can be computed from a little amount of local information provided by its nodes. The distributed model we consider is a restriction of the classical CONGEST (distributed) model and it is close to the simultaneous messages (communication complexity) model defined by Babai, Kimmel and Lokam. More precisely, each of these n nodes -which only knows its own ID and the IDs of its neighbors- is allowed to send a message of O(log n) bits to some central entity, called the referee. Is it possible for the referee to decide some basic structural properties of the network topology G? We show that simple questions like, "does G contain a square?", "does G contain a triangle?" or "Is the diameter of G at most 3? cannot be solved in general. On the other hand, the referee can decode the messages in order to have full knowledge of G when G belongs to many graph classes such as planar graphs, bounded treewidth graphs and, more generally, bounded degeneracy graphs. We leave open questions related to the connectivity of arbitrary graphs.

cs.CC↗