4,663 research outputs found
Building a Nest by an Automaton
A robot modeled as a deterministic finite automaton has to build a structure from material available to it. The robot navigates in the infinite oriented grid Z x Z. Some cells of the grid are full (contain a brick) and others are empty. The subgraph of the grid induced by full cells, called the field, is initially connected. The (Manhattan) distance between the farthest cells of the field is called its span. The robot starts at a full cell. It can carry at most one brick at a time. At each step it can pick a brick from a full cell, move to an adjacent cell and drop a brick at an empty cell. The aim of the robot is to construct the most compact possible structure composed of all bricks, i.e., a nest. That is, the robot has to move all bricks in such a way that the span of the resulting field be the smallest.
Our main result is the design of a deterministic finite automaton that accomplishes this task and subsequently stops, for every initially connected field, in time O(sz), where s is the span of the initial field and z is the number of bricks. We show that this complexity is optimal
Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure
String languages recognizable in (deterministic) log-space are characterized
either by two-way (deterministic) multi-head automata, or following Immerman,
by first-order logic with (deterministic) transitive closure. Here we elaborate
this result, and match the number of heads to the arity of the transitive
closure. More precisely, first-order logic with k-ary deterministic transitive
closure has the same power as deterministic automata walking on their input
with k heads, additionally using a finite set of nested pebbles. This result is
valid for strings, ordered trees, and in general for families of graphs having
a fixed automaton that can be used to traverse the nodes of each of the graphs
in the family. Other examples of such families are grids, toruses, and
rectangular mazes. For nondeterministic automata, the logic is restricted to
positive occurrences of transitive closure.
The special case of k=1 for trees, shows that single-head deterministic
tree-walking automata with nested pebbles are characterized by first-order
logic with unary deterministic transitive closure. This refines our earlier
result that placed these automata between first-order and monadic second-order
logic on trees.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur
Some remarks on A_1^{(1)} soliton cellular automata
In this short note, we describe the A_1^{(1)} soliton cellular automata as an
evolution of a poset. This allows us to explain the conservation laws for the
A_1^{(1)} soliton cellular automata, one given by Torii, Takahashi and Satsuma,
and the other given by Fukuda, Okado and Yamada, in terms of the stack
permutations of states in a very natural manner. As a biproduct, we can prove a
conjectured formula relating these laws.Comment: 10 pages, LaTeX2
Inflow process of pedestrians to a confined space
To better design safe and comfortable urban spaces, understanding the nature
of human crowd movement is important. However, precise interactions among
pedestrians are difficult to measure in the presence of their complex
decision-making processes and many related factors. While extensive studies on
pedestrian flow through bottlenecks and corridors have been conducted, the
dominant mode of interaction in these scenarios may not be relevant in
different scenarios. Here, we attempt to decipher the factors that affect human
reactions to other individuals from a different perspective. We conducted
experiments employing the inflow process in which pedestrians successively
enter a confined area (like an elevator) and look for a temporary position. In
this process, pedestrians have a wider range of options regarding their motion
than in the classical scenarios; therefore, other factors might become
relevant. The preference of location is visualized by pedestrian density
profiles obtained from recorded pedestrian trajectories. Non-trivial patterns
of space acquisition, e.g., an apparent preference for positions near corners,
were observed. This indicates the relevance of psychological and anticipative
factors beyond the private sphere, which have not been deeply discussed so far
in the literature on pedestrian dynamics. From the results, four major factors,
which we call flow avoidance, distance cost, angle cost, and boundary
preference, were suggested. We confirmed that a description of decision-making
based on these factors can give a rise to realistic preference patterns, using
a simple mathematical model. Our findings provide new perspectives and a
baseline for considering the optimization of design and safety in crowded
public areas and public transport carriers.Comment: 23 pages, 6 figure
Solutions of a two-particle interacting quantum walk
We study the solutions of the interacting Fermionic cellular automaton
introduced in Ref. [Phys Rev A 97, 032132 (2018)]. The automaton is the
analogue of the Thirring model with both space and time discrete. We present a
derivation of the two-particles solutions of the automaton, which exploits the
symmetries of the evolution operator. In the two-particles sector, the
evolution operator is given by the sequence of two steps, the first one
corresponding to a unitary interaction activated by two-particle excitation at
the same site, and the second one to two independent one-dimensional Dirac
quantum walks. The interaction step can be regarded as the discrete-time
version of the interacting term of some Hamiltonian integrable system, such as
the Hubbard or the Thirring model. The present automaton exhibits scattering
solutions with nontrivial momentum transfer, jumping between different regions
of the Brillouin zone that can be interpreted as Fermion-doubled particles, in
stark contrast with the customary momentum-exchange of the one dimensional
Hamiltonian systems. A further difference compared to the Hamiltonian model is
that there exist bound states for every value of the total momentum, and even
for vanishing coupling constant. As a complement to the analytical derivations
we show numerical simulations of the interacting evolution.Comment: 16 pages, 6 figure
Fuel Efficient Computation in Passive Self-Assembly
In this paper we show that passive self-assembly in the context of the tile
self-assembly model is capable of performing fuel efficient, universal
computation. The tile self-assembly model is a premiere model of self-assembly
in which particles are modeled by four-sided squares with glue types assigned
to each tile edge. The assembly process is driven by positive and negative
force interactions between glue types, allowing for tile assemblies floating in
the plane to combine and break apart over time. We refer to this type of
assembly model as passive in that the constituent parts remain unchanged
throughout the assembly process regardless of their interactions. A
computationally universal system is said to be fuel efficient if the number of
tiles used up per computation step is bounded by a constant. Work within this
model has shown how fuel guzzling tile systems can perform universal
computation with only positive strength glue interactions. Recent work has
introduced space-efficient, fuel-guzzling universal computation with the
addition of negative glue interactions and the use of a powerful non-diagonal
class of glue interactions. Other recent work has shown how to achieve fuel
efficient computation within active tile self-assembly. In this paper we
utilize negative interactions in the tile self-assembly model to achieve the
first computationally universal passive tile self-assembly system that is both
space and fuel-efficient. In addition, we achieve this result using a limited
diagonal class of glue interactions
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