64 research outputs found

### Properties of Pseudo-Primitive Words and their Applications

A pseudo-primitive word with respect to an antimorphic involution \theta is a
word which cannot be written as a catenation of occurrences of a strictly
shorter word t and \theta(t). Properties of pseudo-primitive words are
investigated in this paper. These properties link pseudo-primitive words with
essential notions in combinatorics on words such as primitive words,
(pseudo)-palindromes, and (pseudo)-commutativity. Their applications include an
improved solution to the extended Lyndon-Sch\"utzenberger equation u_1 u_2 ...
u_l = v_1 ... v_n w_1 ... w_m, where u_1, ..., u_l \in {u, \theta(u)}, v_1,
..., v_n \in {v, \theta(v)}, and w_1, ..., w_m \in {w, \theata(w)} for some
words u, v, w, integers l, n, m \ge 2, and an antimorphic involution \theta. We
prove that for l \ge 4, n,m \ge 3, this equation implies that u, v, w can be
expressed in terms of a common word t and its image \theta(t). Moreover,
several cases of this equation where l = 3 are examined.Comment: Submitted to International Journal of Foundations of Computer Scienc

### Transfer matrix analysis of one-dimensional majority cellular automata with thermal noise

Thermal noise in a cellular automaton refers to a random perturbation to its
function which eventually leads this automaton to an equilibrium state
controlled by a temperature parameter. We study the 1-dimensional majority-3
cellular automaton under this model of noise. Without noise, each cell in this
automaton decides its next state by majority voting among itself and its left
and right neighbour cells. Transfer matrix analysis shows that the automaton
always reaches a state in which every cell is in one of its two states with
probability 1/2 and thus cannot remember even one bit of information. Numerical
experiments, however, support the possibility of reliable computation for a
long but finite time.Comment: 12 pages, 4 figure

### Proving the Turing Universality of Oritatami Co-Transcriptional Folding (Full Text)

We study the oritatami model for molecular co-transcriptional folding. In
oritatami systems, the transcript (the "molecule") folds as it is synthesized
(transcribed), according to a local energy optimisation process, which is
similar to how actual biomolecules such as RNA fold into complex shapes and
functions as they are transcribed. We prove that there is an oritatami system
embedding universal computation in the folding process itself.
Our result relies on the development of a generic toolbox, which is easily
reusable for future work to design complex functions in oritatami systems. We
develop "low-level" tools that allow to easily spread apart the encoding of
different "functions" in the transcript, even if they are required to be
applied at the same geometrical location in the folding. We build upon these
low-level tools, a programming framework with increasing levels of abstraction,
from encoding of instructions into the transcript to logical analysis. This
framework is similar to the hardware-to-algorithm levels of abstractions in
standard algorithm theory. These various levels of abstractions allow to
separate the proof of correctness of the global behavior of our system, from
the proof of correctness of its implementation. Thanks to this framework, we
were able to computerize the proof of correctness of its implementation and
produce certificates, in the form of a relatively small number of proof trees,
compact and easily readable and checkable by human, while encapsulating huge
case enumerations. We believe this particular type of certificates can be
generalized to other discrete dynamical systems, where proofs involve large
case enumerations as well

### Freezing 1-Tag Systems with States

We study 1-tag systems with states obeying the freezing property that only
allows constant bounded number of rewrites of symbols. We look at examples of
languages accepted by such systems, the accepting power of the model, as well
as certain closure properties and decision problems. Finally we discuss a
restriction of the system where the working alphabet must match the input
alphabet.Comment: In Proceedings AFL 2023, arXiv:2309.0112

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