arXiv subjects

Shinnosuke Seki

Publications and source records attributed to Shinnosuke Seki.

At least 19 recordsLinked to original sources

arXiv2026

A Congestion Parameter for Depth-First Graph Traversals

We explore a new graph parameter, the KLX number, which quantifies the minimum edge congestion of depth-first search (DFS) traversals of a given graph. Originally motivated by a problem in RNA nanostructure design, this parameter is also of independent theoretical interest. Informally, the KLX number of a graph is defined as the minimum, over all its DFS traversals, of the maximum number of back edges that are simultaneously open during the traversal. We provide full characterisations and linear-time recognition algorithms for graphs with KLX numbers 0, 1 and 2. We also relate KLX to tree-width, proving that any graph satisfies $\mathrm{TW} \le \mathrm{KLX}+1$. Furthermore, we show that the property $\mathrm{KLX} \le k$ is $\mathrm{MSO}_2$-expressible for every fixed $k$. Combined with the tree-width bound, this result implies that determining whether a graph has KLX number at most $k$ can be achieved in linear time for any constant $k$.

cs.DM↗

arXiv2026

Bandwidth of Nondeterministic Finite Automata

Co-transcriptional splicing generates RNA sequences from a DNA template by deleting subsequences nondeterministically. Recent work showed how to encode an NFA into such a template, but the construction requires deleting subsequences whose length grows with the distance between states, which makes such deletions unlikely under the local nature of co-transcriptional splicing. We introduce $k$-bandwidth NFAs, in which transitions span at most $k$ states. These automata form a strict hierarchy of language classes. For finite languages, bandwidth $2$ suffices, and bandwidth $1$ can be decided in polynomial-time when the language is presented as a list of words. Minimizing the bandwidth is NP-hard even for fixed $k \geq 2$.

cs.FL↗

arXiv2025

Programmable Co-Transcriptional Splicing: Realizing Regular Languages via Hairpin Deletion

RNA co-transcriptionality, where RNA is spliced or folded during transcription from DNA templates, offers promising potential for molecular programming. It enables programmable folding of nano-scale RNA structures and has recently been shown to be Turing universal. While post-transcriptional splicing is well studied, co-transcriptional splicing is gaining attention for its efficiency, though its unpredictability still remains a challenge. In this paper, we focus on engineering co-transcriptional splicing, not only as a natural phenomenon but as a programmable mechanism for generating specific RNA target sequences from DNA templates. The problem we address is whether we can encode a set of RNA sequences for a given system onto a DNA template word, ensuring that all the sequences are generated through co-transcriptional splicing. Given that finding the optimal encoding has been shown to be NP-complete under the various energy models considered, we propose a practical alternative approach under the logarithmic energy model. More specifically, we provide a construction that encodes an arbitrary nondeterministic finite automaton (NFA) into a circular DNA template from which co-transcriptional splicing produces all sequences accepted by the NFA. As all finite languages can be efficiently encoded as NFA, this framework solves the problem of finding small DNA templates for arbitrary target sets of RNA sequences. The quest to obtain the smallest possible such templates naturally leads us to consider the problem of minimizing NFA and certain practically motivated variants of it, but as we show, those minimization problems are computationally intractable.

cs.FL↗

arXiv2025

A Formalization of Co-Transcriptional Splicing as an Operation on Formal Languages

RNA co-transcriptionality is the process where RNA sequences are spliced while being transcribed from DNA templates. This process holds potential as a key tool for molecular programming. Co-transcriptional folding has been shown to be programmable for assembling nano-scale RNA structures, and recent advances have proven its Turing universality. While post-transcriptional splicing has been extensively studied, co-transcriptional splicing is gaining attention for its potential to save resources and space in molecular systems. However, its unpredictability has limited its practical applications. In this paper, we focus on engineering co-transcriptional splicing, moving beyond natural occurrences to program RNA sequences that produce specific target sequences through DNA templates. We introduce contextual lariat deletion operations under three energy models - linear loop penalty, logarithmic loop penalty, and constantly bounded loop length - as well as bracketed contextual deletion, where deletion occurs solely based on context matching, without any structural constraints from hairpin loops. We examine the complexity of the template constructability problem associated with these operations and study the closure properties of the languages they generate, providing insights for RNA template design in molecular programming systems.

cs.FL↗

arXiv2023

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.

cs.FL↗

arXiv2019

A general architecture of oritatami systems for simulating arbitrary finite automata

In this paper, we propose an architecture of oritatami systems with which one can simulate an arbitrary nondeterministic finite automaton (NFA) in a unified manner. The oritatami system is known to be Turing-universal but the simulation available so far requires 542 bead types and $O(t^4 \log^2 t)$ steps in order to simulate $t$ steps of a Turing machine. The architecture we propose employs only 329 bead types and requires just $O(t |Q|^4 |Σ|^2)$ steps to simulate an NFA over an input alphabet $Σ$ with a state set $Q$ working on a word of length $t$.

cs.DM↗

arXiv2018

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.

cs.CG↗

arXiv2018

Know When to Fold 'Em: Self-Assembly of Shapes by Folding in Oritatami

An oritatami system (OS) is a theoretical model of self-assembly via co-transcriptional folding. It consists of a growing chain of beads which can form bonds with each other as they are transcribed. During the transcription process, the $δ$ most recently produced beads dynamically fold so as to maximize the number of bonds formed, self-assemblying into a shape incrementally. The parameter $δ$ is called the delay and is related to the transcription rate in nature. This article initiates the study of shape self-assembly using oritatami. A shape is a connected set of points in the triangular lattice. We first show that oritatami systems differ fundamentally from tile-assembly systems by exhibiting a family of infinite shapes that can be tile-assembled but cannot be folded by any OS. As it is NP-hard in general to determine whether there is an OS that folds into (self-assembles) a given finite shape, we explore the folding of upscaled versions of finite shapes. We show that any shape can be folded from a constant size seed, at any scale n >= 3, by an OS with delay 1. We also show that any shape can be folded at the smaller scale 2 by an OS with unbounded delay. This leads us to investigate the influence of delay and to prove that, for all δ > 2, there are shapes that can be folded (at scale 1) with delay δ but not with delay δ'<δ. These results serve as a foundation for the study of shape-building in this new model of self-assembly, and have the potential to provide better understanding of cotranscriptional folding in biology, as well as improved abilities of experimentalists to design artificial systems that self-assemble via this complex dynamical process.

cs.DS↗

arXiv2017

The Complexity of Fixed-Height Patterned Tile Self-Assembly

We characterize the complexity of the PATS problem for patterns of fixed height and color count in variants of the model where seed glues are either chosen or fixed and identical (so-called non-uniform and uniform variants). We prove that both variants are NP-complete for patterns of height 2 or more and admit O(n)-time algorithms for patterns of height 1. We also prove that if the height and number of colors in the pattern is fixed, the non-uniform variant admits a O(n)-time algorithm while the uniform variant remains NP-complete. The NP-completeness results use a new reduction from a constrained version of a problem on finite state transducers.

cs.FL↗

arXiv2014

A manually-checkable proof for the NP-hardness of 11-color pattern self-assembly tile set synthesis

Patterned self-assembly tile set synthesis (PATS) aims at finding a minimum tile set to uniquely self-assemble a given rectangular (color) pattern. For k >= 1, k-PATS is a variant of PATS that restricts input patterns to those with at most $k$ colors. A computer-assisted proof has been recently proposed for 2-PATS by Kari et al. [arXiv:1404.0967 (2014)]. In contrast, the best known manually-checkable proof is for the NP-hardness of 29-PATS by Johnsen, Kao, and Seki [ISAAC 2013, LNCS 8283, pp.~699-710]. We propose a manually-checkable proof for the NP-hardness of 11-PATS.

cs.DM↗

arXiv2014

Computing Minimum Tile Sets to Self-Assemble Colors Patterns

Patterned self-assembly tile set synthesis (PATS) aims at finding a minimum tile set to uniquely self-assemble a given rectangular color pattern. For $k \ge 1$, $k$-PATS is a variant of PATS that restricts input patterns to those with at most $k$ colors. We prove the {\bf NP}-hardness of 29-PATS, where the best known is that of 60-PATS.

cs.CC↗

arXiv2014

Binary pattern tile set synthesis is NP-hard

In the field of algorithmic self-assembly, a long-standing unproven conjecture has been that of the NP-hardness of binary pattern tile set synthesis (2-PATS). The $k$-PATS problem is that of designing a tile assembly system with the smallest number of tile types which will self-assemble an input pattern of $k$ colors. Of both theoretical and practical significance, $k$-PATS has been studied in a series of papers which have shown $k$-PATS to be NP-hard for $k = 60$, $k = 29$, and then $k = 11$. In this paper, we close the fundamental conjecture that 2-PATS is NP-hard, concluding this line of study. While most of our proof relies on standard mathematical proof techniques, one crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof of the four color theorem by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and Thomas) or the recent important advance on the Erdős discrepancy problem by Konev and Lisitsa using computer programs. We utilize a massively parallel algorithm and thus turn an otherwise intractable portion of our proof into a program which requires approximately a year of computation time, bringing the use of computer-assisted proofs to a new scale. We fully detail the algorithm employed by our code, and make the code freely available online.

cs.CC↗

arXiv2013

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.

cond-mat.stat-mech↗

arXiv2013

3-color Bounded Patterned Self-assembly

Patterned self-assembly tile set synthesis PATS is the problem of finding a minimal tile set which uniquely self-assembles into a given pattern. Czeizler and Popa proved the NP-completeness of PATS and Seki showed that the PATS problem is already NP-complete for patterns with 60 colors. In search for the minimal number of colors such that PATS remains NP-complete, we introduce multiple bound PATS (mbPATS) where we allow bounds for the numbers of tile types of each color. We show that mbPATS is NP-complete for patterns with just three colors and, as a byproduct of this result, we also obtain a novel proof for the NP-completeness of PATS which is more concise than the previous proofs.

cs.CC↗

arXiv2013

Combinatorial Optimization in Pattern Assembly

Pattern self-assembly tile set synthesis (PATS) is a combinatorial optimization problem which aim at minimizing a rectilinear tile assembly system (RTAS) that uniquely self-assembles a given rectangular pattern, and is known to be NP-hard. PATS gets practically meaningful when it is parameterized by a constant c such that any given pattern is guaranteed to contain at most c colors (c-PATS). We first investigate simple patterns and properties of minimum RTASs for them. Then based on them, we design a 59-colored pattern to which 3SAT is reduced, and prove that 59-PATS is NP-hard.

cs.CC↗

arXiv2012

Converting Nondeterministic Automata and Context-Free Grammars into Parikh Equivalent One-Way and Two-Way Deterministic Automata

We investigate the conversion of one-way nondeterministic finite automata and context-free grammars into Parikh equivalent one-way and two-way deterministic finite automata, from a descriptional complexity point of view. We prove that for each one-way nondeterministic automaton with $n$ states there exist Parikh equivalent one-way and two-way deterministic automata with $e^{O(\sqrt{n \ln n})}$ and $p(n)$ states, respectively, where $p(n)$ is a polynomial. Furthermore, these costs are tight. In contrast, if all the words accepted by the given automaton contain at least two different letters, then a Parikh equivalent one-way deterministic automaton with a polynomial number of states can be found. Concerning context-free grammars, we prove that for each grammar in Chomsky normal form with h variables there exist Parikh equivalent one-way and two-way deterministic automata with $2^{O(h^2)}$ and $2^{O(h)}$ states, respectively. Even these bounds are tight.

cs.FL↗

arXiv2012

On the behavior of tile assembly system at high temperatures

Behaviors of Winfree's tile assembly systems (TASs) at high temperatures are investigated in combination with integer programming of a specific form called threshold programming. First, we propose a way to build bridges from the Boolean satisfiability problem (SAT) to threshold programming, and further to TAS's behavior, in order to prove the NP-hardness of optimizing temperatures of TASs that behave in a way given as input. These bridges will take us further to two important results on the behavior of TASs at high temperatures. The first says that arbitrarily high temperatures are required to assemble some shape by a TAS of "reasonable" size. The second is that for any temperature at least 4 given as a parameter, it is NP-hard to find the minimum size TAS that self-assembles a given shape and works at the given temperature or below.

cs.CC↗

arXiv2011

Iterated Hairpin Completions of Non-crossing Words

Iterated hairpin completion is an operation on formal languages that is inspired by the hairpin formation in DNA biochemistry. Iterated hairpin completion of a word (or more precisely a singleton language) is always a context-sensitive language and for some words it is known to be non-context-free. However, it is unknown whether regularity of iterated hairpin completion of a given word is decidable. Also the question whether iterated hairpin completion of a word can be context-free but not regular was asked in literature. In this paper we investigate iterated hairpin completions of non-crossing words and, within this setting, we are able to answer both questions. For non-crossing words we prove that the regularity of iterated hairpin completions is decidable and that if iterated hairpin completion of a non-crossing word is not regular, then it is not context-free either.

cs.FL↗