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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:2309.01126

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 $O(t |Q|^4 |\Sigma|^2)$ steps to simulate an NFA over an input alphabet $\Sigma$ with a state set $Q$ working on a word of length $t$.

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 $\delta$ most recently produced beads dynamically fold so as to maximize the number of bonds formed, self-assemblying into a shape incrementally. The parameter $\delta$ 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 {\delta} > 2, there are shapes that can be folded (at scale 1) with delay {\delta} but not with delay {\delta}'<{\delta}. 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.

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., Comment: An abstract version appears in the proceedings of CIAA 2016

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.

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.

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.

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 Erd\H{o}s 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.

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 figures

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.

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.

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., Comment: 30 pages, 2 figure. A preliminary version has been presented at DLT 2012, LNCS 7410, pp. 284-295. Version 2: an example has been added in Section 3

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., Comment: This paper is an extended version of the following paper: S. Seki and Y. Okuno. On the behavior of tile assembly system at high temperatures. In CiE 2012: How the World Computes - Turing Centenary Conference and 8th Conference on Computability in Europe, LNCS 7318, pages 549-559, Springer, 2012

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.

Mateescu, Salomaa, and Yu asked: is it decidable whether a given subword history assumes only non-negative values for all words over a given alphabet. In this paper, we solve this open problem by proving that this problem is undecidable even under stronger conditions than supposed originally.

Hairpin completion is an abstract operation modeling a DNA bio-operation which receives as input a DNA strand $w = x\alpha y \calpha$, and outputs $w' = x \alpha y \bar{\alpha} \bar{x}$, where $\bar{x}$ denotes the Watson-Crick complement of $x$. In this paper, we focus on the problem of finding conditions under which the iterated hairpin completion of a given word is regular. According to the numbers of words $\alpha$ and $\calpha$ that initiate hairpin completion and how they are scattered, we classify the set of all words $w$. For some basic classes of words $w$ containing small numbers of occurrences of $\alpha$ and $\calpha$, we prove that the iterated hairpin completion of $w$ is regular. For other classes with higher numbers of occurrences of $\alpha$ and $\calpha$, we prove a necessary and sufficient condition for the iterated hairpin completion of a word in these classes to be regular., Comment: 17 pages, 1 figure, submitted to Fundamenta Informaticae

Winfree's abstract Tile Assembly Model (aTAM) is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing "seed" assembly based on specific binding sites on their four sides. We show that there is a polynomial-time algorithm that, given an n x n square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self-assembles the square, answering an open question of Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanes, and Rothemund ("Combinatorial Optimization Problems in Self-Assembly", STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its "temperature" (the binding strength threshold required for a tile to attach)., Comment: The previous version contained more sections, but we have split that paper into two. The other half will be posted as a separate paper

We investigate the role of nondeterminism in Winfree's abstract Tile Assembly Model (aTAM), which was conceived to model artificial molecular self-assembling systems constructed from DNA. Of particular practical importance is to find tile systems that minimize resources such as the number of distinct tile types, each of which corresponds to a set of DNA strands that must be custom-synthesized in actual molecular implementations of the aTAM. We seek to identify to what extent the use of nondeterminism in tile systems affects the resources required by such molecular shape-building algorithms. We first show a "molecular computability theoretic" result: there is an infinite shape S that is uniquely assembled by a tile system but not by any deterministic tile system. We then show an analogous phenomenon in the finitary "molecular complexity theoretic" case: there is a finite shape S that is uniquely assembled by a tile system with c tile types, but every deterministic tile system that uniquely assembles S has more than c tile types. In fact we extend the technique to derive a stronger (classical complexity theoretic) result, showing that the problem of finding the minimum number of tile types that uniquely assemble a given finite shape is Sigma-P-2-complete. In contrast, the problem of finding the minimum number of deterministic tile types that uniquely assemble a shape was shown to be NP-complete by Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espan\'es, and Rothemund (Combinatorial Optimization Problems in Self-Assembly, STOC 2002). The conclusion is that nondeterminism confers extra power to assemble a shape from a small tile system, but unless the polynomial hierarchy collapses, it is computationally more difficult to exploit this power by finding the size of the smallest tile system, compared to finding the size of the smallest deterministic tile system., Comment: Accepted to SODA 2011. The previous version of this paper (which appears in the SODA proceedings) had open questions about computing the minimum number of tile types to weakly self-assemble a set. The answer to these questions is "no", by a very simple imitation of the proof that Kolmogorov complexity is uncomputable based on the Berry paradox. These open questions have been removed

We propose a novel theoretical biomolecular design to implement any Boolean circuit using the mechanism of DNA strand displacement. The design is scalable: all species of DNA strands can in principle be mixed and prepared in a single test tube, rather than requiring separate purification of each species, which is a barrier to large-scale synthesis. The design is time-responsive: the concentration of output species changes in response to the concentration of input species, so that time-varying inputs may be continuously processed. The design is digital: Boolean values of wires in the circuit are represented as high or low concentrations of certain species, and we show how to construct a single-input, single-output signal restoration gate that amplifies the difference between high and low, which can be distributed to each wire in the circuit to overcome signal degradation. This means we can achieve a digital abstraction of the analog values of concentrations. Finally, the design is energy-efficient: if input species are specified ideally (meaning absolutely 0 concentration of unwanted species), then output species converge to their ideal concentrations at steady-state, and the system at steady-state is in (dynamic) equilibrium, meaning that no energy is consumed by irreversible reactions until the input again changes. Drawbacks of our design include the following. If input is provided non-ideally (small positive concentration of unwanted species), then energy must be continually expended to maintain correct output concentrations even at steady-state. In addition, our fuel species - those species that are permanently consumed in irreversible reactions - are not "generic"; each gate in the circuit is powered by its own specific type of fuel species. Hence different circuits must be powered by different types of fuel. Finally, we require input to be given according to the dual-rail convention, so that an input of 0 is specified not only by the absence of a certain species, but by the presence of another. That is, we do not construct a "true NOT gate" that sets its output to high concentration if and only if its input's concentration is low. It remains an open problem to design scalable, time-responsive, digital, energy-efficient molecular circuits that additionally solve one of these problems, or to prove that some subset of their resolutions are mutually incompatible., Comment: version 2: the paper itself is unchanged from version 1, but the arXiv software stripped some asterisk characters out of the abstract whose purpose was to highlight words. These characters have been replaced with underscores in version 2. The arXiv software also removed the second paragraph of the abstract, which has been (attempted to be) re-inserted. Also, although the secondary subject is "Soft Condensed Matter", this classification was chosen by the arXiv moderators after submission, not chosen by the authors. The authors consider this submission to be a theoretical computer science paper.

We discuss the self-assembly system of triangular tiles instead of square tiles, in particular right triangular tiles and equilateral triangular tiles. We show that the triangular tile assembly system, either deterministic or non-deterministic, has the same power to the square tile assembly system in computation, which is Turing universal. By providing counter-examples, we show that the triangular tile assembly system and the square tile assembly system are not comparable in general. More precisely, there exists square tile assembly system S such that no triangular tile assembly system is a division of S and produces the same shape; there exists triangular tile assembly system T such that no square tile assembly system produces the same compatible shape with border glues. We also discuss the assembly of triangles by triangular tiles and obtain results similar to the assembly of squares, that is to assemble a triangular of size O(N^2), the minimal number of tiles required is in O(log N/log log N).

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 Science