arXiv subjects

Lila Kari

Publications and source records attributed to Lila Kari.

29 records · Page 2Linked to original sources

arXiv2011

On the regularity of iterated hairpin completion of a single word

Hairpin completion is an abstract operation modeling a DNA bio-operation which receives as input a DNA strand $w = xαy \calpha$, and outputs $w' = x αy \barα \bar{x}$, 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 $α$ 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 $α$ and $\calpha$, we prove that the iterated hairpin completion of $w$ is regular. For other classes with higher numbers of occurrences of $α$ and $\calpha$, we prove a necessary and sufficient condition for the iterated hairpin completion of a word in these classes to be regular.

cs.FL↗

arXiv2011

Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings

This paper provides a bridge between the classical tiling theory and the complex neighborhood self-assembling situations that exist in practice. The neighborhood of a position in the plane is the set of coordinates which are considered adjacent to it. This includes classical neighborhoods of size four, as well as arbitrarily complex neighborhoods. A generalized tile system consists of a set of tiles, a neighborhood, and a relation which dictates which are the "admissible" neighboring tiles of a given tile. Thus, in correctly formed assemblies, tiles are assigned positions of the plane in accordance to this relation. We prove that any validly tiled path defined in a given but arbitrary neighborhood (a zipper) can be simulated by a simple "ribbon" of microtiles. A ribbon is a special kind of polyomino, consisting of a non-self-crossing sequence of tiles on the plane, in which successive tiles stick along their adjacent edge. Finally, we extend this construction to the case of traditional tilings, proving that we can simulate arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving some of their essential properties.

cs.CC↗

arXiv2010

The Power of Nondeterminism in Self-Assembly

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 Espanés, 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.

cs.CC↗

arXiv2010

Ciliate Gene Unscrambling with Fewer Templates

One of the theoretical models proposed for the mechanism of gene unscrambling in some species of ciliates is the template-guided recombination (TGR) system by Prescott, Ehrenfeucht and Rozenberg which has been generalized by Daley and McQuillan from a formal language theory perspective. In this paper, we propose a refinement of this model that generates regular languages using the iterated TGR system with a finite initial language and a finite set of templates, using fewer templates and a smaller alphabet compared to that of the Daley-McQuillan model. To achieve Turing completeness using only finite components, i.e., a finite initial language and a finite set of templates, we also propose an extension of the contextual template-guided recombination system (CTGR system) by Daley and McQuillan, by adding an extra control called permitting contexts on the usage of templates.

cs.FL↗

arXiv2010

State Complexity of Catenation Combined with Star and Reversal

This paper is a continuation of our research work on state complexity of combined operations. Motivated by applications, we study the state complexities of two particular combined operations: catenation combined with star and catenation combined with reversal. We show that the state complexities of both of these combined operations are considerably less than the compositions of the state complexities of their individual participating operations.

cs.FL↗

arXiv2010

State Complexity of Two Combined Operations: Reversal-Catenation and Star-Catenation

In this paper, we show that, due to the structural properties of the resulting automaton obtained from a prior operation, the state complexity of a combined operation may not be equal but close to the mathematical composition of the state complexities of its component operations. In particular, we provide two witness combined operations: reversal combined with catenation and star combined with catenation.

cs.FL↗

arXiv2010

Scalable, Time-Responsive, Digital, Energy-Efficient Molecular Circuits using DNA Strand Displacement

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.

cs.OH↗

arXiv2010

Triangular Self-Assembly

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).

cs.DM↗

arXiv2010

Properties of Pseudo-Primitive Words and their Applications

A pseudo-primitive word with respect to an antimorphic involution θis a word which cannot be written as a catenation of occurrences of a strictly shorter word t and θ(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-Schützenberger equation u_1 u_2 ... u_l = v_1 ... v_n w_1 ... w_m, where u_1, ..., u_l \in {u, θ(u)}, v_1, ..., v_n \in {v, θ(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 θ. 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 θ(t). Moreover, several cases of this equation where l = 3 are examined.

cs.CC↗

arXiv2010

Negative Interactions in Irreversible Self-Assembly

This paper explores the use of negative (i.e., repulsive) interaction the abstract Tile Assembly Model defined by Winfree. Winfree postulated negative interactions to be physically plausible in his Ph.D. thesis, and Reif, Sahu, and Yin explored their power in the context of reversible attachment operations. We explore the power of negative interactions with irreversible attachments, and we achieve two main results. Our first result is an impossibility theorem: after t steps of assembly, Omega(t) tiles will be forever bound to an assembly, unable to detach. Thus negative glue strengths do not afford unlimited power to reuse tiles. Our second result is a positive one: we construct a set of tiles that can simulate a Turing machine with space bound s and time bound t, while ensuring that no intermediate assembly grows larger than O(s), rather than O(s * t) as required by the standard Turing machine simulation with tiles.

cs.DS↗

arXiv2009

Pseudo-Power Avoidance

Repetition avoidance has been studied since Thue's work. In this paper, we considered another type of repetition, which is called pseudo-power. This concept is inspired by Watson-Crick complementarity in DNA sequence and is defined over an antimorphic involution $ϕ$. We first classify the alphabet $Σ$ and the antimorphic involution $ϕ$, under which there exists sufficiently long pseudo-$k$th-power-free words. Then we present algorithms to test whether a finite word $w$ is pseudo-$k$th-power-free.

cs.FL↗