Single Length Cycle Research Articles

Identification of ECA rules forming MACA in periodic boundary condition

Cellular automaton (CA) is a computing model which is emerging rapidly. It is largely used in different types of scientific applications and simulations due to its ability to solve complex problems using simple rule(s). Cellular automata (CAs) are used in different types of applications like cryptography, VLSI systems, fault detection, etc. Typically, most of these applications utilize one-dimensional, 2-state, 3-neighborhood CAs. This paper explores the concept of Next State RMT Transition Diagram (NSRTD) for characterization of all the Elementary Cellular Automata (ECA) rules in periodic boundary condition leading to the identification of all ECA rules forming more than two fixed points (referred to as Single Length Cycle Multi-Attractor CA (MACA)) for an arbitrary CA length (n). For this, the 88 Wolfram classification rules and their equivalent rules have been utilized to reduce the search complexity by avoiding exhaustive searching on all the 256 ECA rules.

Characterization of Single Length Cycle Two-Attractor Cellular Automata Using Next-State Rule Minterm Transition Diagram

Cellular automata (CAs) are simple mathematical models that are effectively being used to analyze and understand the behavior of complex systems. Researchers from a wide range of fields are interested in CAs due to their potential for representing a variety of physical, natural and real-world phenomena. Three-neighborhood one-dimensional CAs, a special class of CAs, have been utilized to develop various applications in the field of very large-scale integration (VLSI) design, error-correcting codes, test pattern generation, cryptography and others. A thorough analysis of a three-neighborhood cellular automaton (CA) with two states per cell is presented in this paper. A graph-based tool called the next-state rule minterm transition diagram (NSRTD) is presented for analyzing the state transition behavior of CAs with fixed points. A linear time mechanism has been proposed for synthesizing a special class of irreversible CAs referred to as single length cycle two-attractor CAs (TACAs), having only two fixed points.

NSRT diagram for identification of SACA and TACA rules in null-boundary

This work reports characterization of one-dimensional null-boundary cellular automata (CA) rules in 3-neighborhood. The outcome of this characterization is the identification of CA rules that are the probable candidates for producing single length cycle attractor (fixed point). It targets synthesis of CA with only one fixed point (referred to as the single length cycle single attractor CA (SACA)) and the CA with only two fixed points (referred to as single length cycle two attractors CA (TACA)) for arbitrary length. A tool referred to as Next State RMT Transition Diagram (NSRTD) is developed to identify the fixed points as well as the cycles, involving multiple states (multi-length cycles) of a CA. The NSRTD provides the theoretical basis to search for the rules that form SACA and TACA, for all lengths, in null-boundary.

Synthesis of Scalable Single Length Cycle, Single Attractor Cellular Automata in Linear Time

This paper proposes the synthesis of single length cycle, single attractor cellular automata (SACAs) for arbitrary length. The n-cell single length cycle, single attractor cellular automaton (SACA), synthesized in linear time O(n), generates a pattern and finally settles to a point state called the single length cycle attractor state. An analytical framework is developed around the graph-based tool referred to as the next state transition diagram to explore the properties of SACA rules for three-neighborhood, one-dimensional cellular automata. This enables synthesis of an (n+1)-cell SACA from the available configuration of an n-cell SACA in constant time and an (n+m)-cell SACA from the available configuration of n-cell and m-cell SACAs also in constant time.

Distinguishing Phylogenetic Networks

Phylogenetic networks are becoming increasingly popular in phylogenetics since they have the ability to describe a wider range of evolutionary events than their tree counterparts. In this paper, we study Markov models on phylogenetic networks and their associated geometry. We restrict our attention to large-cycle networks, networks with a single undirected cycle of length at least four. Using tools from computational algebraic geometry, we show that the semidirected network topology is generically identifiable for Jukes--Cantor large-cycle network models.

A cellular automata based self-correcting protocol processor for scalable CMPs

This work reports an effective design of protocol processor (PP), the key component of coherence controller, in a Chip Multiprocessors (CMPs) cache system. It caters to the need for determining the state of a data block in processors' private caches with high accuracy. An insignificant defect in the PP can introduce major inconsistencies in computing the states of cached copies of a data block. A realizable PP, therefore, is a necessity for accurate computation of cache line states. The proposed design is the outcome of radical change in the design approach of PP. The modeling tool of cellular automata (CA) is considered for the design. The special class of cellular automata with single length cycle attractors (SLCA) is tuned to replicate the PP for computing cache line states in CMPs. Theory has been developed to empower the SLCA based PP to sense its malfunctioning and that leads to the design of a self-correcting PP.

A Cache System Design for CMPs with Built-In Coherence Verification

This work reports an effective design of cache system for Chip Multiprocessors (CMPs). It introduces built-in logic for verification of cache coherence in CMPs realizing directory based protocol. It is developed around the cellular automata (CA) machine, invented by John von Neumann in the 1950s. A special class of CA referred to as single length cycle 2-attractor cellular automata (TACA) has been planted to detect the inconsistencies in cache line states of processors’ private caches. The TACA module captures coherence status of the CMPs’ cache system and memorizes any inconsistent recording of the cache line states during the processors’ reference to a memory block. Theory has been developed to empower a TACA to analyse the cache state updates and then to settle to an attractor state indicating quick decision on a faulty recording of cache line status. The introduction of segmentation of the CMPs’ processor pool ensures a better efficiency, in determining the inconsistencies, by reducing the number of computation steps in the verification logic. The hardware requirement for the verification logic points to the fact that the overhead of proposed coherence verification module is much lesser than that of the conventional verification units and is insignificant with respect to the cost involved in CMPs’ cache system.

A cellular automata based highly accurate memory test hardware realizing March C−

This work reports a highly accurate test structure for high speed memories. The theoretical bases of the design are the March algorithm and cellular automata (CA) proposed by von Neumann in 1950s. Theory of 3 and 5-neighborhood CA, employed for the current application, has been developed to enhance the self-testability of memory test logic. The special class of single length cycle attractor cellular automata, introduced in this work, accepts status of each memory word and evaluates it to decide on the faults in the memory. The extension of CA neighborhood to 5 enables propagation of the effect of faults in memory or in the test logic to the error line of the test structure. This overcomes the inability of classical memory test hardware designed with the ex−or and or logic.

Permuting operations on strings and the distribution of their prime numbers

Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation X gives rise to a family {Xn}n≥2 of similar permutations. We call an integer nX-prime if Xn consists of a single cycle of length n (n≥2). For some instances of X–such as shuffle, twist, operations based on the Archimedes’ spiral and on the Josephus problem–we investigate the distribution of X-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures on the density of certain sets of prime numbers.

Permuting operations on strings and their relation to prime numbers

Some length-preserving operations on strings only permute the symbol positions in strings; such an operation X gives rise to a family { X n } n ≥ 2 of similar permutations. We investigate the structure and the order of the cyclic group generated by X n . We call an integer n X - prime if X n consists of a single cycle of length n ( n ≥ 2 ). Then we show some properties of these X -primes, particularly, how X -primes are related to X ′ -primes as well as to ordinary prime numbers. Here X and X ′ range over well-known examples (reversal, cyclic shift, shuffle, twist) and some new ones based on the Archimedes spiral and on the Josephus problem.

The spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups

AbstractA ternary quasigroup (or 3‐quasigroup) is a pair (N, q) where N is an n‐set and q(x, y, z) is a ternary operation on N with unique solvability. A 3‐quasigroup is called 2‐idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x)=y. A conjugation of a 3‐quasigroup, considered as an OA(3, 4, n), \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document}, is a permutation of the coordinate positions applied to the 4‐tuples of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\mathcal{B}}$\end{document}. The subgroup of conjugations under which \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document} is invariant is called the conjugate invariant subgroup of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document}. In this article, we determined the existence of 2‐idempotent 3‐quasigroups of order n, n≡7 or 11 (mod 12) and n≥11, with conjugate invariant subgroup consisting of a single cycle of length three. This result completely determined the spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triple system of order n exists if and only if n≡0, 1 (mod 3) and n≠6. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 292–304, 2010

Counting Matchings and Tree-Like Walks in Regular Graphs

The number of closed tree-like walks in a graph is closely related to the moments of the roots of the matching polynomial for the graph. Thus, by counting these walks up to a given length it is possible to find approximations for the matching polynomial. This approach has been used in two separate problems involving asymptotic enumerations of 1-factorizations of regular graphs. Nevertheless, a systematic way to count the required walks had not previously been found.In this paper we give an algorithm to count closed tree-like walks in a regular graph up to a given length. For smallm, this provides expressions for the number ofm-matchings in the graph in terms of the numbers of copies of certain small subgraphs that appear in the graph. The simplest of these expressions were already known, having been rediscovered by numerous authors usingad hocmethods. We offer the first general method for producing the expressions. We also find generating functions that isolate the contribution from the simplest kind of subgraph – namely a single cycle of arbitrary length.

Characterization of Single Cycle CA and its Application in Pattern Classification

The special class of irreversible cellular automaton (CA) with multiple attractors is of immense interest to the CA researchers. Characterization of such a CA is the necessity to devise CA based solutions for diverse applications. This work explores the essential properties of CA attractors towards characterization of the 1-dimensional cellular automata with point states (single length cycle attractors). The concept of Reachability Tree is introduced for such characterization. It enables identification of the pseudo-exhaustive bits (PE bits) of a CA defining its point states. A theoretical framework has been developed to devise schemes for synthesizing a single length cycle multiple attractor CA with the specific set of PE bits. It also results in a linear time solution while synthesizing a CA for the given set of attractors and its PE bits. The experimentation establishes that the proposed CA synthesis scheme is most effective in designing the efficient pattern classifiers for wide range of applications.

2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four

A ternary quasigroup (or 3-quasigroup) is a pair (N, q) where N is an n-set and q(x, y, z) is a ternary operation on N with unique solvability. A 3-quasigroup is called 2-idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x) = y. A conjugation of a 3-quasigroup, considered as an OA(3, 4, n), $${(N, \mathcal{B})}$$ , is a permutation of the coordinate positions applied to the 4-tuples of $${\mathcal{B}}$$ . The subgroup of conjugations under which $${(N, \mathcal{B})}$$ is invariant is called the conjugate invariant subgroup of $${(N, \mathcal{B})}$$ . In this paper, we will complete the existence proof of the last undetermined infinite class of 2-idempotent 3-quasigroups of order n, n ? 1 (mod 4) and n > 9, with a conjugate invariant subgroup consisting of a single cycle of length four.

The spectrum for quasigroups with cyclic automorphisms and additional symmetries

We determine necessary and sufficient conditions for the existence of a quasigroup of order n having an automorphism consisting of a single cycle of length m and n − m fixed points, and having any combination of the additional properties of being idempotent, unipotent, commutative, semi-symmetric or totally symmetric. Quasigroups with such additional properties and symmetries are equivalent to various classes of triple systems.

1‐Rotational Steiner triple systems over arbitrary groups

AbstractPhelps and Rosa introduced the concept of 1‐rotational Steiner triple system, that is an STS(ν) admitting an automorphism consisting of a fixed point and a single cycle of length ν − 1 [Discrete Math. 33 (1981), 57–66]. They proved that such an STS(ν) exists if and only if ν ≡ 3 or 9 (mod 24). Here, we speak of a 1‐rotational STS(ν) in a more general sense. An STS(ν) is 1‐rotational over a group G when it admits G as an automorphism group, fixing one point and acting regularly on the other points. Thus the STS(ν)'s by Phelps and Rosa are 1‐rotational over the cyclic group. We denote by 𝒜1r, 𝒞1r, 𝒬1r, 𝒢1r, the spectrum of values of ν for which there exists a 1‐rotational STS(ν) over an abelian, a cyclic, a dicyclic, and an arbitrary group, respectively. In this paper, we determine 𝒜1r and find partial answers about 𝒬1r and 𝒢1r. The smallest 1‐rotational STSs have orders 9, 19, 25 and are unique up to isomorphism. In particular, the only 1‐rotational STS(25) is over SL2(3), the special linear group of dimension 2 over Z3. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 215–226, 2001

1‐Rotational Steiner triple systems over arbitrary groups

Phelps and Rosa introduced the concept of 1-rotational Steiner triple system, that is an STS(ν) admitting an automorphism consisting of a fixed point and a single cycle of length ν − 1 [Discrete Math. 33 (12), 57–66]. They proved that such an STS(ν) exists if and only if ν ≡ 3 or 9 (mod 24). Here, we speak of a 1-rotational STS(ν) in a more general sense. An STS(ν) is 1-rotational over a group G when it admits G as an automorphism group, fixing one point and acting regularly on the other points. Thus the STS(ν)'s by Phelps and Rosa are 1-rotational over the cyclic group. We denote by 𝒜1r, 𝒞1r, 𝒬1r, 𝒢1r, the spectrum of values of ν for which there exists a 1-rotational STS(ν) over an abelian, a cyclic, a dicyclic, and an arbitrary group, respectively. In this paper, we determine 𝒜1r and find partial answers about 𝒬1r and 𝒢1r. The smallest 1-rotational STSs have orders 9, 19, 25 and are unique up to isomorphism. In particular, the only 1-rotational STS(25) is over SL2(3), the special linear group of dimension 2 over Z3. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 215–226, 2001

Cyclic and rotational hybrid triple systems

A hybrid triple system of order ν, denoted HTS(ν), is said to be cyclic if it admits an automorphism consisting of a single cycle of length ν. A HTS(ν) admitting an automorphism consisting of a fixed point and a cycle of length ν − 1 is said to be rotational. Necessary and sufficient conditions are given for the existence of a cyclic HTS(ν) and a rotational HTS(ν).

Cyclic and rotational hybrid triple systems

A hybrid triple system of order v, denoted HTS( v), is said to be cyclic if it admits an automorphism consisting of a single cycle of length v. A HTS( v) admitting an automorphism consisting of a fixed point and a cycle of length v − 1 is said to be rotational. Necessary and sufficient conditions are given for the existence of a cyclic HTS( v) and a rotational HTS( v).

Cyclic antiautomorphisms of directed triple systems

A transitive triple, (a,b,c), is defined to be the set {(a,b), (b,c), (a,c)} of ordered pairs. A directed triple system of order v, DTS(v), is a pair (D,β), where D is a set of v points and β is a collection of transitive triples of pairwise distinct points of D such that any ordered pair of distinct points of D is contained in precisely one transitive triple of β. An antiautomorphism of a Directed triple system, (D,β), is a permutation of D that maps β to β−1, where β −1 = {(c,b,a)|(a,b,c) E β}. In this article we give necessary and sufficient conditions for the existence of a Directed triple system of order v admitting an antiautomorphism consisting of a single cycle of length d and having v − d fixed points. Further, we give a more general result for partial Directed triple systems in which the missing ordered pairs are precisely those containing two fixed points. © 1996 John Wiley & Sons, Inc.