In this article, we study the impact of applying simple reduction rules to random syntactic formulas encoded as trees. We assume that there is an operator that has an absorbing pattern and prove that if we use this property to simplify a uniform random expression with n nodes, then the expected size of the result is bounded by a constant. The same holds for higher moments, establishing the lack of

A language over an alphabet B=A∪A¯ of opening (A) and closing (A¯) brackets, is balanced if it is a subset of the Dyck language 𝔻B over B, and it is well-formed if all words are prefixes of words in 𝔻B. We show that well-formedness of a context-free language is decidable in polynomial time, and that the longest common reduced suffix can be computed in polynomial time. With this at a hand we decide

After an apparent hiatus of roughly 30 years, we revisit a seemingly neglected subject in the theory of (one-dimensional) cellular automata: sublinear-time computation. The model considered is that of ACAs, which are language acceptors whose acceptance condition depends on the states of all cells in the automaton. We prove a time hierarchy theorem for sublinear-time ACA classes, analyze their intersection

We prove an inequality for the number of periods in a word x in terms of the length of x and its initial critical exponent. Next, we characterize all periods of the length-n prefix of a characteristic Sturmian word in terms of the lazy Ostrowski representation of n, and use this result to show that our inequality is tight for infinitely many words x. We propose two related measures of periodicity for

The diagnosability of a multiprocessor system or an interconnection network plays an important role in measuring the fault tolerance of the network. In 2016, Zhang et al. proposed a new measure for fault diagnosis of the system, namely, the r-extra diagnosability, which restrains that every fault-free component has at least (r+1) fault-free nodes. As a famous topology structure of interconnection networks

For a formal language L, the problem of language enumeration asks to compute the length-lexicographically smallest word in L larger than a given input w∈Σ∗ (henceforth called the L-successor of w). We investigate this problem for regular languages from a computational complexity and state complexity perspective. We first show that if L is recognized by a DFA with n states, then 2Θ(nlogn) states are

Splicing systems are generative mechanisms introduced by Tom Head in 1987 to model the biological process of DNA recombination. The computational engine of a splicing system is the “splicing operation”, a cut-and-paste binary string operation defined by a set of “splicing rules”, quadruples r=(u1,u2;u3,u4) where u1,u2,u3,u4 are words over an alphabet Σ. For two strings x1u1u2y1 and x2u3u4y2, applying

In this paper, we present a new interconnection topology, called the(n,k)-modified-bubble-sort graph, which is a generalization of the modified-bubble-sort graph. Additionally, we show many of its properties, such as its hierarchical structure, vertex transitivity, connectivity, edge-connectivity, super connectivity and super edge-connectivity. The (n,k)-modified-bubble-sort graph presents more flexibility

We introduce the affine OBDD model and show that zero-error affine OBDDs can be exponentially narrower than bounded-error unitary and probabilistic OBDDs on certain problems. Moreover, we show that Las-Vegas unitary and probabilistic OBDDs can be quadratically narrower than deterministic OBDDs. We also obtain the same results for the automata counterparts of these models.

A reconstruction problem of words from scattered factors asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely determine a word. We show that a word w∈{a,b}∗ can be reconstructed from the number of occurrences of at most min(|w|a,|w|b)+1 scattered factors of the form aib, where

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts Sβ and its relation to direct topological factorizations. We show that any reversible CA F:Sβ→Sβ has an almost equicontinuous direction whenever Sβ is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations

We examine different generalizations of checking stack automata by allowing multiple input heads and multiple stacks, and characterize their computing power in terms of two-way multi-head finite automata and space-bounded Turing machines. For various models, we obtain hierarchies in terms of their computing power. Our characterizations and hierarchies expand or tighten some previously known results

We study the NFA reductions by invariant equivalences and preorders. It is well-known that the NFA minimization problem is PSPACE-complete. Therefore, there have been many approaches to reduce the size of NFAs in low polynomial time by computing invariant equivalence or preorder relation and merging the states within same equivalence class. Here we consider the nondeterminism reduction of NFAs by invariant

Union-free expressions are regular expressions without using the union operation. Consequently, (nondeterministic) union-free languages are described by regular expressions using only concatenation and Kleene star. The language class is also characterised by a special class of finite automata: 1CFPAs have exactly one cycle-free accepting path from each of their states. Obviously such an automaton has

Oritatami is a computational model of RNA cotranscriptional folding, in which an RNA transcript is folding upon itself while being synthesized from its template DNA. This model is known to be Turing universal. Under the restriction on its parameters delay and arity both being 1, however, any deterministically foldable conformation is known to be at most ten times as large as its initial conformation

Computationally lightweight and unconditionally secure key agreement protocols are very useful for secure communication in public networks. Recently, Guan et al. proposed a key agreement protocol whose security is based on the unpredictability of channel noise rather than computationally hard problems. These protocols are efficient, computationally lightweight, and unconditionally secure. However,

The k-prefix-free, k-suffix-free and k-infix-free languages generalize the prefix-free, suffix-free and infix-free languages by allowing marginal errors. For example, a string x in a k-prefix-free language L can be a prefix of up to k different strings in L. We also define finitely prefix-free languages in which a string x can be a prefix of finitely many strings. We present efficient algorithms that

Let R=𝔽q+u𝔽q, where q is a power of a prime number p and u2=0. A triple cyclic code of length (r,s,t) over R is a set that can be partitioned into three parts that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as R[x]-submodules of R[x]/〈xr−1〉×R[x]/〈xs−1〉×R[x]/〈xt−1〉. In this paper, we study the generator polynomials and the minimum generating

Reliability of interconnection networks is important to design multiprocessor systems. The extra edge connectivity and component edge connectivity are two parameters for the reliability evaluation. The k-extra edge connectivity λk(G) is the cardinality of the minimum extra edge cut F such that G−F is not connected and each component of G−F has at least k vertices. The t-component edge connectivity

Let G be an undirected graph. An H-structure-cut (resp. H-substructure-cut) of G is a set of subgraphs of G, if any, whose deletion disconnects G, where the subgraphs deleted are isomorphic to a certain graph H (resp. where for any T′ of the subgraphs deleted, there is a subgraph T of G, isomorphic to H, such that T′ is a subgraph of T). G is superH|M-connected (resp. super sub-H|M-connected) if the

In this paper, we give a proof of Parikh’s semilinear theorem via Dickson’s lemma. It is notable that our proof provides a clear separation between properties derived from Dickson’s lemma and tree decomposition for context-free grammars.

The inclusion problem is one of the common problems in real-time systems. The general form of this problem is undecidable; however, the time-bounded verification of inclusion problem is decidable for timed automata. In this study, we propose a new discretization technique to verify the inclusion problem. The proposed technique is applied to a non-Zeno timed automaton with an upper bound that does not

We present a short proof of Cobham’s theorem without using Kronecker’s approximation theorem, making it suitable for generalization beyond automatic sequences.

The goal of an outdegree-constrained edge-modification problem is to find a spanning subgraph or supergraph H of an input undirected graph G such that either: (Type I) the number of edges in H is minimized or maximized and H can be oriented to satisfy some specified constraints on the vertices’ resulting outdegrees; or: (Type II) among all subgraphs or supergraphs of G that can be constructed by deleting

Linearizability is a commonly accepted consistency condition for concurrent objects. Filipović et al. show that linearizability is equivalent to observational refinement. However, linearizability does not permit concurrent objects to share memory spaces with their client programs. We show that linearizability (or observational refinement) can be broken even though a client program of an object accesses

Complementary prism graphs arise from the disjoint union of a graph G and its complement Ḡ by adding the edges of a perfect matching joining pairs of corresponding vertices of G and Ḡ. Classical graph problems such as Clique and Independent Set were proved to be NP-complete on such a class of graphs. In this work, we study the complexity of both problems on complementary prism graphs from the parameterized

In this paper, we study the fault-tolerant capability of hypercubes with respect to the hamiltonian property based on the concept of forbidden faulty sets. We show, with the assumption that each vertex is incident with at least three fault-free edges, that an n-dimensional hypercube contains a fault-free hamiltonian cycle, even if there are up to (4n−13) edge faults. Moreover, we give an example to

The Cramer-Shoup (CS) like cryptosystem based on index exchangeable family (IEF) construction is a novel scheme introduced in Asiaccs 2016 by Li et al. Its versatility was illustrated by building two public key encryption (PKE) schemes, a cramer-shoup encryption scheme based on IEFs, as well as an outsourcing technique based on non-abelian analog. However, the two schemes are not secure over the recommended

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is wwell-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted as WCW(G). Deciding whether an input graph G is well-covered is

In this article, we study some properties of random regular expressions of size n, when the cardinality of the alphabet also depends on n. For this, we revisit and improve the classical Transfer Theorem from the field of analytic combinatorics. This provides precise estimations for the number of regular expressions, the probability of recognizing the empty word and the expected number of Kleene stars

We consider the problem of partitioning effectively a given irreflexive (and possibly symmetric) rational relation R into two asymmetric rational relations. This problem is motivated by a recent method of embedding an R-independent language into one that is maximal R-independent, where the method requires to use an asymmetric partition of R. We solve the problem when R is length-separable, which means

For a regular language L, let Var(L) be the minimal number of nonterminals necessary to generate L by right linear grammars. Moreover, for natural numbers k1,k2,…,kn and an n-ary regularity preserving operation f, let the range gfVar(k1,k2,…,kn) be the set of all numbers k such that there are regular languages L1,L2,…,Ln with Var(Li)=ki for 1≤i≤n and Var(f(L1,L2,…,Ln))=k. We show that, for the circular

As one of the most fundamental networks for parallel and distributed computation, cycle is suitable for developing simple algorithms with low communication cost. A graph G is called k-fault-tolerant edge-pancyclic if after deleting any faulty set F of k vertices and/or edges from G, every correct edge in the resulting graph lies in a cycle of every length from g to |V(G−F)|, inclusively, where g is

We are interested in regular expressions and transducers that represent word relations in an alphabet-invariant way — for example, the set of all word pairs u,v where v is a prefix of u independently of what the alphabet is. Current software systems of formal language objects do not have a mechanism to define such objects. We define transducers in which transition labels involve what we call set specifications

It is well known that reversible finite automata do not accept all regular languages, that reversible pushdown automata do not accept all deterministic context-free languages, and that reversible queue automata are less powerful than deterministic real-time queue automata. It is of significant interest from both a practical and theoretical point of view to close these gaps. We here extend these reversible

The conversion of a given regular tree expression into a tree automaton has been widely studied. However, classical interpretations are based upon a top-down interpretation of tree automata. In this paper, we propose new constructions based on Gluskov’s one and on the one by Ilie and Yu using a bottom-up interpretation. One of the main goals of this technique is to consider as a next step the links

We define a new strict and computable hierarchy for the family of automaton semigroups, which reflects the various asymptotic behaviors of the state-activity growth. This hierarchy extends that given by Sidki for automaton groups, and also gives new insights into the latter. Its exponential part coincides with a notion of entropy for some associated automata. We prove that the Order Problem is decidable

In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on n−1 states, and the others as permutations. We prove that this

The state complexity of a regular language Lm is the number m of states in a minimal deterministic finite automaton (DFA) accepting Lm. The state complexity of a regularity-preserving binary operation on regular languages is defined as the maximal state complexity of the result of the operation where the two operands range over all languages of state complexities ≤m and ≤n, respectively. We determine

Non-self-embedding grammars are a restriction of context-free grammars which does not allow to describe recursive structures and, hence, which characterizes only the class of regular languages. A double exponential gap in size from non-self-embedding grammars to deterministic finite automata is known. The same size gap is also known from constant-height pushdown automata and 1-limited automata to deterministic

We examine the accepting state complexity, i.e., the minimal number of accepting states of deterministic finite automata for languages resulting from unary and binary operations on languages with accepting state complexity given as a parameter. This is a continuation of the work of [J. Dassow: On the number of accepting states of finite automata, J. Autom., Lang. Comb., 21, 2016]. We solve most of

Techniques are developed for creating new and general language families of only semilinear languages, and for showing families only contain semilinear languages. It is shown that for language families ℒ that are semilinear full trios, the smallest full AFL containing ℒ that is also closed under intersection with languages in NCM (where NCM is the family of languages accepted by NFAs augmented with

Optimal codebooks meeting the Welch bound with equality are desirable in many areas, such as direct spread code division multiple access communications, compressed sensing and so on. However, it is difficult to construct such optimal codebooks. There have been a number of attempts to construct codebooks nearly meeting the Welch bound. In this paper, we introduce a generic construction of codebooks

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial t-designs have attracted lots of research interest for decades. The interplay between coding theory and t-designs started many years ago. It is generally known that t-designs can be used to derive linear codes over any finite field, and that the supports of all codewords with a fixed weight

For a simple, undirected graph G, a function h:V(G)→{0,1,2} which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of G with weight h(V)=∑v∈Vh(v). (C1) For all q∈V with h(q)=0 there exists a vertex r such that qr∈E and h(r)=2, (C2) The induced subgraph with vertex set {p:h(p)≥1} has no isolated vertices and (C3) The induced subgraph with vertex

The interconnetion network plays an important role in a parallel system. To avoid the edge number of the interconnect network scaling rapidly with the increase of dimension and achieve a good balance of hardware costs and properties, this paper presents a new interconnection network called exchanged 3-ary n-cube (E3C). Compared with the 3-ary n-cube structures, E3C shows better performance in terms

The problem of the commutative equivalence of context-free and regular languages is studied. Conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated.

We consider the problem of fault-tolerant parallel search on an infinite line by n robots. Starting from the origin, the robots are required to find a target at an unknown location. The robots can move with maximum speed 1 and can communicate wirelessly among themselves. However, among the n robots, there are f robots that exhibit byzantine faults. A faulty robot can fail to report the target even

Evaluating the importance of nodes for complex networks is an important part of invulnerability research. In this paper, node importance analysis is conducted in wheel-related networks by a method of evaluating node importance by node contraction based on network agglomeration in communication networks. Both degrees and positions of nodes are considered with this method. This method was also proved

Large scale multiprocessor systems or multicomputer systems, taking interconnection networks as underlying topologies, have been widely used in the big data era. Fault tolerance is becoming an essential attribute in multiprocessor systems as the number of processors is getting larger. A connected graph G is called strong Menger (edge) connected if, for any two distinct vertices u and v, there are min{dG(u)

Watson–Crick jumping finite automata work on tapes which are double stranded sequences of symbols similar to that of Watson–Crick automata. The double stranded sequence is scanned in a discontinuous manner. That is, after reading a double stranded string, the automata can jump over some subsequence and continue scanning depending on the rule. Some variants of such automata are 1-limited, No state,

A double occurrence word (DOW) is a word in which every symbol appears exactly twice. We define the symbol separation of a DOW w to be the number of letters between the two copies of a symbol, and the separation of w to be the sum of separations over all symbols in w. We then analyze relationship among size, reducibility and separation of DOWs. Specifically, we provide tight bounds of separations of

Diagnosability is an important factor in multiple-processor systems defined as the maximum number of faulty nodes that a system can recognize. In this paper, we propose a new form of diagnosability called non-inclusive diagnosability that requires all faulty sets to be non-inclusive. Furthermore, we study the non-inclusive diagnosability of hypercubes under the MM* model for n≥1.

The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization. In this paper we study this problem in the CONGESTED CLIQUE model (CCM) [29] of distributed computing. For the Steiner tree problem in the CCM, we consider that each vertex of the input graph is uniquely mapped to a processor and edges are naturally mapped to the links between the corresponding

The identification of node importance in complex networks is of theoretical and practical significance for improving network robustness and invulnerability. In this paper, the importance of each node is evaluated and important nodes are identified in cycles and related networks by node contraction method based on network agglomeration. This novel method considers both the degree and the position of

We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number >25 is the sum of at most three natural numbers whose base-2 representation has an equal number of 0’s and 1’s.

We consider one of the weakest variants of cost register automata over a tropical semiring, namely copyless cost register automata over ℕ with updates using min and increments. We show that this model can simulate, in some sense, the runs of counter machines with zero-tests. We deduce that a number of problems pertaining to that model are undecidable, namely equivalence, upperboundedness, and semilinearity

We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric. We then consider three particular automata models (finite automata, transducers and automata weighted by integers here called sum-automata) and instantiate the generic logic for each of them. We

We consider an extension of tree automata on infinite trees that can use equality and disequality constraints between direct subtrees of a node. Recently, it has been shown that the emptiness problem for these kind of automata with a parity acceptance condition is decidable and that the corresponding class of languages is closed under Boolean operations. In this paper, we show that the class of languages

Motivated by recent results relating synchronizing DFAs and primitive sets, we tackle the synchronization process and the related longstanding Černý conjecture by studying the primitivity phenomenon for sets of nonnegative matrices having neither zero-rows nor zero-columns. We formulate the primitivity process in the setting of a two-player probabilistic game and we make use of convex optimization