Linear Automata Research Articles

Синтез обнаруживающих тестов для нечетких автоматов с конечной памятью

The problem of test synthesis for fuzzy automata with finite memory is considered. The paper is based on the results obtained by the author earlier. They study linear automata (LA), a subclass of which are automata with finite memory (AFM). Both types of automata are used as models in diagnostics of discrete devices and in other applications. The advantages of LA are simplicity of realization and availability of developed mathematical apparatus for their study. AFMs allow describing the dependencies between inputs and outputs in the past and present moments of time without using states of the automata. This property is used in the development of finite-automation cryptosystems, in the generation of exhaustive tests for discrete devices, in the formulation of fuzzy rules of logical inference and others. The paper proposes a method for synthesis of detecting tests for fuzzy AFMs, based on reducing it to the case of its analog for crisp AFMs. Fuzzy AFM (as a subclass of LA) can be described using transition and output functions represented as equations with characteristic matrices. Fuzzy AFM is realized using matrices containing some fuzzy elements. These elements obtain values randomly from alternative sets given for them. The other (crisp) elements of the matrices are fixed values from the field GF(p), over which the AFM is defined. After all possible substitutions in the matrices of fuzzy AFM of all fuzzy elements (from alternative sets) we obtain a set of crisp automata describing the "behavior" of the considered fuzzy automaton under all possible "scenarios" of substitutions. Each faulty AFM is obtained by changing the values of the elements of its characteristic matrices. As a set of admissible faults, all possible changes in these matrices obtained by substitutions and insertion of fault introduction are considered. Now we can use the method of test synthesis proposed earlier by the author of the submitted paper, organizing the procedure of sequential analysis and comparison of "behavior" of crisp pairs (faultless AFM, faulty AFM). The AFMs obtained after substitutions should be used sequentially as faultless AFMs and as faulty AFMs after faults have been introduced. Conclusions about the faultless AFM or the presence of a fault in it are made on the basis of analyzing the totality of the results obtained. A generalization of the synthesis method proposed earlier by the author of the submitted paper the is described for µ-defined (and synchronizable) AFMs. The method is based on obtaining tests as solutions of systems of linear equations, for which there is a well-developed mathematical apparatus in algebra.

Decomposition of polynomials using the shift-composition operation

Abstract V. I. Solodovnikov had employed the shift-composition operation to investigate homomorphisms of shift registers into linear automata; in his papers, conditions for the absence of nontrivial inner homomorphisms of shift registers were derived. An essential role was played by the condition of linearity of the left component of the shift-composition operation in the corresponding polynomial decomposition. In this paper we consider the case where the left component is a function belonging to a wider class, which includes the class of linear functions.

Linear weighted tree automata with storage and inverse linear tree homomorphisms

We introduce linear weighted tree automata with storage and show that this model generalizes linear pushdown tree automata. We prove that the class of weighted tree languages recognizable by our linear automaton model is closed under inverse linear tree homomorphisms.

The simplification of computationals in error correction coding

The object of research is the processes of error correction transformation of information in automated systems. The research is aimed at reducing the complexity of decoding cyclic codes by combining modern mathematical models and practical tools. The main prerequisite for the complication of computations in deterministic linear error-correcting codes is the use of the algebraic representation as the main mathematical apparatus for these types of codes. Despite the universalism of the algebraic approach, its main drawback is the impossibility of taking into account the characteristic features of all subclasses of linear codes. In particular, the cyclic property is not taken into account at all for cyclic codes. Taking this property into account, one can go to a fundamentally different mathematical representation of cyclic codes – the theory of linear automata in Galois fields (linear finite-state machine). For the automaton representation of cyclic codes, it is proved that the problem of syndromic decoding of these codes in the general case is an NP-complete problem. However, if to use the proposed hierarchical approach to problems of complexity, then on its basis it is possible to carry out a more accurate analysis of the growth of computational complexity. Correction of single errors during one time interval (one iteration) of decoding has a linear decoding complexity on the length of the codeword, and error correction during m iterations of permutations of codeword bits has a polynomial complexity. According to three subclasses of cyclic codes, depending on the complexity of their decoding: easy decoding (linear complexity), iteratively decoded (polynomial complexity), complicate decoding (exponential complexity). Practical ways to reduce the complexity of computations are considered: alternate use of probabilistic and deterministic linear codes, simplification of software and hardware implementation by increasing the decoding time, use of interleaving. A method of interleaving is proposed, which makes it possible to simultaneously generate the burst errors and replace them with single errors. The mathematical apparatus of linear automata allows solving together the indicated problems of error correction coding.

Conditions of A-completeness for linear automata over dyadic rationals

Abstract We consider the problem of A-completeness in the class of linear automata such that the sets of inputs, outputs and states are Cartesian products of dyadic rationals; systems checked for completeness are comprised of a variable finite set and a fixed additional set. We obtain conditions of A-completeness in terms of maximal subclasses in the cases when the additional set is the set of all unary automata and when the additional set consists of the adder.

ВАРИАНТ АЛГОРИТМА ГЕНЕРАЦИИ ПСЕВДОСЛУЧАЙНЫХ ДВОИЧНЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ, ОСНОВАННЫЙ НА СВОЙСТВАХ ЛИНЕЙНЫХ КЛЕТОЧНЫХ АВТОМАТОВ

The article is devoted to the development of a linear system for generating a pseudo-random sequence based on cellular automata, the development of a model for several generators of nonlinear pseudo-random sequences with practical applications in symmetric data transformation systems. Such a model generates all solutions to linear binary difference equations. It is important to note that many of these solutions are pseudo-random keystream sequences. In the process of developing a linear system based on cellular automata, two main structures are considered: linear difference equations and one-dimensional linear hybrid cellular automata. This article shows that all solutions to linear binary difference equations can be implemented using linear models based on cellular automata using the "rule 90" and "rule 150". A model for generating pseudo-random bit sequences has been developed, which is most applicable in communication systems with a high transmission rate. The main difference of this model is that it is built on an exclusively sequential concatenation of a basic linear automaton, which determines the simplicity of the proposed model. Also, an algorithm for controlling the integrity and authenticity of block data was proposed based on the proposed algorithm for generating a pseudo-random sequence. The practical significance lies in the fact that the proposed model is simple and can be applied in information security systems in practice, including in authentication and data integrity control systems.

Maximum subclasses in classes of linear automata over finite fields

AbstractThe sets of all maximum subclasses in classes of linear automata over finite fields are found. An algorithm that decides completeness of finite subsets based on maximum subclasses is formulated.

Тестирование систем с нечеткими дискретными компонентами.

Modern complex systems are build based on heterogeneous components with various interrela tionships, fuzziness, and uncertainty of the laws of functioning of the components and the system. An important class of such systems comprises hybrid intelligent systems, where the components are represented by analytical models of fuzzy objects, artifi cial neural networks, expert systems, etc. The article considers fuzzy discrete devices being, for example, part of hybrid systems. Fuzzy linear automata (FLA) introduced in the article are used as a mathematical model of such components. The problem of test synthesis for FLA used to detect faults in them is discussed. Normal single-stuck faults are permissible faults in FLA. The faults originating from the replacement of some elements of the FLA characteristic matrices with others (from a given set of alternative ones) are also permissible. Test synthesis methods for FLA belonging to the class of m-deterministic and synchronized automata, as well as arbitrary linear automata have been developed. The fi rst two methods are based on reducing the considered problem of solving systems of linear algebraic equations. It should be noted that there is a well-developed mathematical apparatus applying a few eff ective methods for searching for such solutions. The tests synthesized by these methods for m-deterministic and synchronized FLA are sufficiently short and do not exceed the memory depth of the corresponding automata. It is shown that the conditions for an FLA referring to the two fi rst classes mentioned above are not too strict. It is noted that the known methods of test synthesis for linear automata require compliance with much more stringent requirements. The synthesis method for arbitrary FLA also builds short tests

Linear automata with translucent letters and linear context-free trace languages

Linear automata with translucent letters are studied. These are finite-state acceptors that have two heads that read the input from opposite sides and for which a set of translucent letters is associated with each state. Thus, head 1, which proceeds from left to right, does not necessarily read the first letter of the current tape content, but it skips a prefix that consists of translucent letters only and reads the first letter after that prefix. Analogously, head 2, which proceeds from right to left, does not necessarily read the last letter, but it skips a suffix that consists of translucent letters only and reads the last letter before that. After such a read operation, the head always returns to its corresponding end of the tape. These linear automata with translucent letters are a generalization of the finite-state acceptors with translucent letters that were studied by the authors in B. Nagy and F. Otto [Finite-state acceptors with translucent letters. In BILC 2011, Proc., edited by G. Bel-Enguix, V. Dahl, and A.O. De La Pente, SciTePress, Portugal (2011) 3-13.] It is shown that these linear automata are strictly more expressive than the model with a single head, but that they still only accept languages that have a semi-linear Parikh image. On the other hand, we obtain a characterization for the class of linear context-free trace languages in terms of a specific class of linear automata with translucent letters.

About the Stability of Fuzzy Linear Automata

The concept of stability of fuzzy linear automaton is introduced. It is assumed that the fuzziness reveals as presence of alternative values of elements in one of matrices of trajectory equation of the automaton. Necessary and sufficient conditions for stability are obtained.

Fuzzy Linear Automata and Some Equivalences

In this paper, we present two new normal forms for fuzzy linear grammars that preserving language. In addition, we introduce two new classes of fuzzy machines, Fuzzy Linear Automata (FLA) and Fuzzy Nondeterministic 2-Tape Automata (2-FNA), we show the equivalence between them and also show that the class of fuzzy languages recognized by these machines is exactly the class of fuzzy linear languages, i.e., FLA and 2-DNA are recognized machines for class of fuzzy linear languages.

Models and Algorithms of Automata Theory for the Control of an Aircraft Group

We develop models and algorithms to control the process of landing aircraft under uncertainty of their positions in space caused by unfavorable, unforeseen external influences. As the objective function we consider the execution time of the dispatcher’s commands, and as constraints we introduce requirements for the safety of aircraft. Based on the developed formalism of generalized synchronization of linear automata, we develop an approach that makes it possible to reduce this problem to an integer linear programming problem. We consider an example of the use of the developed models and methods for controlling the aircraft landing process.

Decision Problems on Unary Probabilistic and Quantum Automata

It is well known that the emptiness problem for binary probabilistic automata and so for quantum automata is undecidable. We present the current status of the emptiness problems for unary probabilistic and quantum automata with connections with Skolem's and positivity problems. We also introduce the concept of linear recurrence automata in order to show the connection naturally. Then, we also give possible generalizations of linear recurrence relations and automata on vectors.

Synchronization of fuzzy linear automata

For fuzzy linear automaton the method constructing of generalized synchronization sequences that are optimized according to various criteria are proposed. It is assumed that the fuzziness inherent in the description of law of the automaton functioning.

Cascade Connections and Triangular Products of Linear Automata

In this note, we focus again on the basics of triangular product of automata construction and to introduce the notion of linear automata complexity. It contains three main results. (1) For any two pure automata we consider the category of their cascade connections. It possesses the universal terminal object. This object is the wreath product of the automata. Hence, every cascade connection admits a natural embedding into the wreath product of automata. (2) A similar theory is developed for linear automata, where we also consider the category of cascade connections. It also has the terminal object. This object is the triangular product of linear automata. (3) Triangular products have various applications. This construction is used in linear automata decomposition theory and in the definition of complexity of a linear automaton. We consider a special linear complexity and give the rule for its calculation.

Nonlinear generalized minimum variance control of linear hybrid automata

This paper investigates the linear hybrid automata(LHA) model. Under some certain conditions, linear hybrid automata(LHA) can be converted into their equivalent state-dependent space models, where equivalence means that the two systems generate the same trajectory. The controller of state-dependent space model can be designed by using a nonlinear generalized minimum variance(NGMV) algorithm. NGMV controller is designed for a very general nonlinear model, which can include a set of delay terms and external disturbances. The controller is easy to be computed and implemented. Simulation results show that the NGMV algorithm can control linear hyprid automata effectively, and satisfying effect is obtained under the condition of time delay, disturbance and noise exsiting in the system.

Распознавание входов линейного автомата по нечетким выходам

Распознавание входов линейного автомата по нечетким выходам

An Application of Linear Automata to Near Rings

In this paper , we have established an intimate connection between near-nings and linear automata，and obtain the following results: 1) For a near-ring N there exists a linear GSA S with N ≌ N(S) iff (a) (N, +) is abelian, (b) N has an identity 1, (c) There is some d ∈ Nd such that N0 is generated by {1,d}; 2) Let h: S → S’ be a GSA- epimorphism. Then there exists a near-ring epimorphism from N(S) to N(S’) with h(qn) = h(q)h(n) for all q ∈ Q and n ∈ N(S); 3) Let A = (Q,A,B,F,G) be a GA. Then (a) Aa:=(Q(N(A)) =: Qa,A,B,F/Qa × A) is accessible, (b) Q = 0N(A), (c) A/~:= (Q/~,A,B,F~), Q~) with F~([q], a):= [F(q,a)] and G~([q], a):= G(q,a) is reduced, (d) Aa/~ is minimal.

Revisiting LFSRs for Cryptographic Applications

Linear finite state machines (LFSMs) are particular primitives widely used in information theory, coding theory and cryptography. Among those linear automata, a particular case of study is linear feedback shift registers (LFSRs) used in many cryptographic applications such as design of stream ciphers or pseudo-random generation. LFSRs could be seen as particular LFSMs without inputs. In this paper, we first recall the description of LFSMs using traditional matrices representation. Then, we introduce a new matrices representation with polynomial fractional coefficients. This new representation leads to sparse representations and implementations. As direct applications, we focus our work on the Windmill generators case, used for example in the E0 stream cipher and on other general applications that use this new representation. In a second part, a new design criterion called diffusion delay for LFSRs is introduced and well compared with existing related notions. This criterion represents the diffusion capacity of an LFSR. Thus, using the matrices representation, we present a new algorithm to randomly pick LFSRs with good properties (including the new one) and sparse descriptions dedicated to hardware and software designs. We present some examples of LFSRs generated using our algorithm to show the relevance of our approach.

Generative replication and the evolution of complexity

This paper identifies generative replication as a form of replication which has the potential to enhance complexity in social and biological evolution, including the wondrous complexity in the biological world, and complex social institutions such as human language and business corporations. We draw inspiration from the literature on self-reproducing automata to clarify the notion of information transfer in replication processes. To enhance complexity, developmental instructions must be part of the information that is transmitted in replication. In addition to the established triple conditions of causality, similarity and information transfer, a generative replicator involves a conditional generative mechanism that can use signals from an environment and create developmental instructions. We develop a simple model, a one-dimensional linear automaton that is consistent with our four proposed conditions for a generative replicator. We show that evolution within this model will indeed approach maximal complexity, but only if our four proposed conditions are not violated.