Universal Automaton Research Articles

Approximate NFA universality and related problems motivated by information theory

In coding and information theory, it is desirable to construct maximal codes that can be either variable length codes or error control codes of fixed length. However deciding code maximality boils down to deciding whether a given NFA is universal, and this is a hard problem (including the case of whether the NFA accepts all words of a fixed length). On the other hand, it is acceptable to know whether a code is ‘approximately’ maximal, which then boils down to whether a given NFA is ‘approximately’ universal. Here we introduce the notion of a (1−ε)-universal automaton and present polynomial randomized approximation algorithms to test NFA universality and related hard automata problems, for certain natural probability distributions on the set of words. We also conclude that the randomization aspect is necessary, as approximate universality remains hard for any fixed polynomially computable ε.

Optimization-based motion primitive automata for autonomous driving

Abstract Trajectory planning for autonomous cars can be addressed by primitive-based methods, which encode nonlinear dynamical system behavior into automata. In this paper, we focus on optimal trajectory planning. Since, typically, multiple criteria have to be taken into account, multiobjective optimization problems have to be solved. For the resulting Pareto-optimal motion primitives, we introduce a universal automaton, which can be reduced or reconfigured according to prioritized criteria during planning. We evaluate a corresponding multi-vehicle planning scenario with both simulations and laboratory experiments.

Development and computer implementation of the motion model and control algorithms of a hybrid multicopter

Among modern unmanned aerial vehicles, a special place is occupied by hybrid multicopters, in which internalcombustion engines are used along with electric motors to create lift. They can carry a larger load for a longertime. But the combination of a more powerful engine with less powerful ones, the variability of the inertialcharacteristics during the flight, significantly complicate the dynamics of such an object and its control. Therefore,there is almost no practical experience in operating such devices, which is the motivation for research anddevelopment in this area. The article deals with a helicopter-type multicopter with five electric motors (4+1 on thetail) and a central internal combustion engine. A mathematical description of the dynamics and kinematics of the
 apparatus has been created for the purpose of computer simulation of a controlled flight. The description takes intoaccount all internal dynamic factors and is given simultaneously with respect to the inertial and associatedcoordinate systems, which simplifies the solution of the flight control problem as much as possible. For the latter, amultiloop motion control scheme with the help of motors has been developed. The algorithms are based on thedecomposition of the problem under the conditions of typical motion modes and the modal control method, whichprovides the desired quality of transient processes. A universal automaton for the interaction of control algorithmsfor each engine is described. A typical flight program is simulated, containing climb, course change, following tothe set hovering point and landing under it. The presented results showed the performance of the algorithms anddetermined the directions for further research. The results in the form of a mathematical model, recommendationsfor task decomposition and specific algorithms can be used in the process of designing a hybrid multicopter controlsystem.

On quotients of formal power series

Quotient is a basic operation of formal languages, which plays a key role in the construction of minimal deterministic finite automata (DFA) and the universal automata. In this paper, we extend this operation to formal power series and systemically investigate its implications in the study of weighted automata. In particular, we define two quotient operations for formal power series which are called (left or right) quotients and (left or right) residuals respectively. Algebraic properties of the quotient operation and closure properties (w.r.t. regular and context-free series) under quotients and residuals are also investigated for formal power series. Using these operations, we define for each formal power series A two weighted automata that accept A. The first is the minimal deterministic weighted automaton of A, and the second is the universal weighted automaton of A. An effective algebraic method to construct the universal automaton is also presented.

Structural Machines as Unconventional Knowledge
 Processors

Knowledge systems often have very sophisticated structures depicting cognitive andstructural entities. For instance, representation of knowledge in the form of a text involves thestructure of this text. This structure is represented by a hypertext, which is networks consisting oflinguistic objects, such as words, phrases and sentences, with diverse links connecting them.Current computational machines and automata such as Turing machines process information inthe form of symbol sequences. Here we discuss based the methods of structural machinesachieving higher flexibility and efficiency of information processing in comparison with regularmodels of computation. Being structurally universal abstract automata, structural machines allowworking directly with knowledge structures formed by knowledge objects and connectionsbetween them.

Elementary Definability of the Class of Universal Planar Automata in the Class of Semigroups

Universal planar automata are universal attracting objects in the category of semigroup automata whose set of states and set of output signals are equipped with algebraic structures of the planes that are invariant under the actions of the transition and output functions. We establish the elementary definability of the class of universal planar automata in the class of semigroups and study the problem of the elementary classification of universal planar automata with the use of first-order theories of input signal semigroups of these automata.

Universal Hypergraphic Automata Representation by Autonomous Input Symbols

Hypergraphic automata are automata with state sets and input symbol sets being hypergraphs which are invariant under actions of transition and output functions. Universally attracting objects of a category of hypergraphic automata are automata Atm(H1,H2). Here,H1is a state hypergraph,H2is classified as an output symbol hypergraph, andS= EndH1× Hom(H1,H2) is an input symbol semigroup. Such automata are called universal hypergraphic automata. The input symbol semigroupSof such an automaton Atm(H1,H2) is an algebra of mappings for such an automaton. Semigroup properties are interconnected with properties of the algebraic structure of the automaton. Thus, we can study universal hypergraphic automata with the help of their input symbol semigroups. In this paper, we investigated a representation problem of universal hypergraphic automata in their input symbol semigroup. The main result of the current study describes a universal hypergraphic automaton as a multiple-set algebraic structure canonically constructed from autonomous input automaton symbols. Such a structure is one of the major tools for proving relatively elementary definability of considered universal hypergraphic automata in a class of semigroups in order to analyze interrelation of elementary characteristics of universal hypergraphic automata and their input symbol semigroups. The main result of the paper is the solution of this problem for universal hypergraphic automata for effective hypergraphs withp-definable edges. It is an important class of automata because such an algebraic structure variety includes automata with state sets and output symbol sets represented by projective or affine planes, along with automata with state sets and output symbol sets divided into equivalence classes. The article is published in the authors' wording.

On problem of concrete characterization of universal automata

In the paper we investigate automata whose state sets and sets of output symbols are endowed with algebraic structures of hypergraphs. Universal hypergraphic automata are universally attracting objects in the category of such automata. The main result is a solving of the problem of concrete characterization of universal hypergraphic automata.

Language-Theoretic and Finite Relation Models for the (Full) Lambek Calculus

We prove completeness for some language-theoretic models of the full Lambek calculus and its various fragments. First we consider syntactic concepts and syntactic concepts over regular languages, which provide a complete semantics for the full Lambek calculus $${\mathbf {FL}}_\perp $$FLź. We present a new semantics we call automata-theoretic, which combines languages and relations via closure operators which are based on automaton transitions. We establish the completeness of this semantics for the full Lambek calculus via an isomorphism theorem for the syntactic concepts lattice of a language and a construction for the universal automaton recognizing the same language. Finally, we use automata-theoretic semantics to prove completeness of relation models of binary relations and finite relation models for the Lambek calculus without and with empty antecedents (henceforth: $$\mathbf L $$L and $$\mathbf L1 $$L1), thus solving a problem left open by Pentus (Ann Pure Appl Log 75:179---213, 1995).

Polynomial inference of universal automata from membership and equivalence queries

A MAT learning algorithm is presented that infers the universal automaton (UA) for a regular target language, using a polynomial number of queries with respect to that automaton. The UA is one of several canonical characterizations for regular languages. Our learner is based on the concept of an observation table, which seems to be particularly fitting for this computational model, and the necessary definitions are adapted from the literature to the case of UA.

FACTORIZATIONS AND UNIVERSAL AUTOMATON OF OMEGA LANGUAGES

We extend the concept of factorization on finite words to ω-rational languages and show how to compute them. We define a normal form for Büchi automata and introduce a universal automaton for Büchi automata in normal form. We prove that, for every ω-rational language, this Büchi automaton, based on factorization, is canonical and that it is the smallest automaton that contains the morphic image of every equivalent Büchi automaton in normal form.

Some more algorithms for Conway’s universal automaton

Abstract In this paper authors propose algorithms for constructing so called automaton COM(L) and prove that this automaton coincides, up to re-denoting states, with Conway’s universal automaton. We give some algorithms of constructing this automaton and consider some examples.

The syntactic concept lattice: Another algebraic theory of the context-free languages?

The syntactic concept lattice is a residuated lattice associated with a given formal language; it arises naturally as a generalization of the syntactic monoid in the analysis of the distributional structure of the language. In this article we define the syntactic concept lattice and present its basic properties, and its relationship to the universal automaton and the syntactic congruence; we consider several different equivalent definitions, as Galois connections, as maximal factorizations and finally using universal algebra to define it as an object that has a certain universal (terminal) property in the category of complete idempotent semirings that recognize a given language, applying techniques from automata theory to the theory of context-free grammars (CFGs). We conclude that grammars that are minimal, in a certain weak sense, will always have non-terminals that correspond to elements of this lattice, and claim that the syntactic concept lattice provides a natural system of categories for representing the denotations of CFGs.

Antichains and compositional algorithms for LTL synthesis

In this paper, we present new monolithic and compositional algorithms to solve the LTL realizability problem. Those new algorithms are based on a reduction of the LTL realizability problem to a game whose winning condition is defined by a universal automaton on infinite words with a k-co-Buchi acceptance condition. This acceptance condition asks that runs visit at most k accepting states, so it implicitly defines a safety game. To obtain efficient algorithms from this construction, we need several additional ingredients. First, we study the structure of the underlying automata constructions, and we show that there exists a partial order that structures the state space of the underlying safety game. This partial order can be used to define an efficient antichain algorithm. Second, we show that the algorithm can be implemented in an incremental way by considering increasing values of k in the acceptance condition. Finally, we show that for large LTL formulas that are written as conjunctions of smaller formulas, we can solve the problem compositionally by first computing winning strategies for each conjunct that appears in the large formula. We report on the behavior of those algorithms on several benchmarks. We show that the compositional algorithms are able to handle LTL formulas that are several pages long.

On concrete characterization of universal hypergraphic automata

In this paper, we consider structured automata without output signals whose state sets are endowed with an algebraic structure of hypergraphs. The main result of the paper is a theorem where we obtain necessary and sufficient conditions for the possibility of defining on the state set of some automaton A a structure of a hypergraph H such that the automaton A will be the universal hypergraphic automaton.

A Universal Cellular Automaton on the Ternary Heptagrid

In this paper, we construct the first weakly universal cellular automaton on the ternary heptagrid. It requires six states only. It provides a universal automaton with less states than in the case of the pentagrid where the best result is nine states, a result also recently established by the authors.

ON TRANSITION MINIMALITY OF BIDETERMINISTIC AUTOMATA

Bideterministic automata are deterministic automata with the property of their reversal automata also being deterministic. Bideterministic automata have previously been shown to be unique (up to an isomorphism) minimal NFAs with respect to the number of states. In this paper, we show that in addition to state minimality, bideterministic automata are also transition-minimal NFAs. However, as this transition minimality is not necessarily unique, we also present the necessary and sufficient conditions for a bideterministic automaton to be uniquely transition-minimal among NFAs. These results are obtained using the notion of the universal automaton. We also present some properties of the universal automaton. Furthermore, we show that bideterministic automata are transition-minimal ∊-NFAs.

Universal automata and NFA learning

The aim of this paper is to develop a new algorithm that, with a complete sample as input, identifies the family of regular languages by means of nondeterministic finite automata. It is a state-merging algorithm. One of its main features is that the convergence (which is proved) is achieved independently from the order in which the states are merged, that is, the merging of states may be done “randomly”.

MINIMALIZATIONS OF NFA USING THE UNIVERSAL AUTOMATON

As is well known, each minimal NFA for a regular language L is isomorphic to a subautomaton of the so-called universal automaton [Formula: see text] for L. We explore and compare various conditions on sets of states of [Formula: see text] which are related to the fact that induced subautomata of [Formula: see text] accept the whole language L. The methods of several previous works on minimalizations of NFA can be modified so that they fit in our approach. We also propose a new algorithm which is easy to implement.

Functional Renewal of Behavior of Systems: Numerical Methods

The principles of functional renewal of the behavior of technical systems based on the functional capabilities in the design are stated. A finite deterministic automaton is used as the model. The functional renewal problem is solved within the framework of the theory of universal automata. A class of automata admitting modeling by a family of polynomials is described. The numerical model of automaton behavior is helpful in applying algebraic tools to solve the functional behavior renewal problem. For this automaton class, a universal enumerator is designed and analyzed, and a method of constructing a renewal sequence is described.