Canonical Automaton Research Articles

Yet another canonical nondeterministic automaton

Several canonical forms of finite automata have been introduced over the decades. In particular, if one considers the minimal deterministic finite automaton (DFA), the canonical residual finite state automaton (RFSA), and the átomaton of a language, then the átomaton can be seen as the dual automaton of the minimal DFA, but no such dual has been presented for the canonical RFSA so far. We fill this gap by introducing a new canonical automaton that we call the maximized prime átomaton, and study its properties. We also describe how these four automata can be extracted from suitable observation tables used in the automata learning context.

Canonical Automata via Distributive Law Homomorphisms

The classical powerset construction is a standard method converting a non-deterministic automaton into a deterministic one recognising the same language. Recently, the powerset construction has been lifted to a more general framework that converts an automaton with side-effects, given by a monad, into a deterministic automaton accepting the same language. The resulting automaton has additional algebraic properties, both in the state space and transition structure, inherited from the monad. In this paper, we study the reverse construction and present a framework in which a deterministic automaton with additional algebraic structure over a given monad can be converted into an equivalent succinct automaton with side-effects. Apart from recovering examples from the literature, such as the canonical residual finite-state automaton and the \'atomaton, we discover a new canonical automaton for a regular language by relating the free vector space monad over the two element field to the neighbourhood monad. Finally, we show that every regular language satisfying a suitable property parametric in two monads admits a size-minimal succinct acceptor.

Automata, reduced words and Garside shadows in Coxeter groups

In this article, we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated Coxeter system (W,S). The definition of these automata is straightforward as it only requires the notion of weak order on (W,S) and the related notion of Garside shadows in(W,S), an analog of the notion of a Garside family. Then we discuss the relations between this class of automata and the canonical automaton built from Brink and Howlett's small roots. We end this article by providing partial positive answers to two conjectures: (1) the automata associated to the smallest Garside shadow is minimal; (2) the canonical automaton is minimal if and only if the support of all small roots is spherical, i.e., the corresponding root system is finite.

A canonical automaton for one-rule length-preserving string rewrite systems

In this work, we use rearrangements in rewriting positions sequence in order to study precisely the structure of the derivations in one-rule length-preserving string rewrite systems. That yields to the definition of a letter-to-letter transducer that computes the relation induced by a one-rule length-preserving string rewrite system. This transducer can be seen as an automaton over an alphabet A×A. We prove that this automaton is finite if and only if the corresponding relation is rational. We also identify a sufficient condition for the context-freeness of the language L recognized by this automaton and, when this condition is satisfied, we construct a pushdown automaton that recognizes L.

LANGUAGES WITH A FINITE ANTIDICTIONARY: SOME GROWTH QUESTIONS

We study FAD-languages, which are regular languages defined by finite sets of forbidden factors, together with their “canonical” recognizing automata. We are mainly interested in the possible asymptotic orders of growth for such languages. We analyze certain simplifications of sets of forbidden factors and show that they “almost” preserve the canonical automata. Using this result and structural properties of canonical automata, we describe an algorithm that effectively lists all canonical automata having a sink strong component isomorphic to a given digraph, or reports that no such automata exist. This algorithm can be used, in particular, to prove the existence of a FAD-language over a given alphabet with a given exponential growth rate. On the other hand, we give an example showing that the algorithm cannot prove non-existence of a FAD-language having a given growth rate. Finally, we provide some examples of canonical automata with a nontrivial condensation graph and of FAD-languages with a “complex” order of growth.

A succinct canonical register automaton model

We present a novel canonical automaton model, based on register automata, that can be used to specify protocol or program behavior. Register automata have a finite control structure and a finite number of registers (variables), and process sequences of terms that carry data values from an infinite domain. We consider register automata that compare data values for equality. A major contribution is the definition of a canonical automaton representation of any language recognizable by a deterministic register automaton, by means of a Nerode congruence. This canonical form is well suited for modeling, e.g., protocols or program behavior. Our model can be exponentially more succinct than previous proposals, since it filters out ‘accidental’ relations between data values. This opens the way to new practical applications, e.g., in automata learning.

COMPOSITIONAL VERIFICATION OF THE GENERALIZED NONBLOCKING PROPERTY USING ABSTRACTION AND CANONICAL AUTOMATA

This paper brings together two methods to compositionally verify the generalized non-blocking property, which is a weak liveness property to express the ability of concurrent systems to terminate under given preconditions. Appropriate notions of equivalence and refinement for the generalized nonblocking property are discussed, and abstraction rules to simplify finite-state automata accordingly are presented. The paper also presents a unique canonical automaton representation for all generalized nonblocking equivalent automata, which gives rise to more general means of abstraction and to effective decision procedures for generalized nonblocking equivalence. Experimental results demonstrate how abstraction rules and canonical automata can be combined to model-check large models of concurrent software.

Four one-shot learners for regular tree languages and their polynomial characterizability

We survey two existing algorithms for the inference of finite-state tree automata from membership queries and a finite positive sample or equivalence queries, and we suggest a reformulation of one of them which we deem necessary to ensure its termination. We present two algorithms for the same two settings when the underlying description is not a deterministic but a residual canonical finite-state automaton. To this end, we adapt all necessary notions to the residual tree case. From our completed perspective, we discuss the terms on which those four algorithms can be considered to be of polynomial complexity, and also where there may be hidden exponentiality.

ENUMERATING NONDETERMINISTIC AUTOMATA FOR A GIVEN LANGUAGE WITHOUT CONSTRUCTING THE CANONICAL AUTOMATON

Our aim is to enumerate all NFAs (nondeterministic finite automata) that recognize a given regular language [Formula: see text]. More precisely, we produce a set 𝔸 of automata such that each automaton A recognizing [Formula: see text] appears in 𝔸 up to the merging of some states and the addition of some transitions, that is, there is a surjective morphism that maps A onto an automaton of 𝔸. We provide a common theoretical framework, based on morphism properties, to previous works of Kameda and Weiner (1970), and of Sengoku (1992), whose issue is the minimization of NFAs. Our paper gives two incomparable enumeration techniques. Both proceed by enumerating a specific class of grid covers of the automaton map. The first one is related to the canonical automaton introduced by Carrez. The second one is based on new outcomes related to the relationship between grid covers and their projections.

Reducing memory requirements in reachability‐based finite automata operations

AbstractNew applications of finite automata, such as computational linguistics and asynchronous circuit simulation, can require automata of millions or even billions of states. All known construction methods (in particular, the most effective reachability‐based ones that save memory, such as the subset construction, and simultaneously minimizing constructions, such as Brzozowski's) have intermediate memory usage much larger than the final automaton, thereby restricting the maximum size of the automata which can be built. In this paper, I present a reachability‐based optimization which can be used in most of these construction algorithms to reduce the intermediate memory requirements. The optimization is presented in conjunction with an easily understood (and implemented) canonical automaton construction algorithm. Copyright © 2003 John Wiley & Sons, Ltd.

Some properties of the basis finite automaton

The present paper discusses non-deterministic finite Rabin-Scott's automata. The majority of works recently dealing with this subject were, in fact, concerned only with properties of a canonical form automata or of some objects equivalent to it. This article continues the series of works in which the authors state a different point of view, describing the finite automata as just another invariant of the given regular language called basis finite automaton. In this article the authors argue on some new properties for the basis finite automaton. One of them is included into basis automaton's table of binary relations. It is stated that this table can not contain either identical strings or identical columns. Another property depicts a possibility to obtain any finite automaton for a given regular language by the process of duplicating or combining some of its states.