In our recent paper we have established close relationships between state reduction of a fuzzy recognizer and resolution of a particular system of fuzzy relation equations. In that paper we have also studied reductions by means of those solutions which are fuzzy equivalences. In this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasi-orders. Generally, fuzzy quasi-orders and fuzzy equivalences are equally good in the state reduction, but we show that right and left invariant fuzzy quasi-orders give better reductions than right and left invariant fuzzy equivalences. We also show that alternate reductions by means of fuzzy quasi-orders give better results than alternate reductions by means of fuzzy equivalences. Furthermore we study a more general type of fuzzy quasi-orders, weakly right and left invariant ones, and we show that they are closely related to determinization of fuzzy recognizers. We also demonstrate some applications of weakly left invariant fuzzy quasi-orders in conflict analysis of fuzzy discrete event systems.
University of Ni s, Faculty of Sciences；University of Kragujevac, Technical faculty in Ca cak, Svetog Save 65, P. O. Box 131, 32000 Ca cak, Serbia
Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata $\cal A$ and $\cal B$ is a forward bisimulation if and only if its kernel and co-kernel are for…
To deal with system uncertainty, finite automata have been generalized into fuzzy automata. Stamenkovic and Ciric proposed an approach using the position automata for the construction of fuzzy automata from fuzzy regular expressions. There exist multifarious methodologies for the construction of finite automata from regular expressions known as Thompson construction, Antimirov partial derivatives, Glushkov automata and follow automata etc. In this paper, we propose an approach for the conversion of fuzzy regular expressions into fuzzy automata using the concept of follow automata. The number o…
University of Niš（尼什大学）；University of Kragujevac（克拉古耶瓦茨大学）
Recently, two types of simulations (forward and backward simulations) and four types of bisimulations (forward, backward, forward-backward, and backward-forward bisimulations) between fuzzy automata have been introduced. If there is at least one simulation/bisimulation of some of these types between the given fuzzy automata, it has been proved that there is the greatest simulation/bisimulation of this kind. In the present paper, for any of the above-mentioned types of simulations/bisimulations we provide an effective algorithm for deciding whether there is a simulation/bisimulation of this typ…
Simulations and bisimulations between two fuzzy automata over a complete residuated lattice were defined by Ćirić et al. (2012) as fuzzy relations between the sets of states of the automata. However, they act as a crisp relationship between the automata. In particular, if there exists a (forward) bisimulation between two fuzzy automata, then the fuzzy languages recognized by them are crisply equal. Approximate simulations and bisimulations introduced by Stanimirović et al. (2020) aim at fuzzifying this phenomenon. However, they are defined only for fuzzy automata over a complete Heyting algebr…
In this paper we adapt the well-known Brzozowski determinization method to fuzzy automata. This method gives better results than all previously known methods for determinization of fuzzy automata developed by Bělohlávek [Inform Sciences 143 (2002) 205--209], Li and Pedrycz [Fuzzy Set Syst 156 (2005) 68--92], Ignjatović et al. [Inform Sciences 178 (2008) 164--180], and Jančić et al. [Inform Sciences 181 (2011) 1358--1368]. Namely, as in the case of ordinary nondeterministic automata, Brzozowski type determinization of a fuzzy automaton results in a minimal crisp-deterministic fuzzy automaton eq…
State minimization is a fundamental problem in automata theory. The problem is also of great importance in the study of fuzzy automata. However, most work in the literature considered only state reduction of fuzzy automata, whereas the state minimization problem is almost untouched for fuzzy automata. Thus in this paper we focus on the latter problem. Formally, the decision version of the minimization problem of fuzzy automata is as follows: \begin{itemize} \item Given a fuzzy automaton $\mathcal{A}$ and a natural number $k$, that is, a pair $\langle \mathcal{A}, k\rangle$, is there a $k$-stat…
We study three notions of directability of fuzzy automata akin to the D1-, D2- and D3-directability of nondeterministic automata. Thus an input word $w$ of a fuzzy automaton is D1-directing if a fixed single state is reachable by $w$ from all states, D2-directing if exactly the same states are reachable by $w$ from every state, and D3-directing if there is a state reachable by $w$ from every state. We study the various sets of directing words of fuzzy automata, prove that the directability properties are decidable, and show how such results can be deduced from the theory of directable nondeter…
We give a new version of fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata over distributive lattices: weights are putting in every leaf node of run trees rather than along with edges from every node to its children. Such settings are great benefit to obtain complement just by taking dual operation and replacing each final weight with its complement. We prove that $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ automata have the same expressive power as $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ ones. A direct construction (without related knowledge about $L$-fuzzy nondeterministic $\ma…
Universidade Federal do Vale do São Francisco（圣弗朗西斯科河谷联邦大学）；Universidade Federal do Rio Grande do Norte（北大河州联邦大学）
The idea of nondeterministic typical hesitant fuzzy automata is a generalization of the fuzzy automata presented by Costa and Bedregal. This paper, presents the sufficient and necessary conditions for a typical hesitant fuzzy language to be computed by nondeterministic typical hesitant fuzzy automata. Besides, the paper introduces a new class of Typical Hesitant Fuzzy Automata with crisp transitions, and we will show that this new class is equivalent to the original class introduced by Costa and Bedregal
We introduce MK-fuzzy automata over a bimonoid K which is related to the fuzzification of the McCarthy-Kleene logic. Our automata are inspired by, and intend to contribute to, practical applications being in development in a project on runtime network monitoring based on predicate logic. We investigate closure properties of the class of recognizable MK-fuzzy languages accepted by MK-fuzzy automata as well as of deterministically recognizable MK-fuzzy languages accepted by their deterministic counterparts. Moreover, we establish a Nivat-like result for recognizable MK-fuzzy languages. We introd…
In this paper we introduce and study fuzzy deterministic top-down (DT) tree automata over a lattice L. The L-fuzzy tree languages recognized by these automata are said to be DT-recognizable, and they form a proper subfamily $DRec_L$ of the family of $Rec_L$ of all regular L-fuzzy tree languages. We prove a Pumping Lemma for $DRec_L$ from which several decidability results follow. The closure properties of $DRec_L$ under various operations are established. We also characterize DT-recognizability in terms of L-fuzzy path languages, and prove that the path closure of any regular L-fuzzy tree lang…
Li and Pedrycz [Y. M. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids, Fuzzy Sets and Systems 156 (2005) 68--92] have proved fundamental results that provide different equivalent ways to represent fuzzy languages with membership values in a lattice-ordered monoid, and generalize the well-known results of the classical theory of formal languages. In particular, they have shown that a fuzzy language over an integral lattice-ordered monoid can be represented by a fuzzy regular expression if and only if it can be recognized by a…
In this paper we combine determinization and state reduction methods into two-in-one algorithms that simultaneously perform determinization and state reduction. These algorithms perform better than all previous determinization algorithms for fuzzy finite automata, developed by Belohlavek [Inform Sciences 143 (2002) 205-209], Li and Pedrycz [Fuzzy Set Syst 156 (2005) 68-92], Ignjatović et al. [Inform Sciences 178 (2008) 164-180], and Jančić et al. [Inform Sciences 181 (2011) 1358-1368], in the sense that they produce smaller automata, while require the same computation time. The only exception …
Determinization of fuzzy finite automata is understood here as a procedure of their conversion into equivalent crisp-deterministic fuzzy automata, which can be viewed as being deterministic with possibly infinitely many states, but with fuzzy sets of terminal states. Particularly significant determinization methods are those that provide a minimal crisp-deterministic fuzzy automaton equivalent to the original fuzzy finite automaton, called canonization methods. One canonization method for fuzzy finite automata, the Brzozowski type determinization, has been developed recently by Jančić and Ćiri…
A deterministic finite automaton is directable if it has a directing word which takes the automaton from every state to the same state. These notions have been extended also to other kinds of automata. Thus, B.~Imreh and M.~Steinby (1999) identified three natural types of directing words, called D1-, D2- and D3-directing words, for nondeterministic finite automata (NFAs). Here we adapt these notions for fuzzy finite automata (FFAs). The D3-directing words obtained this way are precisely the directing words introduced by V.~Karthikeyan and M. Rajasekar (2015). With any FFA F we associate an NFA…
This paper is the second part of a series of two papers dealing with bulking: a way to define quasi-order on cellular automata by comparing space-time diagrams up to rescaling. In the present paper, we introduce three notions of simulation between cellular automata and study the quasi-order structures induced by these simulation relations on the whole set of cellular automata. Various aspects of these quasi-orders are considered (induced equivalence relations, maximum elements, induced orders, etc) providing several formal tools allowing to classify cellular automata.
This work considers weak deterministic Büchi automata reading encodings of non-negative $d$-vectors of reals in a fixed base. A saturated language is a language which contains all encoding of elements belonging to a set of $d$-vectors of reals. A Real Vector Automaton is an automaton which recognizes a saturated language. It is explained how to decide in quasi-linear time whether a minimal weak deterministic Büchi automaton is a Real Vector Automaton. The problem is solved both for the two standard encodings of vectors of numbers: the sequential encoding and the parallel encoding. This algorit…
In 1964 Černý conjectured that each $n$-state synchronizing automaton posesses a reset word of length at most $(n-1)^2$. From the other side the best known upper bound on the reset length (minimum length of reset words) is cubic in $n$. Thus the main problem here is to prove quadratic (in $n$) upper bounds. Since 1964, this problem has been solved for few special classes of \sa. One of this result is due to Kari \cite{Ka03} for automata with Eulerian digraphs. In this paper we introduce a new approach to prove quadratic upper bounds and explain it in terms of Markov chains and Perron-Frobenius…
Watson-Crick (WK) finite automata are working on a Watson-Crick tape, that is, on a DNA molecule. A double stranded DNA molecule contains two strands, each having a 5' and a 3' end, and these two strands together form the molecule with the following properties. The strands have the same length, their 5' to 3' directions are opposite, and in each position, the two strands have nucleotides that are complement of each other (by the Watson-Crick complementary relation). Consequently, WK automata have two reading heads, one for each strand. In traditional WK automata both heads read the whole input…
We focus on the family of $(k,l)$-unambiguous automata that encompasses the one of deterministic $k$-lookahead automata introduced by Han and Wood. We show that this family presents nice theoretical properties that allow us to compute quasi-deterministic structures. These structures are smaller than DFAs and can be used to solve the membership problem faster than NFAs.
In this paper we introduce the notion of quasi-expansivity for 2D CA and we show that it shares many properties with expansivity (that holds only for 1D CA). Similarly, we introduce the notions of quasi-sensitivity and prove that the classical dichotomy theorem holds in this new setting. Moreover, we show a tight relation between closingness and openness for 2D CA. Finally, the undecidability of closingness property for 2D CA is proved.
We study \emph{multiplicity equivalence} testing of automata over partially commutative monoids (pc monoids) and show efficient algorithms in special cases, exploiting the structure of the underlying non-commutation graph of the monoid. Specifically, if the clique cover number of the non-commutation graph (the minimum number of cliques covering the graph) of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm. As a consequence, we also obtain the first deterministic quasi-polynomial time algorithms for multiplicity equivalence testing of $k$-tape automata and…
An improved translation from alternating parity automata on infinite words to alternating weak automata is given. The blow-up of the number of states is related to the size of the smallest universal ordered trees and hence it is quasi-polynomial, and only polynomial if the asymptotic number of priorities is logarithmic in the number of states. This is an exponential improvement on the translation of Kupferman and Vardi (2001) and a quasi-polynomial improvement on the translation of Boker and Lehtinen (2018). Any slightly better such translation would (if---like all presently known such transla…
This work considers weak deterministic Büchi automata reading encodings of non-negative reals in a fixed base. A Real Number Automaton is an automaton which recognizes all encoding of elements of a set of reals. It is explained how to decide in linear time whether a set of reals recognized by a given minimal weak deterministic RNA is ${FO}[\mathbb R;+,<,1]$-definable. Furthermore, it is explained how to compute in quasi-quadratic (respectively, quasi-linear) time an existential (respectively, existential-universal) ${FO}[\mathbb R;+,<,1]$-formula which defines the set of reals recognized by th…
Several distinct techniques have been proposed to design quasi-polynomial algorithms for solving parity games since the breakthrough result of Calude, Jain, Khoussainov, Li, and Stephan (2017): play summaries, progress measures and register games. We argue that all those techniques can be viewed as instances of the separation approach to solving parity games, a key technical component of which is constructing (explicitly or implicitly) an automaton that separates languages of words encoding plays that are (decisively) won by either of the two players. Our main technical result is a quasi-polyn…
Recently, five quasi-polynomial-time algorithms solving parity games were proposed. We elaborate on one of the algorithms, by Lehtinen (2018). Czerwiński et al. (2019) observe that four of the algorithms can be expressed as constructions of separating automata (of quasi-polynomial size), that is, automata that accept all plays decisively won by one of the players, and rejecting all plays decisively won by the other player. The separating automata corresponding to three of the algorithms are deterministic, and it is clear that deterministic separating automata can be used to solve parity games.…
The memory of a deque (double ended queue) automaton is more general than a queue or two stacks; to avoid overgeneralization, we consider quasi-real-time operation. Normal forms of such automata are given. Deque languages form an AFL but not a full one. We define the characteristic deque language, CDL, which combines Dyck and AntiDyck (or FIFO) languages, and homomorphically characterizes the deque languages. The notion of deque graph, from graph theory, well represents deque computation by means of a planar hamiltonian graph on a cylinder, with edges visualizing producer-consumer relations fo…
We introduce and study cellular automata whose cell spaces are left-homogeneous spaces. Examples of left-homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; uniform tilings acted on by symmetries; vertex-transitive graphs, in particular, Cayley graphs, acted on by automorphisms; groups acting on themselves by multiplication; and integer lattices acted on by translations. For such automata and spaces, we prove, in particular, generalisations of topological and uniform variants of the Curtis-Hedlund-Lyndon theorem, of the Tarski-Følner theorem, …
