Cybernetics and Systems Analysis, Vol. 39, No. 5, 2003
REPRESENTATIONS OF REGULAR IDEALS IN FINITE AUTOMATA
I. K. Rystsov
UDC 519.713.4
Regular ideals of a free monoid are shown to be characterized by weak (noninitial) representations in finite automata. The class of kernel ideals is also shown to be invariant under the action of several automata functors and to be equal to the class of zero ideals. Keywords: finite automata, representation of events, linear automata, Cerny problem. INTRODUCTION In this article, distinctive features of representation of regular ideals, i.e., regular events that are simultaneously ideals of a free semigroup, are investigated. Regular ideals were found to be characterized by the so-called weak representations in which an event is represented in a compact form by a noninitial automaton but, to obtain its initial representation, a globalization procedure should be executed in some form or other. This fact can be illustrated by the class of kernel ideals that coincides, as is shown, with the class of reset and zero ideals. A functor is also constructed that transforms any finite automaton into an automaton with zero in which the same kernel event is represented. This article can be considered as a development of ideas of V. M. Glushkov on algebraic theory of automata and as a continuation of the work [1] on kernel (terminal) ideals and ranks of automata, which was performed in connection with investigations of the well-known Cerny problem on the minimal reset word length in a finite automaton [2]. BASIC CONCEPTS A finite deterministic automaton (without output) A over an input alphabet X = [1, m] is understood to be a collection of basic functions { f1 , . . . , fm } each of which is a transformation fi : S → S on a finite set of states S. By the tradition established in the automata theory, basic transformations are specified by a transition function A : S × X → S by the formula fi ( s ) = A( s , i) for all s ∈ S and 1 ≤ i ≤ m. And vice versa, as is obvious, a collection of basic transformations defines a transition function of this form. Therefore, an automaton is frequently defined as a triple consisting of a set of states, an input alphabet, and a transition function. In what follows, an automaton is identified with its transition function and this function is understood to be of the form A: X → S S , where S S is the set of all the transformations (one-place operations) on the set of states. Then basic transformations are specified as follows: fi = A(i) for all 1 ≤ i ≤ m. Note that the set S S contains n n transformations, where n = | S | is the number of states of the automaton and this set is isomorphic to the nth Cartesian degree of the set of states. We recall that a monoid is understood to be a semigroup with identity [3] and note that the set of all transformations S of S is a finite monoid with respect to the operation of superposition of transformations. On the other hand, the set of all input sequences (words) w = i1 . . . ik over an alphabet X = [1, m] forms a free (infinite) monoid X * with respect to the
Interregional Academy of Personnel Management, Kiev, Ukraine,
[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 48-58, September-October, 2003. Original article submitted February 18, 2002.
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1060-0396/03/3905-0668$25.00
©
2003 Plenum Publishing Corporation
operation of concatenation of words with one another. Therefore, the transition function of the automaton can naturally be extended to the following canonical (finite-state) homomorphism of these monoids: A: X * → S S ,
(1)
where A( w) = A(i1 ) o . . . o A(ik ) for a word w = i1 . . . ik . The image of this homomorphism A( X * ) is called the monoid of transformations (transitions) of the automaton A. Note that the monoid itself of transformations of a finite automaton will be finite since it is a submonoid of the finite monoid S S and is obtained by the closure of the basic collection of transformations A( X ) with respect to the superposition operation. In this sense, the monoid of transformations of the automaton A can also be denoted by A( X ) * = A( X * ) and can be considered as the semigroup closure of the basis of the original automaton. We denote the image of a state s under the action of a transformation A( w) by A( s , w). A state t is called reachable from a state s in the automaton A if there exists an input word w such that we have t = A( s , w). An automaton is called initial (monogenic) if, in it, a state can be selected that is initial and from which all the other states are achievable. An automaton is called transitive (strongly connected) if, from each of its state, any other state is achievable in it. An automaton can be considered as a monadic algebra (an algebra with one-place operations) and, hence, the concepts of a subautomaton, a homomorphic image (a quotient automaton), and direct product are defined for automata as in universal algebra [4]. For example, a subautomaton is defined by a subset of states that is invariant under the actions of all basic transformations and a homomorphism of automata is defined as a mapping between the sets of states of automata that is permutable or matched with all the basic transformations. The monoid of the automaton A can also be considered as an automaton (denoted by A* ) over the same input alphabet X = [1, m] if, for any i , 1 ≤ i ≤ m, by a transformation A* (i) on the monoid A( X ) * we understand the multiplication of A* ( f , i) = f o A(i) from the right (the right shift) by the basic transformation A(i). In what follows, the automaton A* is called the (monoidal) global automaton for A. In contrast to the original automaton, the global automaton is initial since the identity of a monoid can be used in the capacity of its initial state and, in this case, this identity coincides with the identical transformation ∆ S on the set S. It remains to note that a monoidal automaton is a subautomaton of the nth Cartesian degree of the original automaton and, hence, its size can be superexponential in relation to the size of the original automaton. Henceforth, we will consider the passage from an automaton to its monoidal global automaton as the first global functor Φ1: A → A* that maps the manifold of all finite automata into the manifold of finite monoids (semigroups). Here, a manifold is understood to be the class of automata that is closed under the taking of subautomata, homomorphic images, and finite direct products (a pseudo-manifold in terms of [3]). REGULAR IDEALS By tradition, the subsets of the free monoid X * are called events. An event P is called regular [5] if it can be represented in a finite initial automaton A from the initial state s 0 by a subset of final states T , i.e., we have P = {w| A( s 0 , w) ∈ T }. A useful characteristic of regular events is that the event P is regular if and only if there exists a finite automaton (a finite-state homomorphism) A such that we have P = A −1 o A( P) [3, Chapter 6, Statement 1.5]. By an ideal I of the free monoid X * we mean an event that satisfies the condition X * ⋅I ⋅ X * ⊆ I.
(2)
A regular ideal is understood to be a regular event that simultaneously is an ideal of the free monoid. The statement formulated below shows that regular ideals are preimages of finite ideals under finite-state homomorphisms. THEOREM 1. An event I ⊆ X * is a regular ideal if and only if a finite automaton A and an ideal J in its monoid A( X ) * can be found such that the condition I = A −1 ( J ) is fulfilled. Proof. If I is a regular ideal in the free monoid X * , then, by virtue of regularity, there must be a finite automaton A that satisfies the condition I = A −1 o A( I ). Moreover, since I is an ideal, we put J = A( I ), apply the homomorphism A to
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condition (2), and obtain the inclusion A( X * ) o J o A( X * ) ⊆ J . Hence, the subset J is an ideal of the monoid A( X ) * and satisfies the condition I = A −1 ( J ). Thus, this condition is necessary. And vice versa, let J be an ideal of the monoid of a finite automaton A, and let I = A −1 ( J ). Then the event I must be an ideal as the preimage of an ideal under the homomorphism of monoids. Moreover, it should be regular since we have I = A −1 ( J ) ⊇ A −1 o A( I ) ⊇ I . Thus, the theorem is proved. Henceforth, of special interest will be the preimage of the minimal ideal under a finite-state homomorphism. We denote the least ideal of a finite monoid M by Ker ( M ) and note that such a monoid exists for any finite monoid and is called its kernel [6]. We recall that the kernel of an automaton is understood to be the (direct) sum of all its transitive (strongly connected) subautomata [5]. We denote the kernel of a finite automaton A by Ker ( A) and, accordingly, the kernel of the global automaton A* by Ker ( A* ). These two concepts that are different at first sight turn out to be matched with each other since, in any finite automaton, the kernel of its global automaton coincides with the kernel of its monoid of transformations. In fact, by the well-known Sushkevich theorem [6], the minimal ideal of the monoid A( X ) * is the union of its minimal right ideals each of which corresponds to a transitive subautomaton of the automaton A* . Definition 1. We call an input word w a kernel (terminal) word for an automaton A if we have A( w) ∈ Ker ( A* ). We denote the set of all the kernel words for a finite automaton A by Ter ( A). By definition, we have the following condition: Ter ( A) = A −1 ( Ker ( A* )). Therefore, by Theorem 1, the event Ter ( A) is a regular ideal, which is called the kernel ideal of the automaton A. Note that the kernel ideal is nonempty in any finite automaton and, hence, any finite automaton has a kernel word of minimum length. A transformation f : S → S is called constant if we have f ( s ) = s1 for all states s and a fixed state s1 . Note that constant transformations are right zeros of the monoid S S and that the collection of all the constant transformations, which is denoted by C S , forms the minimal ideal (kernel) of this monoid. An input word w is called a reset (synchronizing) word for an automaton A if the transformation A( w) is constant. We denote the set of all the reset words of a finite automaton A by Res ( A). By definition, we have the following condition: Res ( A) = A −1 (C S ). Hence, by Theorem 1, the event Res ( A) is the regular ideal called the reset ideal of the automaton A. In contrast to a kernel ideal, a reset ideal can be empty. A finite automaton is called a reset one if it has a reset word, i.e., if its reset ideal is nonempty. In the statement, which is given below and is presented without proof, a simple criterion of reflexivity of an automaton is given. Statement 1. A finite automaton A is a reset automaton if and only if, for any of two states s and t, an input word w can be found such that we have A( s , w) = A(t , w). Note that, for any finite automaton A, its reset ideal is empty or the condition Ter ( A) = Res ( A) is satisfied since, in this case, the kernel of its global automaton consists of only constant transformations. In this sense, reset ideals can be considered as a special case of kernel ideals. In fact, as will be shown below, these two classes of regular ideals coincide with each other. We denote by 2 S the set of all the subsets from S and consider a mapping Im : S S → 2 S that to each transformation f : S → S assigns the domain of its values Im ( f ). This mapping transforms the monoid of the automaton A( X ) * into a finite family of subsets Im ( A( X ) * ). On this family, the automaton A induces the action of a new automaton denoted by 2 A by the formula 2 A ( Im ( f ), i) = Im( f o A(i)), 1 ≤ i ≤ m. We call the automaton 2 A the Boolean global automaton of the automaton A or the Boolean of the automaton A. The following equality immediately follows from the definitions of the global automata A* and 2 A : 2 A ( Im ( f ), i) = Im( A* ( f , i)),
1 ≤ i ≤ m.
From this, we obtain the statement formulated below. Statement 2. The mapping Im is a homomorphism of the global automaton A* on the Boolean 2 A .
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Under a homomorphism, the automaton 2 A inherits the property of being initial from the global automaton A* and, as a result, its initial state is the state S = Im( ∆ S ). By induction on the length of its input word w, we can easily obtain the equality Im( A( w)) = 2 A ( S , w). Of course, the Boolean 2 A is noticeably smaller than the global monoid and can be constructed without explicit construction of the automaton A* on the basis of the formula 2 A ( T , w) = {A( s , w)| s ∈ T } for a subset T ⊆ S . Next, since the transitivity property is retained under homomorphisms of automata, Statement 2 implies that the kernel Ker ( A ) under the homomorphism Im is mapped into the kernel Ker (2 A ), which was earlier called the automaton of *
minimal images [3]. The states of this kernel are subsets of minimal cardinality from the family Im( A( X ) * ). Thus, all the subsets of the kernel Ker (2 A ) have the same cardinality, which is called the rank of the automaton A [1]. Moreover, it follows from the well-known Sushkevich theorem on the minimal ideal in a finite semigroup of transformations [3, Vol. 2, Statement 1.3] that the preimage of the kernel Ker (2 A ) under the homomorphism Im coincides with the kernel Ker ( A* ). This allows us to prove the theorem given below in which an important property of the Boolean of a finite automaton is described. THEOREM 2. For any finite automaton A, the condition Ter ( A) = Res (2 A ) is satisfied. Proof. If A( w) ∈ Ker ( A* ), then the subset Im( A( w)) = 2 A ( S , w) must belong to the kernel of the automaton 2 A and, hence, its cardinality must be equal to the rank of the automaton A. Next, for any word x ∈ X * , the word xw will also be a kernel word and, hence, the cardinality of the subset Im( A( xw)) = 2 A ( S , xw) must also be equal to the rank of the automaton A. Then we put T = 2 A ( S , x ) and obtain the following inclusion: 2 A ( S , xw) = 2 A ( T , w) ⊆ 2 A ( S , w). Since these subsets have the same cardinality, they must be equal to each other. Hence, for any T ∈ Im( A( X ) * ), the equality 2 A ( T , w) = 2 A ( S , w) is true and, hence, the word w is a reset word in the automaton 2 A . Thus, the inclusion Ter ( A) ⊆ Res (2 A ) is proved. And vice versa, if w ∈ Res (2 A ), then the equality 2 A ( T , w) = 2 A ( S , w) must be fulfilled for all T ∈ Im( A( X ) * ). Hence, the cardinality of the subset 2 A ( S , w) must be equal to the rank of the automaton A and, hence, the subset Im( A( w)) must belong to the kernel of the automaton 2 A . Then it follows from the Sushkevich theorem that A( w) ∈ Ker ( A* ). Thus, the theorem is proved. As a consequence of this theorem, we obtain the statement, which is formulated below and is useful as a source of numerous examples of transitive reset automata. COROLLARY 1. The kernel Ker (2 A ) of the global automaton 2 A is a transitive reset automaton for any automaton A. Proof. In fact, the kernel Ker (2 A ) must be a reset automaton since it is a subautomaton of the reset automaton 2 A . On the other hand, the kernel must be the direct sum of transitive automata but the number of transitive components of a reset automaton cannot have more than one. Hence, the kernel Ker (2 A ) must be a transitive reset automaton. The statement is proved. The passage from an automaton to its Boolean can be considered as the second global functor Φ 2 = Im o Φ1 that maps the manifold of all the finite automata into the manifold of reset automata; in this case, kernel ideals are transformed into reset ideals. Thus, the kernel and reset ideals represent the same class of regular ideals. In the next section, one more characteristic of this class will be given. AUTOMATA WITH ZERO A monoid can be factored modulo any ideal and, in this way, a monoid with zero can be obtained [6]. We will show a way of applying such a construction to automata. Moreover, we will show here that reset ideals can be represented in 671
automata with zero and, as a result, the distinction between weak (noninitial) and strong (initial) representations of regular ideals will be obvious. A state s 0 ∈ S of an automaton A is called stable (persistent) if we have A( s 0 , i) = s 0 for all i , 1 ≤ i ≤ m. An automaton with zero or a 0-automaton is understood to be a finite automaton that has a unique persistent state, which is called the zero state (denoted by zero) and is reachable from all the other states of the automaton. It is obvious that the kernel of a 0-automaton consists of a unique (zero) state and this property completely characterizes 0-automata. Any 0-automaton is a reset automaton, but the converse is false. The simplest counterexample is a trigger, i.e., an automaton with two states whose transitions are specified with the help of the transformations f1 = (1, 1) and f 2 = (2 , 2). If A is a 0-automaton, then we consider the transformation f 0 ( s ) = 0 for all s ∈ S . It is obvious that f 0 is the bilateral zero of the monoid of the automaton A and, hence, the monoid of the 0-automaton has zero that is, in this case, its least ideal. Definition 2. We call a word w a zero word for the 0-automaton A if we have A( w) = f 0 . We denote the set of all the zero words of the finite 0-automaton A by Nul ( A) and call it the zero ideal of the automaton A. It follows from Theorem 1 that the event Nul ( A) is a regular ideal. We now consider a construction that connects reset and zero ideals with each other. We denote by A 2 = A × A the square of the automaton A that operates on the set of ordered pairs of states S × S as follows: A 2 (( s , t ), i) = ( A( s , i), A(t , i)), i ∈ X . Then the subautomaton that operates on the diagonal ∆ S is isomorphic, as is obvious, to the automaton A and, hence, the quotient automaton A2 = A 2 / A can be constructed, which is called the automaton of pairs for A. Note that, after factorization, a subautomaton that is isomorphic to the automaton A is replaced by one stable state, which is denoted by zero. Moreover, we will not distinguish henceforth between the pairs that differ only in the order of arrangement of states, i.e., we paste together all such pairs. As a result, we obtain a reduced form of the automaton of pairs A2 that has ( n 2 − n) / 2 + 1 states, where n is the number of states of the automaton A. THEOREM 3. For any reset automaton A, its automaton of pairs A2 is a 0-automaton that satisfies the condition Res ( A) = Nul ( A2 ). Proof. If w ∈ Res ( A), then we have A( s , w) = A(t , w) for any different states s , t ∈ S . It follows from this that, in the automaton of pairs, the equality A2 ({s , t}, w) = 0 is fulfilled for all the pairs of states {s , t} and, hence, w ∈ Nul ( A2 ). The reverse inclusion Nul ( A2 ) ⊆ Res ( A) is also obviously follows from the definition of an automaton of pairs. The theorem is proved. The passage from a reset automaton to its automaton of pairs can be considered as the third functor Φ 3 : A → A2 that maps the manifold of all the reset automata into the manifold of 0-automata; in this case, reset ideals are transformed into zero ideals. Thus, the class of zero ideals coincides with the class of reset ideals and, hence, with the class of kernel ideals. We now denote by Nul j ( A) the set of words in the 0-automaton A that transform a state s j , 1 ≤ j ≤ n, into the zero state. Then it is obvious that the event Nul j ( A) is the right ideal and the ideal Nul ( A) is the intersection of these right ideals, i.e., we have Nul ( A) = Nul1 ( A) ∩ Nul 2 ( A) ∩ . . . ∩ Nul n ( A). This shows the distinction between this implicit (weak) representation of the zero ideal and its usual initial representation with the help of the global automaton 2 A with exponential growth of the number of states: Nul ( A) = {w | 2 A ( S , w) = 0}. The cause of this growth also becomes evident since, to represent the intersection of regular events, the Cartesian product of automata must be constructed. Thus, the globalization in some form or other have to be realized, which is also confirmed by the statement presented below and proved in [7]. THEOREM 4. The problem of construction of the minimal zero word in a 0-automaton is NP-complete. Nevertheless, as is proved in [7], any 0-automaton with n states contains a zero word whose length does not exceed ( n 2 − n) / 2 and this estimate is exact [8]. But this fact was found to be insufficient to solve the Cerny problem formulated below. 672
Hypothesis (J. Cerny, 1964). Any reset automaton with n states has a reset word whose length is no more than ( n − 1) 2 . In fact, the pair functor Φ 3 approximately squares the number of states of the original reset automaton and, hence, to prove the Cerny hypothesis, a linear estimate of the zero word length in a 0-automaton of pairs should be obtained. Thus, the Cerny hypothesis can be related to distinctive features of the manifold of 0-automata that is obtained as a result of actions of the pair functor. An example of such a distinctive feature, namely, the presence of linear dependences between states of an automaton of pairs is discussed in the next section. LINEAR AUTOMATA Let us consider linear automata that contain zero states by definition and there are linear dependences between states of an automaton. This makes it possible to realize a natural embedding of the automaton of pairs described in the previous section into a linear automaton. We call a linear automaton L over the binary field F 2 = {0 , 1} with an input alphabet X = [1, m] a function L: X → End (V )
(3)
that assigns to each input index i ∈ X a linear operator L (i): V → V on a vector space V over the field F 2 . Of course, linear automata of this type can be considered over any field but the simplest field is sufficient for our purposes. Note that, for any states u and υ from V and any i , 1 ≤ i ≤ m, the equality L (u ⊕ υ , i) = L (u , i) ⊕ L ( υ , i) is true. In other words, these automata are linear in states but the function L itself can be nonlinear. Therefore, linear automata of this type were called general (generalized) linear automata [9] since they generalize ordinary linear automata that are linear both in states and in inputs [10]. The dimension of a linear automaton is understood to be the dimension of its state space. As in the case of abstract automata, mapping (3) can be extended to a homomorphism of monoids L: X * → End (V ). The image of the homomorphism L ( X ) * = L ( X * ) is a finite multiplicative monoid of linear operators, which is called the monoid of the automaton L. When the basis in the space V is fixed, this monoid is isomorphic to the multiplicative monoid of n × n matrices over the binary field, where n is the dimension of the space V. A subset U ⊆ V that is invariant under the action of all operators L (i), 1 ≤ i ≤ m, determines a subautomaton L U of the automaton L. We call a subautomaton closed if the subset that determines it is a subspace. It is obvious that a closed subautomaton itself is a linear automaton and any subautomaton of a linear automaton can be extended to a closed subautomaton by the linear closure of the set of its states. If the invariant subset U contains the basis of the space V, then we call L U the total subautomaton of the automaton L. Note that any n-dimensional vector space over the binary field consists of 2 n vectors and is isomorphic to the space F 2n of binary vectors of length n with componentwise addition modulo two. In particular, the set of all the subsets of a set S with n elements is isomorphic to the space F 2n if the subsets are summed modulo two. In other words, their symmetric difference T ⊕ U = ( T \ U ) ∪ (U \ T ) must be used in the capacity of the sum of the subsets. The isomorphism of these spaces is established by assigning the characteristic binary vector χ( T ) of length n to each subset of T . This allows us to embed any finite automaton A: X → S S with n states into a linear automaton L A : X → End ( F 2n ) as its subautomaton if we define the transition function for any subset T = {s1 , . . . , s k } and all i ∈ X by the following formula: L A ( χ( T ), i) = χ( A( s1 , i)) ⊕ . . . ⊕ χ( A( s k , i)). As is obvious, the automaton that is a subautomaton of the above automaton and operates on one-element subsets is isomorphic to the automaton A; therefore, we call the automaton L A the standard linear extension of the automaton A over the field F 2 . Thus, we obtain the statement formulated below. Statement 3. Any finite automaton with n states can be embedded into a linear automaton of dimension n as its total subautomaton. We now consider linear 0-automata. The zero state is persistent in any linear automaton by virtue of the linearity of its transition function. But a linear automaton is a 0-automaton only if the zero state is reachable from all the states of the automaton. We assume that a linear operator is zeroth if it maps all the vectors (states) into zero. By the tradition accepted in linear algebra, we also denote the zero operator by zero. 673
We denote the zero ideal in the linear 0-automaton L by Nul ( L ) = {w| L ( w) = 0} and the set of zero words in its subautomaton L U by Nul ( L U ), accordingly. The obvious statement formulated below will be useful in what follows. Statement 4. For any total subautomaton L U of the linear 0-automaton L, the equality Nul ( L U ) = Nul ( L ) is true. Proof. In fact, the inclusion Nul ( L ) ⊆ Nul ( L U ) is obvious from the definition. On the other hand, if an input word transforms all the states of a subautomaton into zero, then it must also transform any of their linear combinations into zero. Therefore, the reverse inclusion must also be fulfilled. The statement is proved. We now pass to the formulation of a linear analog of Theorem 3. If A is a reset automaton, then, as is easily seen, the 0-automaton of pairs A2 is isomorphically mapped into the linear extension L A of the automaton A over the field F 2 . To this end, the corresponding vector χ( s ⊕ t ) must be assigned to each pair of different states {s , t} and then the entire set of pairs is mapped on a subset of n( n − 1) / 2 vectors D2 = {χ( s j ⊕ s k ) |1 ≤ j < k ≤ n}. The zero state of the linear automaton must be assigned to the zero state of the automaton A2 . Taking the linear closure of the subset D2 in the space F 2n , we obtain a linear automaton L ( A2 ) of dimension n − 1 since the set of vectors {χ( s1 ⊕ s k ) | 2 ≤ k ≤ n} forms the basis of its state space. We call the automaton L ( A2 ) the linear automaton of pairs for the automaton A. As a result, we obtain the statement formulated below. THEOREM 5. For any reset automaton A with n states, its linear automaton of pairs L ( A2 ) is a 0-automaton of dimension n − 1 and satisfies the condition Res ( A) = Nul ( L ( A2 )). Proof. It follows from Theorem 3 that we have Res ( A) = Nul ( A2 ). On the other hand, the automaton of pairs A2 is isomorphic to the total subautomaton of the automaton L ( A2 ) and, hence, according to Statement 4, the equality Nul ( A2 ) = Nul ( L ( A2 )) is true. Thus, the theorem is proved. This theorem can be used to solve the Cerny problem but, to this end, the construction of the global linear automaton should be refined. Since the image Im( L (i)) of a linear operator L (i) is a subspace, the global automaton 2 L operates on the finite collection of subspaces {U |U ∈ Im( L ( X ) * )} as follows: 2 L (U , i) = {L ( υ , i) | υ ∈U} for all i ∈ X . This operation yields, in particular, the initial representation of the zero ideal Nul ( L ) = {w | 2 L (V , w) = 0}.
(4)
We now prove a statement that can be considered as the first step in the linear treatment of the Cerny problem. THEOREM 6. For any two-dimensional linear 0-automaton over the field F 2 , there exists a zero word whose length is no more than four. Proof. In fact, in a two-dimensional linear automaton over the field F 2 , only the following five subspaces exist: zeroth, three one-dimensional subspaces, and the entire space. It follows from this and representation (4) that a zero word whose length is not more than four must exist in a two-dimensional 0-automaton. The theorem is proved. When n = 3, the validity of the Cerny hypothesis follows from the above statement and Theorem 5. COROLLARY 2. For any reset automaton with three states, there exists a reset word whose length is no more than four. At the next step, the methods used in the above theorem cannot be used in an automaton of dimension three. In this case, the linear automaton has 8 states and the number of subspaces can be up to 16. Therefore, it is no wonder that there are automata whose zero word length is more than nine. In fact, let us consider a three-dimensional linear 0-automaton that is specified over the field F 2 by the following transition operators: L (1) = [(000), (100), (110)], L (2) = [(010), (100), (011)]. Here, the images of the basic vectors (001), (010), and (100) are only specified that, by virtue of linearity, completely determine the automaton. By direct construction of the global automaton, one can make sure that the input words 12221212221 and 12222212221 are the shortest zero words of length 11 for this automaton. This example, in particular, refutes the author’s hypothesis that, in a linear 0-automaton over the field F 2 , there exists a zero word whose length does not exceed the square of the dimension of the automaton [1].
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CONCLUSIONS Thus, as is shown in this article, regular ideals can be specified implicitly (a weak representation) when an ideal is specified as the intersection of regular events represented by different states of the same automaton. In the case of its explicit initial representation, the number of states exponentially increases. As has been shown, the class of ideals is invariant to the actions of many automaton functors and is eventually proved to be equal to the class of zero ideals. We also note that, in reducing the Cerny problem to 0-automata, the distinctive features of the pair functor specified in Theorem 3 must obviously be taken into account. These distinctive features will be considered in a special work.
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