arXiv subjects

A. N. Trahtman

Publications and source records attributed to A. N. Trahtman.

At least 19 recordsLinked to original sources

arXiv2022

Row monomial matrices and Černy conjecture, short proof

The class of row monomial matrices (one unit and rest zeros in every row) with some non-standard operations of summation and usual multiplication is our main object. These matrices generate a space with respect to the mentioned operations. A word w of letters on edges of underlying graph of deterministic finite automaton (DFA) is called synchronizing if w sends all states of the automaton to a unique state J. Černy discovered in 1964 a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n-1)(n-1). The hypothesis, well known today as the Černy conjecture, claims that (n-1)(n-1) is also precise upper bound on the length of such a word for a complete DFA. The hypothesis was formulated in 1966 by Starke. The problem has motivated great and constantly growing number of investigations and generalizations. We present the proof of the Černy-Starke conjecture: the deterministic complete n-state synchronizing automaton has synchronizing word of length at most (n-1)(n-1). The proof used connection between dimension of the space and the length of words on paths of edges in underlying graph of automaton.

cs.FL↗

arXiv2022

The algebra of row monomial matrices

We consider an algebra with non-standard operations on the class of row monomial matrices (having one unit and rest of zeros in every row). The class of row monomial matrices is closed under multiplication, but not closed under ordinary matrix addition. The paper considers a kind of summation operation on row monomial matrices and the necessary conditions to be closed under the operation in this class. The most significant difference between the algebra of row monomial matrices and linear algebra is the summation operation, with respect to which the class of row monomial matrices is closed. The operation of summation in the algebra can be considered also as an algebra of subsets of any set. The class of subsets of given set is closed under considered operation of summation. The deterministic finite automaton (DFA) can be presented by a complete underlying graph of the automaton with edges labelled by letters of an alphabet. Row monomial matrices can be induced by words in the alphabet of labels on edges of underlying graph of DFA and present a mapping of the set of states. A word $s$ of letters on edges of underlying graph $Γ$ of deterministic finite automaton (DFA) is called synchronizing if $s$ sends all states of the automaton to a unique state. J. Černy discovered in 1964 a sequence of $n$-state complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. The hypothesis, mostly known today as Černy conjecture, claims that $(n-1)^2$ is a precise upper bound on the length of such a word over alphabet $Σ$ of letters on edges of $Γ$ for every complete $n$-state DFA. The hypothesis was formulated in 1966 by Starke. The following proof of the conjecture uses the algebra of row monomial matrices.

cs.FL↗

arXiv2022

A Partially Synchronizing Coloring

Given a finite directed graph, a coloring of its edges turns the graph into a finite-state automaton. A k-synchronizing word of a deterministic automaton is a word in the alphabet of colors at its edges that maps the state set of the automaton at least on k-element subset. A coloring of edges of a directed strongly connected finite graph of a uniform outdegree (constant outdegree of any vertex) is k-synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a k-synchronizing word. For k=1 one has the well known road coloring problem. The recent positive solution of the road coloring problem implies an elegant generalization considered first by Beal and Perrin: a directed finite strongly connected graph of uniform outdegree is k-synchronizing iff the greatest common divisor of lengths of all its cycles is k. Some consequences for coloring of an arbitrary finite digraph are presented. We describe a subquadratic algorithm of the road coloring for the k-synchronization implemented in the package TESTAS. A new linear visualization program demonstrates the obtained coloring. Some consequences for coloring of an arbitrary finite digraph and of such a graph of uniform outdegree are presented.

cs.FL↗

arXiv2022

A polynomial time algorithm for local testability and its level

A locally testable semigroup S is a semigroup with the property that for some nonnegative integer k, called the order or level of local testability, two words u and v in some set of generators for semigroup S are equal in the semigroup if (1) the prefix and suffix of the words of length k coincide, and (2) the set of intermediate substrings of length k of the words coincide. The local testability problem for semigroups is, given a finite semigroup, to decide, if the semigroup is locally testable or not. Recently, we introduced a polynomial time algorithm for the local testability problem and to find the level of local testability for semigroups based on our previous description of identities of $k$-testable semigroups and the structure of locally testable semigroups. The first part of the algorithm we introduce solves the local testability problem. The second part of the algorithm finds the order of local testability of a semigroup. The algorithm is of quadratic order where n is the order of the semigroup.

cs.FL↗

arXiv2022

The Černy conjecture

A word $w$ of letters on edges of underlying graph $Γ$ of deterministic finite automaton (DFA) is called synchronizing if $w$ sends all states of the automaton to a unique state. J. Černy discovered in 1964 a sequence of $n$-state complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. The hypothesis, well known today as the Černy conjecture, claims that it is also precise upper bound on the length of such a word for a complete DFA. The hypothesis was formulated in 1966 by Starke. The problem has motivated great and constantly growing number of investigations and generalizations. To prove the conjecture, we use algebra w on a special class of row monomial matrices (one unit and rest zeros in every row), induced by words in the alphabet of labels on edges. These matrices generate a space with respect to the mentioned operation. The proof is based on connection between length of words $u$ and dimension of the space generated by solutions $L_x$ of matrix equation $M_uL_x=M_s$ for synchronizing word $s$, as well as on the relation between ranks of $M_u$ and $L_x$.

cs.DM↗

arXiv2021

The length of a minimal synchronizing word and the Černy conjecture

A word w of letters on edges of underlying graph Gamma of deterministic finite automaton (DFA) is called the synchronizing word if w sends all states of the automaton to a unique state. J. Cerny discovered in 1964 a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n-1)^2. The hypothesis, well known today as the Cerny conjecture, claims that it is also precise upper bound on the length of such a word for a complete DFA. This simple looking conjecture is arguably the most fascinating and longstanding open problem in the theory of finite automaton. The hypothesis was formulated in 1966 by Starke. The problem has motivated great and constantly growing number of investigations and generalizations. To prove the conjecture, we use algebra with non-standard operations on a special class of matrices (row monomial), induced by words in the alphabet of labels on edges. These matrices generate a space with respect to the mentioned operation. The proof is based on connection between length of words u and dimension of the space generated by solutions Lx of matrix equation MuLx =Ms for synchronizing word s, as well as on the relation between ranks of Mu and Lx.

cs.FL↗

arXiv2021

Verification Tools for Checking some kinds of Testability

A locally testable language L is a language with the property that for some non negative integer k, called the order of local testability, whether or not a word u is in the language L depends on (1) the prefix and suffix of the word u of length k + 1 and (2) the set of intermediate substrings of length k of the word u. For given k the language is called k-testable. The local testability has a wide spectrum of generalizations. A set of procedures for deciding whether or not a language given by its minimal automaton or by its syntactic semigroup is locally testable, right or left locally testable, threshold locally testable, strictly locally testable, or piecewise testable was implemented in the package TESTAS written in C=C++. The bounds on order of local testability of transition graph and order of local testability of transition semigroup are also found. For given k, the k-testability of transition graph is verified. We consider some approaches to verify these procedures and use for this aim some auxiliary programs. The approaches are based on distinct forms of presentation of a given finite automaton and on algebraic properties of the presentation. New proof and fresh wording of necessary and sufficient conditions for local testability of deterministic finite automaton is presented.

cs.FL↗

arXiv2021

A package TESTAS for checking some kinds of testability

We implement a set of procedures for deciding whether or not a language given by its minimal automaton or by its syntactic semigroup is locally testable, right or left locally testable, threshold locally testable, strictly locally testable, or piecewise testable. The bounds on order of local testability of transition graph and order of local testability of transition semigroup are also found. For given k, the k-testability of transition graph is verified. Some new effective polynomial time algorithms are used. These algorithms have been implemented as a C/C++ package.

cs.FL↗

arXiv2021

An algorithm to verify local threshold testability of deterministic finite automata

A locally threshold testable language L is a language with the property that for some non negative integers k and l, whether or not a word u is in the language L depends on (1) the prefix and suffix of the word u of length k > 1 and (2) the set of intermediate substrings of length k of the word u where the sets of substrings occurring at least j times are the same, for j <= L. For given k and L the language is called l-threshold k-testable. A finite deterministic automaton is called l-threshold k-testable if the automaton accepts a l-threshold k-testable language. In this paper, the necessary and sufficient conditions for an automaton to be locally threshold testable are found. We introduce the first polynomial time algorithm to verify local threshold testability of the automaton based on this characterization. New version of polynomial time algorithm to verify the local testability will be presented too.

cs.FL↗

arXiv2021

The Černy Conjecture for aperiodic automata

A word w is called a synchronizing (recurrent, reset) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some state; a DFA that has a synchronizing word is said to be synchronizing. Cerny conjectured in 1964 that every n-state synchronizing DFA possesses a synchronizing word of length at most (n -1)2. We consider automaton with aperiodic transition monoid (such automaton is called aperiodic). We show that every synchronizing n-state aperiodic automaton has a synchronizing word of length at most n(n-2)+1. Thus, for aperiodic automaton as well as for automatons accepting only star-free languages, the Cerny conjecture holds true.

cs.FL↗

arXiv2021

Cerny-Starke conjecture from the sixties of XX century

A word $s$ of letters on edges of underlying graph $Γ$ of deterministic finite automaton (DFA) is called synchronizing if $s$ sends all states of the automaton to a unique state. J. Černy discovered in 1964 a sequence of $n$-state complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. The hypothesis, mostly known today as Černy conjecture, claims that $(n-1)^2$ is a precise upper bound on the length of such a word over alphabet $Σ$ of letters on edges of $Γ$ for every complete $n$-state DFA. The hypothesis was formulated in 1966 by Starke. Algebra with nonstandard operation over special class of matrices induced by words in the alphabet of labels on edges is used to prove the conjecture. The proof is based on the connection between length of words $u$ and dimension of the space generated by solution $L_x$ of matrix equation $M_uL_x=M_s$ for synchronizing word $s$, as well as on relation between ranks of $M_u$ and $L_x$. Important role below placed the notion of pseudo inverseL matrix, sometimes reversible.

cs.FL↗

arXiv2020

Precise estimation on the order of local testability of deterministic finite automaton

A locally testable language L is a language with the property that for some non negative integer k, called the order or the level of local testable, whether or not a word u in the language L depends on (1) the prefix and the suffix of the word u of length k-1 and (2) the set of intermediate partial strings of length k of the word u. For given k the language is called k-testable. We give necessary and sufficient conditions for the language of an automaton to be k-testable in the terms of the length of paths of a related graph. Some estimations of the upper and of the lower bound of testable order follow from these results. We improve the upper bound on the testable order of locally testable deterministic finite automaton with n states to n(n-2)+1 This bound is the best possible. We give an answer on the following conjecture of Kim, McNaughton and Mac-CLoskey for deterministic finite locally testable automaton with n states: \Is the local testable order of no greater than n in power 1.5 when the alphabet size is two?" Our answer is negative. In the case of size two the situation is the same as in general case.

cs.FL↗

arXiv2020

Reducing the time complexity of testing for local threshold testability

A locally threshold testable language L is a language with the property that for some non negative integers k and l and for some word u from L, a word v belongs to L if and only if (1) the prefixes [suffixes] of length k-1 of words u and v coincide, (2) the numbers of occurrences of every factor of length k in both words u and v are either the same or greater than l-1. A deterministic finite automaton is called locally threshold testable if the automaton accepts a locally threshold testable language for some l and k. New necessary and sufficient conditions for a deterministic finite automaton to be locally threshold testable are found. On the basis of these conditions, we modify the algorithm to verify local threshold testability of the automaton and to reduce the time complexity of the algorithm. The algorithm is implemented as a part of the $C/C ^{++}$ package TESTAS. \texttt{http://www.cs.biu.ac.il/$\sim$trakht/Testas.html}.

cs.FL↗

arXiv2020

Polynomial time algorithm for left [right] local testability

A right [left] locally testable language S is a language with the property that for some non negative integer k two words u and v in alphabet S are equal in the semi group if (1) the prefix and suffix of the words of length k coincide, (2) the set of segments of length k of the words as well as 3) the order of the first appearance of these segments in prefixes [suffixes] coincide. We present necessary and sufficient condition for graph [semi group] to be transition graph [semi group] of the deterministic finite automaton that accepts right [left] locally testable language and necessary and sufficient condition for transition graph of the deterministic finite automaton with locally idempotent semi group. We introduced polynomial time algorithms for the right [left] local testable problem for transition semi group and transition graph of the deterministic finite automaton based on these conditions. Polynomial time algorithm verifies transition graph of automaton with locally idempotent transition semi group.

cs.FL↗

arXiv2019

Matrix approach to synchronizing automata

cs.FL↗

arXiv2014

A Note on a Recent Attempt to Improve the Pin-Frankl Bound

We provide a counterexample to a lemma used in a recent tentative improvement of the the Pin-Frankl bound for synchronizing automata. This example naturally leads us to formulate an open question, whose answer could fix the line of proof, and improve the bound.

cs.FL↗

arXiv2014

Modifying the upper bound on the length of minimal synchronizing word

A word $w$ is called synchronizing (recurrent, reset, magic, directable) word of deterministic finite automaton (DFA) if $w$ sends all states of the automaton to a unique state. In 1964 Jan Černy found a sequence of n-state complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. He conjectured that it is an upper bound on the length of such words for complete DFA. Nevertheless, the best upper bound $(n^3-n)/6$ was found almost 30 years ago. We reduce the upper bound on the length of the minimal synchronizing word to $n(7n^2+6n-16)/48$. An implemented algorithm for finding synchronizing word with restricted upper bound is described. The work presents the distribution of all synchronizing automata of small size according to the length of an almost minimal synchronizing word.

cs.DM↗

arXiv2010

An Algorithm for Road Coloring

A coloring of edges of a finite directed graph turns the graph into finite-state automaton. The synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph of uniform outdegree (constant outdegree of any vertex) is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph of uniform outdegree if the greatest common divisor of the lengths of all its cycles is one. The problem posed in 1970 had evoked a noticeable interest among the specialists in the theory of graphs, automata, codes, symbolic dynamics as well as among the wide mathematical community. A polynomial time algorithm of $O(n^3)$ complexity in the most worst case and quadratic in majority of studied cases for the road coloring of the considered graph is presented below. The work is based on recent positive solution of the road coloring problem. The algorithm was implemented in the package TESTAS

cs.DM↗