1 Introduction↩︎

Synchronization is an important concept in various domains of computer science that consists in regaining control over a system by applying (or observing) a specific set of input instructions. These instructions are usually required to lead the system to a fixed state no matter in which state it was at the beginning. This idea has been studied for automata (deterministic [1], [2], nondeterministic [3], unambiguous [4], weighted and timed [5], partially observable [6], register [7], nested word [8]), parts orienting in manufacturing [9], [10], testing of reactive systems [11], variable length codes [12], and Markov Decision Processes [13], [14].

In this paper, we study the synchronization of partial DFAs, which are a generalization of complete DFAs and a special case of unambiguous NFAs. We are motivated by applications of this model and its connections with others, as well as the need for new techniques applied to partial DFAs. The problems for strongly connected partial DFAs are a motivation for further development and generalization of the methods applied for complete DFAs, since, as we show, these models are closely related. We also hope that our methods will serve as a step toward studying a wider class of strongly connected unambiguous NFAs.

1.1 Observing a reactive system↩︎

Consider a finite-state reactive system modeled by a partial DFA (by partial we mean that for some states there can be no outgoing transitions corresponding to some letters). The observer knows the structure of the DFA but does not know its current state. At every step, the DFA reads a letter (also known to the observer) and transits to another state. The observer wants to eventually learn the actual state of the DFA. Since the DFA is deterministic, once a state is known, it will be known forever.

In this setting, the actual state is known if and only if the system reads a reset word – a word that transits a non-empty set of states to a single state and is undefined for all other states. The presence of undefined transitions indicates that certain actions cannot be performed from certain states, which can be essential for synchronization.

For several identical systems running in parallel and receiving the same input (but possibly starting from different states), the presence of a reset word in the input guarantees that all systems end up in the same state. This idea can be used in robotics, where a sequence of passive obstacles is used for orienting a large number of arbitrarily rotated parts arriving simultaneously on a conveyor belt ([9], [10], see also [15] for an illustrative example).

Reactive systems (such as Web servers, communication protocols, operating systems and processors) are systems developed to run without termination and interact through visible events, so it is natural to assume that the system can return to any state from any other state (NFAs with this property are called strongly connected). The probabilistic version of the described problem for strongly connected partial DFAs has been considered in the context of \(\varepsilon\)-machines [16]. In particular, the observer knows the state of an \(\varepsilon\)-machine precisely if and only if a reset word for the underlying partial DFA was applied. Some experimental results on finding shortest reset words for partial DFAs were recently presented in [17].

1.2 Synchronizing automata↩︎

There exist several definitions that generalize the notion of a synchronizing complete DFA to larger classes of NFAs. In this subsection, we describe the notion which preserves most of the properties of the complete DFAs case, and in 1.5 we briefly describe alternative notions.

An NFA is called unambiguous if for every two states \(p, q\) and every word \(w\), there is at most one path from \(p\) to \(q\) labeled by \(w\) [4]. In the strongly connected case, this is equivalent to a more classical definition of an unambiguous NFA with chosen initial and final states, if there is a unique initial state and a unique final state. An unambiguous NFA is called synchronizing if there exist two non-empty subsets \(C, R\) of its states and a word \(w\) (called a reset word) such that its action maps every state in \(C\) exactly to the whole set \(R\), and is undefined for all states outside \(C\) [4]. For partial DFAs, the set \(R\) has size one [12], and, for complete DFAs, the set \(C\) is also the whole set of states [2].

Partial DFAs are thus a natural intermediate class between unambiguous NFAs and complete DFAs. The bounds on the length of shortest reset words in strongly connected partial DFAs have not been studied before. The famous Černý conjecture, which is one of the most longstanding open problems in automata theory, states that for an \(n\)-state complete DFA we can always find a reset word of length at most \((n-1)^2\), unless there are no reset words. The best known upper bound is cubic in \(n\) [18], [19], and the problem of deciding whether a complete DFA is synchronizing is solvable in time quadratic in \(n\) [2]. For an \(n\)-state strongly connected unambiguous NFA, the best known upper bound on the length of a shortest reset word is \(n^5\), and the existence of a reset word is verifiable in polynomial time [20]. The same upper bound holds for the length of the shortest mortal words in strongly connected unambiguous NFAs [21], whereas partial DFAs admit a tight quadratic bound [22].

1.3 Synchronizing codes↩︎

A variable-length code \(X\) (which we call a code) is a set of finite words over a finite alphabet \(\Sigma\), such that no word over \(\Sigma\) can be written as a concatenation of codewords of \(X\) in two different ways. Such codes (especially Huffman codes [23]) are widely used for lossless data compression. Since the lengths of codewords can be different, one transmission error can spoil the whole decoding process, causing a major data loss. Also, for general codes, decoding a part of a message (e.g., a segment of a compressed video stream) is not possible without decoding the whole message.

These issues can be addressed by using synchronizing codes. A code \(X\) is called synchronizing if there exists a synchronizing word \(w \in X^*\) such that for every \(uwv \in X^*\) we have \(uw, wv \in X^*\). The occurrence of the word \(ww\) thus stops error propagation and allows parallel decoding of the two parts of the message. More generally, each appearance of the word \(ww\) in a coded message allows to run the decoding independently from the position after the first \(w\).

A code is called prefix if none of its codewords is a prefix of another codeword. Such codes allow to obtain the correct partition of a message into codewords one by one by going from left to right. Even if a code is synchronizing, there are no guarantees that a synchronizing word will appear in a message. Codes where every long enough concatenation of codewords is synchronizing are called uniformly synchronizing [12], [24]. A prefix code is called maximal if it is not a subset of another prefix code. All non-trivial uniformly synchronizing finite prefix codes are non-maximal [12].

1.4 Automata for the Kleene star of codes↩︎

A code recognized by an NFA as a language is called recognizable. In particular, every finite code is recognizable. To argue about synchronization properties of a recognizable code \(X\), special NFAs recognizing \(X^*\) are studied. These NFAs have a unique initial and final state \(r\) such that the set of words labeling paths from \(r\) to itself coincides with \(X^*\), thus they are also strongly connected. Provided a recognizable code \(X\), an NFA with the described properties can be chosen to be unambiguous [12]. Moreover, this NFA can be chosen to be a partial (respectively, a complete) DFA if and only if \(X\) is a recognizable prefix (respectively, recognizable maximal prefix) code [12].

For such an unambiguous NFA with the properties as above, \(X\) is synchronizing if and only if the NFA is synchronizing, and the length of a shortest synchronizing word for \(X\) is at most the length of a shortest reset word of the NFA plus twice its number of states [12].

Finite prefix codes admit a direct construction of partial DFAs with the described properties, called literal (or prefix) automata. Let \(X\) be a finite prefix code over an alphabet \(\Sigma\). The literal automaton \(\mathrsfs{A}_X = (Q,\Sigma,\delta)\) is constructed as follows. The set of states \(Q\) is the set of all proper prefixes of the words in \(X\), and the transition function is defined as follows: \(\delta(q,x) = qx\) if \(qx \notin X\) and \(qx\) is a proper prefix of a word in \(X\), \(\delta(q,x) = \varepsilon\) if \(qx \in X\), and \(\delta(q,x) = \bot\) otherwise. The state corresponding to the empty prefix \(\varepsilon\) is called the root state. The height of a literal automaton is the length of a longest path of its transitions without repetition of states; equivalently, this is the length of the longest word in \(X\) minus one. Note that the number of states of \(\mathrsfs{A}_X\) is at most the total length of all codewords of \(X\), which allows to directly transfer upper bounds from literal automata to finite prefix codes. An example of a literal automaton is shown in 1 (right). The literal automaton of a prefix code can be used as a decoder for this code by adding output labels to the transitions [12].

1.5 Carefully synchronizing DFAs↩︎

For general NFAs, synchronizability can be generalized to D\(i\)-directability for \(i = 1, 2, 3\) [3]. As discussed in [25], for partial DFAs the notions of D\(1\)- and D\(3\)-directing words both coincide with carefully synchronizing words. These are words sending every state of a partial DFA to the same state, without using any undefined transitions. A D\(2\)-directing word for a partial DFA is either carefully synchronizing or mortal (undefined for every state). The definitions of carefully synchronizing and D\(2\)-directing words are different from our definition of synchronizing words for partial DFAs.

A carefully synchronizing word can be applied to a partial DFA at any moment without the risk of using an undefined transition. This comes at a high cost: even for strongly connected partial DFAs, the shortest carefully synchronizing words can have exponential length [25], and the problem of checking the existence of such a word is PSPACE-complete [25], in contrast with the case of complete DFAs. On the contrary, the notion of a synchronizing partial DFA preserves most of the properties of a synchronizing complete DFA, at least in the strongly connected case. Note that every carefully synchronizing word is synchronizing, but the converse is not true.

While for complete DFAs the property of being strongly connected is not essential for many synchronization properties [2], the situation changes dramatically for partial DFAs. Partial DFAs that are not strongly connected can have exponentially long shortest reset words, and the problem of checking the existence of a reset word is PSPACE-complete [26]. Thus, strong connectivity is indeed necessary to obtain good bounds and algorithms. As explained above, for reactive systems and prefix codes this requirement comes naturally.

1.6 Our contribution and organization of the paper↩︎

We prove a number of results for strongly connected partial DFAs connected with the Černý conjecture and its generalizations. Where possible, we suggest methods that allow to relate the partial case with the complete case, instead of directly reproving known results in this more general setting. In this way, we do not have to go into the existing proofs, and future findings concerning the complete case should be often immediately transferable to the partial case.

We start from basic properties and introduce more advanced techniques along with their applications. First, we investigate the rank conjecture, which is a generalization of the Černý conjecture from the case of synchronizing automata to the case of all automata. We show that the rank conjecture for complete DFAs implies an analogous statement for partial DFAs (2). For this, we introduce our first basic tool called a fixing automaton, which is a complete DFA obtained from a partial one and sharing some of its properties. Our result shows a general way for transferring upper bounds from the case of complete DFAs to partial DFAs, e.g., we immediately get that the rank conjecture holds true for partial Eulerian automata (3).

To connect the Černý conjecture for the cases of complete and partial DFAs, we need more involved techniques, since the construction developed for the rank conjecture does not preserve the property of being synchronizing. We introduce a collecting automaton, which extends the concept of the fixing automaton. We use it to show that all upper bounds on the length of the shortest reset words, up to a subquadratic additive component (linear in the case of the Černý bound), are equivalent for partial and complete DFAs (5). We also use it to prove that the problems of deciding synchronizability and finding a reset word of a strongly connected partial DFA can be effectively reduced to the same problems for a complete DFA (3.5). This also means that possible improvements of the complexity of the best-known algorithms for these problems for complete DFAs should directly apply to partial DFAs.

As discussed in 1.3 and 1.4, one of the main motivations for studying synchronization of strongly connected partial DFAs is a direct correspondence with synchronization of recognizable prefix codes. An important special case is when the prefix code is finite. We investigate it by studying literal automata of finite prefix codes and obtain stronger upper bounds than those for the general case of strongly connected partial (or complete) DFAs. We show that the length of the shortest reset words for literal automata of finite prefix codes is at most \(\mathcal{O}(n \log^3 n)\), where \(n\) is the number of states of the automaton (5). This upper bound asymptotically matches the strongest known upper bound for maximal prefix codes (which becomes now a special case), but it is not transferred directly, as key statements do not hold in the same way for non-maximal prefix codes. To prove it, we first show that the literal automaton of a finite prefix codes admits a word of linear length whose action sends all the states to a non-empty subset of small size (7). It establishes a natural combinatorial property of finite prefix codes and constitutes the most involved proof in this paper. Once we show the existence of such a word, we use one more construction called the induced automaton, which is a generalization of linear algebraic techniques to the case of partial DFAs (3.6). This particular construction extends the existing techniques originally developed for complete DFAs but simultaneously comes with a new simpler and more general proof.

Finally, we show that the lower bounds for strongly connected partial DFAs are asymptotically the same even if we ensure the existence of undefined transitions (4). In other words, undefined transitions do not help in general, as we cannot significantly improve upper bounds for such automata without doing that for the complete case.

This paper is the full version of a conference paper [27].

2 Preliminaries↩︎

A partial deterministic finite automaton \(\mathrsfs{A}\) (a partial DFA for short) is a triple \((Q,\Sigma,\delta)\), where \(Q\) is a set of states, \(\Sigma\) is an input alphabet, and \(\delta\) is partial function \(Q \times \Sigma \rightharpoonup Q\) called the transition function. Note that the automata we consider do not have any initial or final states. We extend \(\delta\) to a partial function \(Q \times \Sigma^* \rightharpoonup Q\) as usual: we set \(\delta(q, wa) = \delta(\delta(q,w), a)\) for \(w \in \Sigma^*\) and \(a \in \Sigma\). For a state \(q \in Q\) and a word \(w \in \Sigma^*\), if the action \(\delta(q,w)\) is undefined, then we write \(\delta(q,w)=\bot\). Note that if \(\delta(q,w) = \bot\) for a word \(w \in \Sigma^*\), then \(\delta(q,wu) = \bot\) for every word \(u \in \Sigma^*\). A DFA is complete if all its transitions are defined, and it is incomplete otherwise. An partial DFA is strongly connected if for every two states \(p,q \in Q\) there is a word \(w \in \Sigma^*\) such that \(\delta(p,w) = q\).

By \(\Sigma^i\) we denote the set of all words over \(\Sigma\) of length exactly \(i\), and by \(\Sigma^{\le i}\) the set of all words over \(\Sigma\) of length at most \(i\). For two sets of words \(W_1, W_2 \subseteq \Sigma^*\), by \(W_1 W_2\) we denote their product \(\{w_1 w_2 \in \Sigma^* \mid w_1 \in W_1, w_2 \in W_2\}\). The empty word is denoted by \(\varepsilon\). Throughout the paper, by \(n\) we always denote the number of states \(|Q|\).

Given \(S \subseteq Q\), the image of \(S\) under the action of \(w\) is \(\delta(S,w) = \{\delta(q,w) \mid q \in S,\;\delta(q,w) \neq \bot\}\). The preimage of \(S\) under the action of \(w\) is \(\delta^{-1}(S,w) = \{q \in Q \mid \delta(q,w) \in S\}\). Since \(\mathrsfs{A}\) is deterministic, for disjoint subsets \(S,T \subseteq Q\), their preimages under the action of every word \(w \in \Sigma^*\) are also disjoint.

The rank of a word \(w\) is the size of the image of \(Q\) under the action of this word, i.e., \(|\delta(Q,w)|\). In contrast with complete DFAs, partial DFAs may admit words of rank zero; these words are called mortal. Words of non-zero rank are called non-mortal. A word of rank \(1\) is called reset, and if the DFA admits such a word then it is called synchronizing. The reset threshold \(\mathop{\mathrm{rt}}(\mathrsfs{A})\) is the length of the shortest reset words of \(\mathrsfs{A}\).

We say that a word \(w\) compresses a subset \(S \subseteq Q\), if \(\delta(S,w) \neq \emptyset\) and \(|\delta(S,w)| < |S|\). A subset that admits a compressing word is called compressible. There are two ways to compress a subset \(S \subseteq Q\) with \(|S| \ge 2\) in a partial DFA. One possibility is the pair compression, which is the same as in the case of a complete DFA, i.e., mapping at least two states \(p,q \in S\) to the same state (but not to \(\bot\)). The other possibility is to map at least one state from \(S\), but not all states from \(S\), to \(\bot\). Sometimes, a subset can be compressed in both ways simultaneously. We say that a word \(w\) synchronizes a set \(S \subseteq Q\) if \(|\delta(S,w)| = 1\).

An example of a strongly connected partial DFA is shown in 1 (left). We have two undefined transitions: \(\delta(q_3,b)=\delta(q_6,b)=\bot\). The unique shortest reset word is \(bab\): \(\delta(Q,b)=\{q_1,q_2,q_5\}\), \(\delta(Q,ba)=\{q_2,q_3,q_6\}\), and \(\delta(Q,bab) = \{q_2\}\). However, in contrast with the case of a complete DFA, the preimage \(\delta^{-1}(\{q_2\},bab) = \{q_1,q_4\}\) is not \(Q\).

3 Upper bounds↩︎

3.1 Inseparability equivalence↩︎

Let \(\mathrsfs{A}=(Q,\Sigma,\delta)\) be a partial DFA. We define the inseparability relation \(\equiv\) on \(Q\). Two states are separable if there is a word whose action is defined for exactly one of them.

The same relation is considered in [12] if all states of the partial DFA are final. Also, if we replace \(\bot\) with a unique final state, then \(\equiv\) is the well-known Myhill-Nerode congruence on words in a complete DFA. Under a different terminology, it also appears in the context of \(\varepsilon\)-machines, where non-equivalent states are called topologically distinct [16].

For a subset \(S \subseteq Q\), let \(\kappa(S)\) be the number of equivalence classes that have a non-empty intersection with \(S\). In the partial DFA from 1 (left), we have three equivalence classes, namely, \(q_1 \equiv q_4\), \(q_2 \equiv q_5\), and \(q_3 \equiv q_6\).

Our first auxiliary lemma states that every subset \(S \subseteq Q\) with \(\kappa(S) \ge 2\) can be compressed by a short word which decreases the number of intersected equivalence classes. This is done by mapping to \(\bot\) all the states of \(S\) from at least one equivalence class, but not the whole set \(S\). A linear upper bound can be inferred from a standard analysis of the corresponding Myhill-Nerode congruence (e.g., [28]), but we will need a more precise bound in terms of \(\kappa(S)\).

By an iterative application of 1, we can easily compress any subset of states to a subset of a single equivalence class.

3.2 Fixing automaton↩︎

The other possibility of compressing a subset in a partial DFA is the classical pair compression. This is the only way to compress a subset with all states in one equivalence class, which is always the case in a complete DFA.

Our tool to deal with this way of compression is the fixing automaton. This is a complete DFA obtained from a partial one, defined as follows.

We list some useful properties of the fixing automaton.

In the case of a partial DFA that is not strongly connected, it can happen that we cannot compress some subset \(S\) even when there exists a word of non-zero rank smaller than \(|S|\). This is the reason why the shortest words of the minimal non-zero rank can be exponentially long and why deciding if there is a word of a given rank is PSPACE-complete [26]. However, in the case of a strongly connected partial DFA, as well as for a (not necessarily strongly connected) complete DFA, every non-mortal word can be extended to a word of the minimal non-zero rank. This is a fundamental difference that allows constructing compressing words iteratively. Note that the fixing automaton of a strongly connected partial DFA is also strongly connected.

3.3 Rank conjecture↩︎

The rank conjecture (sometimes called Černý-Pin conjecture) is a well-known generalization of the Černý conjecture to non-synchronizing DFAs (e.g., [29]). The rank conjecture is a weaker version of the conjecture originally stated by Pin that was not restricted to the minimal rank and turned out to be false [30]. Some further results on the rank conjecture for strongly connected complete DFAs are provided in [31].

For partial DFAs, the rank conjecture is analogous with the exception that \(r\) is the minimal non-zero rank.

The theorem implies that, in the strongly connected case, the rank conjecture is true for complete DFAs if and only if it is true for partial DFAs. It also immediately implies the following result for the class of Eulerian DFAs. A partial DFA is Eulerian if it is strongly connected and the numbers of outgoing and incoming transitions are the same at every state, i.e., for every \(q \in Q\), we have \(|\{a \in \Sigma \mid \delta(q,a) \ne \bot\}| = |\{(p,a) \in Q \times \Sigma \mid \delta(p,a)=q\}|\). The following corollary follows from the facts that the rank conjecture holds for complete Eulerian DFAs [31] and that the fixing automaton of a partial Eulerian DFA is also Eulerian.

3.4 Collecting automaton↩︎

The fixing automaton allows us to analyze the behavior of words in a partial DFA by studying a complete DFA. However, its main disadvantage is that the fixing automaton of a synchronizing partial DFA is not necessarily synchronizing. Therefore, we will need one more tool, called collecting automaton. It is an extension of the fixing automaton by an additional letter that allows quick synchronization into one inseparability class, while it does not affect the length of a shortest synchronizing word for any particular inseparability class.

By \(\mathrsfs{A}/_{\equiv} = (Q_{\mathrsfs{A}/_{\equiv}},\Sigma,\delta_{\mathrsfs{A}/_{\equiv}})\), we denote the quotient DFA by the inseparability relation (clearly, this relation is a congruence). \(\mathrsfs{A}/_{\equiv}\) is also a partial DFA, and if \(\mathrsfs{A}\) is strongly connected, then so is \(\mathrsfs{A}/_{\equiv}\). By \([p] \in Q_{\mathrsfs{A}/_{\equiv}}\), we denote the equivalence class of a state \(p \in Q\) of the original DFA \(\mathrsfs{A}\).

A collecting tree of \(\mathrsfs{A}\) is a tree \(T\) with the set of vertices \(Q_{\mathrsfs{A}/_{\equiv}}\) and directed edges labeled by letters from \(\Sigma\) in the following way:

1. edges and their labels correspond to transitions in \(\mathrsfs{A}/_{\equiv}\): each edge \(([p],a,[p'])\) is such that \(\delta_{\mathrsfs{A}/_{\equiv}}([p],a) = [p']\);

2. there is a root \([r]\) such that the tree is directed toward it.

See 2 for an example. Equivalently, a collecting tree can be seen as a specific partial DFA which is a sub-automaton of \(\mathrsfs{A}/_{\equiv}\) whose underlying digraph is a tree directed toward one state. A DFA can have many collecting trees, even for the same \([r]\), and every strongly connected DFA has a collecting tree for every class \([r]\).

A collecting automaton of a strongly connected partial DFA is always strongly connected, as it contains all transitions of the original partial DFA. It is also always complete. We prove several properties connecting partial DFAs and their collecting automata. They are preliminary steps toward relating the Černý conjecture for strongly connected partial and complete DFAs.

6 implies in particular that the choice of \(T\) does not matter: \(\mathrsfs{A}=(Q,\Sigma,\delta)\) is synchronizing if and only if all \(\mathrsfs{A}^{\mathrm{C}(T)}\) are synchronizing.

3.5 Algorithmic aspects↩︎

Checking if a strongly connected partial DFA is synchronizing and finding a word of minimum non-zero rank can be done similarly as for a complete DFA, by a suitable generalization of the well-known Eppstein algorithm [9]. The same algorithm for checking synchronizability, under different terminology, was described in the context of \(\varepsilon\)-machines [16].

Furthermore, there exists an even more efficient reduction from the problem of checking if a strongly connected partial DFA is synchronizing to the problem of checking if a complete DFA is synchronizing. Hence, the existence of a more efficient algorithm for the latter would imply the existence of a more efficient algorithm for the former.

3.6 Induced automaton↩︎

We develop an algebraic technique applied to partial DFAs. It will allow us to derive upper bounds on reset thresholds, in particular, in the cases when there exists a short word of a small rank, which is the case, for example, for the literal automaton of a prefix code. We also use the results of this subsection in the next subsection to translate upper bounds on the reset threshold from complete DFAs to partial DFAs. We base on the results from [33] for complete DFAs and generalize them to be applied to partial DFAs. The existing linear algebraic proofs for complete DFAs do not work for partial ones, because the transition matrices may not have a constant sum of the entries in each row. Furthermore, our generalization simplifies the previous proof, which, to show that an induced automaton is synchronizing, uses the stationary distribution of a Markov chain defined by the automaton and the extension method (applying words that yield a larger preimage of a subset).

We need to introduce a few definitions from linear algebra for automata (see, e.g., [19], [33]–[35]). Let \(\mathrsfs{A} = (Q,\Sigma,\delta)\) be a partial DFA. Without loss of generality we assume that \(Q = \{1,\ldots,n\}\). By \(\mathbb{Q}^n\), we denote the rational¹ \(n\)-dimensional linear space of row vectors. For a vector \(v \in \mathbb{Q}^n\) and an \(i \in Q\), we denote the vector’s value at the \(i\)-th position by \(v(i)\). Similarly, for a matrix \(M\), we denote its value in the \(i\)-th row and the \(j\)-th column by \(M(i,j)\). A vector \(g\) is non-negative if \(g(i)\ge 0\) for all \(i\), and it is non-zero if \(g(i) \neq 0\) for some \(i\). For a word \(w \in \Sigma^*\), by \(\mathrm{M}(w)\) we denote the \(n \times n\) matrix of the transformation of \(w\) in \(\delta\): \(\mathrm{M}(w)(p,q) = 1\) if \(\delta(p,w)=q\), and \(\mathrm{M}(w)(p,q)=0\) otherwise. Note that if \(\delta(p,w)=\bot\), then we have \(\mathrm{M}(w)(p,q)=0\) for all \(q \in Q\). The usual scalar product of two vectors \(u,v\) is denoted by \(u \odot v\). The linear subspace spanned by a set of vectors \(V\) is denoted by \(\mathop{\mathrm{span}}(V)\).

Given a transition function \(\delta\) (which defines matrices \(\mathrm{M}(w)\)), call a set of words \(W \subseteq \Sigma^*\) complete for a subspace \(V \subseteq \mathbb{Q}^n\) with respect to a vector \(g \in V\), if \(V \subseteq \mathop{\mathrm{span}}(\{g \mathrm{M}(w) \mid w \in W\})\). A set of words \(W \subseteq \Sigma^*\) is complete for a subspace \(V \subseteq \mathbb{Q}^n\) if for every non-negative non-zero vector \(g \in V\), \(W\) is complete for \(V\) with respect to \(g\). Let \(\chi(p)\) denote the characteristic (unitary) vector of \(\{p\}\). For a subset \(S \subseteq Q\), we define \(\mathbb{V}(S) = \mathop{\mathrm{span}}(\{\chi(p) \mid p \in S\}) \subseteq \mathbb{Q}^n\).

For example, consider the DFA from 1 (left). Let \(V = \mathbb{V}(\{q_1,q_2,q_5\})\) and let \(W = \{ab,aab\}a^{\le 5}\). Let \(g \in V\) be a non-negative non-zero vector, and let \(i\) be such that \(g(i) \neq 0\). If \(i=1\) then let \(u=ab\), and otherwise let \(u=aab\); then \(g \mathrm{M}(u)\) has exactly one non-zero entry. Then, for each \(j \in \{1,2,5\}\), the vector \(g \mathrm{M}(u a^{j'})\) for some \(j'\) has the unique non-zero entry at \(q_j\). These vectors generate \(V\), thus \(W\) is complete for \(V\) with respect to \(g\).

The induced automaton of a partial DFA is a partial DFA acting on a subset of states \(R \subseteq Q\). It is built from two sets of words. Let \(W_1\) be a set of words such that \(R = \bigcup_{w \in W_1} \delta(Q,w)\). Thus, for each state in \(R\), there is some state mapped to it by a word from \(W_1\). Intuitively, the second set \(W_2\) is any non-empty set of words that enriches the actions of words from \(W_1\). The induced automaton is \(\mathrsfs{A}\) restricted to \(R\) with alphabet \(W_2 W_1\). Note that its transition function is well defined, which is ensured by the fact that every word of the form \(w_2 w_1 \in W_2 W_1\) has the action mapping every state \(q \in Q\) into \(R\) or to \(\bot\).

We can analyze an induced automaton as a separate one, and synchronize the whole DFA using it, which is particularly profitable when \(R\) is small. Following our previous example, for 1 (left) with \(W_1 = \{b\}\) and \(W_2 = \{ab,aab\}a^{\le 5}\) we obtain the induced automaton on \(R=\{q_1,q_2,q_5\}\). Furthermore, it is synchronizing already by a letter from \(W_2 W_1\) (e.g., \(abb\)), and each reset word corresponds to a reset word of the original \(\mathrsfs{A}\).

The following lemma states that the completeness of a set of words together with the synchronizability and strong connectivity of the whole DFA transfer to the induced automaton. It generalizes [33] to partial DFAs, and the proof uses a recursion instead of an augmenting argument.

The following corollary directly follows from 7, since \(\Sigma^{\leq n-1}\) is always complete for \(\mathbb{V}(Q)\) in the case of a strongly connected synchronizing partial DFA.

The corollary is useful for deriving upper bounds for DFAs with a word of small rank. Having such a word \(w\), we can further synchronize \(R\) through the induced automaton instead of trying to do this directly. Although every letter of \(\mathrsfs{A}^{\mathrm{I}(W_1,W_2)}\) corresponds to a word of length \(|w|+n-1\) in the original \(\mathrsfs{A}\), if \(R\) is small enough, this yields a better upper bound. We show its application in the next subsection.

3.7 The Černý conjecture↩︎

The famous Černý conjecture is a special case of the rank conjecture for complete DFAs for \(r=1\). Let \(\mathfrak{C}(n)\) be the maximum length of the shortest reset words of all \(n\)-state synchronizing complete DFAs. It is well known that \(\mathfrak{C}(n) \ge (n-1)^2\) [1]. The Černý conjecture states that \(\mathfrak{C}(n) = (n-1)^2\), but the best proved upper bound is cubic [18], [19].

Let \(\mathfrak{C}_\mathrm{P}(n)\) be the maximal length of the shortest reset words of all \(n\)-state synchronizing strongly connected partial DFAs. We show that an improvement of the upper bound on \(\mathfrak{C}(n)\) implies a similar (but slightly weaker) improvement on \(\mathfrak{C}_\mathrm{P}(n)\). In particular, the Černý conjecture implies that \(\mathfrak{C}_\mathrm{P}(n) \le n^2 + \mathcal{O}(n)\).

To prove the following theorem, we combine several techniques, in particular, the inseparability equivalence, the collecting automaton, and an algebraic upper bound on the reset threshold of a complete DFA with a short word of small rank [33].

Note that, in particular, if \(\mathfrak{C}(n) = (n-1)^2\) then the extra component is linear. For \(\mathfrak{C}(n) \le n^3\), it is \(+ \mathcal{O}(n^{4/3})\).

From 5, it follows that all upper bounds on the reset threshold of a complete DFA transfer to upper bounds for partial DFAs, up to a subquadratic component. Thus \(\mathfrak{C}_\mathrm{P}(n) \le 0.1654 n^3 + \mathcal{O}(n^2)\) [18]. The extra component is likely not needed, but it is difficult to completely get rid of it in general, as for that we could not lengthen by any means the reset word assumed for a complete DFA. However, it is easy to omit it when reproving particular bounds for complete DFAs, both combinatorial [29] and based on avoiding words [18], [19]. We conjecture that \(\mathfrak{C}_\mathrm{P}(n) = \mathfrak{C}(n)\) for all \(n\).

3.8 The literal automaton of a finite prefix code↩︎

We now use the obtained results about induced automata to get better bounds for partial literal automata of finite prefix codes (defined in 1.4). To do so, we first need to prove that such automata admit short enough words of small rank. It is known that every complete literal automaton over an alphabet \(\Sigma\) has a word of rank and length at most \(\lceil\log_{|\Sigma|} n\rceil\) ([33], cf.[36]). However, this is no longer true for non-mortal words in partial literal automata [37] and no similar statement was known for any wider class than complete literal automata. We prove that there exist \(\mathcal{O}(\log n)\)-rank non-mortal words of length \(\mathcal{O}(n)\) in such automata, excluding the case of a code with only one word. Then we use this result to provide an \(\mathcal{O}(n \log^3 n)\) upper bound on the reset threshold of \(n\)-state synchronizing partial literal automata, asymptotically matching the known upper bound for complete literal automata [33].

We start with a special case of one-word codes. A non-empty word \(w\) is called primitive if it is not a power of a shorter word, i.e., \(w \neq u^k\) for every word \(u\) and \(k \ge 2\). The upper bound on \(\mathop{\mathrm{rt}}(\mathrsfs{A}_X)\) follows from a result of Weinbaum ([38], [39]).

For the literal automaton of a finite prefix codes consisting of at least two words, there always exists a word of linear length and logarithmic rank.

4 Lower bounds for properly incomplete DFAs↩︎

We conclude with observations for transferring lower bounds from the complete case to the partial case. Of course, in general, this is trivial, since a complete DFA is a special case of a partial DFA. On the other hand, letters with all transitions undefined cannot be used for synchronization. Hence we need to add a restriction to exclude these cases and see the effect of usable incomplete transitions. A partial DFA is properly incomplete if there is at least one letter whose transition is defined for some state and is undefined for some other state.

For an a partial DFA \(\mathrsfs{A}\), let the length of the shortest non-mortal words of rank at most \(r\) be called the rank threshold \(\mathop{\mathrm{rt}}(\mathrsfs{A},r)\). We show that bounding the rank/reset threshold of a strongly connected properly incomplete DFA is related to bounding the corresponding threshold of a complete DFA. A general construction for this is as follows.

The duplicating automaton turns out to be a partial DFA counterpart to the recent Volkov’s construction of complete DFAs [40]. The duplicating automaton \(\mathrsfs{A}^\mathrm{D}\) has twice the number of states of \(\mathrsfs{A}\) and is properly incomplete. Also, it is strongly connected if \(\mathrsfs{A}\) is.

From 8, it follows that we cannot expect an upper bound on the reset threshold of a properly incomplete strongly connected DFA better than \(0.04135 n^3 + \mathcal{O}(n^2)\), unless we can improve the best general upper bound on the reset threshold of a complete DFA, which currently is roughly \(0.1654 n^3 + \mathcal{O}(n^2)\) [18]. We can also show a lower bound on the largest possible reset threshold, using the Rystsov’s construction of DFAs with long shortest mortal words [41].

Acknowledgments↩︎

This work was supported by the National Science Centre, Poland under project numbers 2022/46/E/ST6/00230 (Andrew Ryzhikov) and 2021/41/B/ST6/03691 (Marek Szykuła).

References↩︎