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A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in an analogous way, where by factor we mean contiguous subsequence, and by subword we mean scattered subsequence. We study the state complexity (which we prefer to call quotient complexity) of o...
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Context 1
... n = 1 and n = 2, the bound 2 is met by L = ∅ and L = ε, respectively. Now let n ≥ 3 and let L be the prefix-closed language defined by the quotient automaton shown in Fig. 5; transitions not depicted in the figure go to state n−1. Construct an ε-nfa for L * by removing state n − 1 and adding an ε-transition from all the remaining states to the initial state. Let us show that 2 n−2 +1 states are reachable and pairwise inequivalent in the corresponding subset ...
Context 2
... Let L be the prefix-closed language defined by the quotient automaton in Fig. 5 on page 8; then L meets the upper bound on star. Add a loop with a new letter d in each state and denote the resulting language by K. Then K is a prefix-closed language with κ(K) = n and κ(K \ ε) = n + 1. Next we have κ(K * ) = κ(L * ) = 2 n−2 + 1 and κ(K * \ ε) = 2 n−2 + ...
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Citations
... Some properties of the languages of the classes mentioned above can be found in [23] (monoids), [11] (nilpotent languages), [13] (combinational and commutative languages), [20] (definite languages), [12] and [2] (suffix-closed languages), [24] (ordered languages), [16] (circular languages), [18] (non-counting and strictly locally testable languages), [25] (power-separating languages), [1] (union-free languages). ...
We continue the research on the generative capacity of contextual grammars where contexts are adjoined around whole words (externally) or around subwords (internally) which belong to special regular selection languages. All languages generated by contextual grammars where all selection languages are elements of a certain subregular language family form again a language family. We investigate the computational capacity of contextual grammars with strictly locally testable selection languages and compare those families to families which are based on finite, monoidal, nilpotent, combinational, definite, suffix-closed, ordered, commutative, circular, non-counting, power-separating, or union-free languages. With these results, also an open problem regarding ordered and non-counting selection languages is solved.
... Operations on unary languages were studied by Yu et al. [17] and Pighizzini and Shallit [15], and on finite languages by Câmpeanu et al. [6]. The classes of prefix-and suffix-free languages were examined by Han et al. [7,8], and the classes of bifix-, factor-, and subword-free, star-free, ideal, and closed languages by Brzozowski et al. [2][3][4][5], Except for reversal on star-free languages, the state complexity of basic regular operations in each of the above mentioned classes was determined. Bordihn, Holzer, and Kutrib [1] considered also some other subclasses, including combinational, finitely generated left ideal, symmetric definite, star, comet, two-sided comet, ordered, and power-separating languages, and investigated the complexity of the conversion of nondeterministic finite automata (NFAs) to deterministic ones. ...
The state complexity of a regular operation is a function that assigns the maximal state complexity of the language resulting from the operation to the sizes of deterministic finite automata recognizing the operands of the operation. We study the state complexity of intersection, union, concatenation, star, and reversal on the classes of combinational, singleton, finitely generated left ideal, symmetric definite, star, comet, two-sided comet, ordered, star-free, and power-separating languages. We get the exact state complexities in all cases. The complexity of all operations on combinational languages is given by a constant function. The state complexity of the considered operations on singleton languages is \(\min \{m,n\}\), \(m+n-3\), \(m+n-2\), \(n-1\), and n, respectively, and on finitely generated left ideals, it is \(mn-2\), \(mn-2\), \(m+n-1\), \(n+1\), and \(2^{n-2}+2\). The state complexity of concatenation, star, and reversal is \(m2^{n}-2^{n-1}-m+1\), n, \(2^n\) and \(m2^{n-1}-2^{n-2}+1\), \(n+1\), \(2^{n-1}+1\) for star and symmetric definite languages, respectively. We also show that the complexity of reversal on ordered and power-separating languages is \(2^n-1\), which proves that the lower bound for star-free languages given by [Brzozowski, Liu, Int. J. Found. Comput. Sci. 23, 1261–1276, 2012] is tight. In all the remaining cases, the complexity is the same as for regular languages. Except for reversal on finitely generated left ideals and ordered languages, all our witnesses are described over a fixed alphabet.
... Some suitable examples include results in the classes of prefix-and suffix-free languages by Han et al. [12,13]. Brzozowski et al. examined the classes of factor-and subword-free languages [3], ideal languages [4], and closed languages [5]. All of the mentioned subclasses have their own unique significance. ...
... In this paper, we study arbitrary binary languages (languages over the alphabet E = {0, 1}) that are subword closed: if a word w 1 u 1 w 2 · · · w m u m w m+1 belongs to a language, then the word u 1 · · · u m belongs to this language. Subword-closed languages have attracted the attention of researchers in the field of formal languages for many years [1][2][3][4][5]. ...
In this paper, we study arbitrary subword-closed languages over the alphabet {0,1} (binary subword-closed languages). For the set of words L(n) of the length n belonging to a binary subword-closed language L, we investigate the depth of the decision trees solving the recognition and the membership problems deterministically and nondeterministically. In the case of the recognition problem, for a given word from L(n), we should recognize it using queries, each of which, for some i∈{1,…,n}, returns the ith letter of the word. In the case of the membership problem, for a given word over the alphabet {0,1} of the length n, we should recognize if it belongs to the set L(n) using the same queries. With the growth of n, the minimum depth of the decision trees solving the problem of recognition deterministically is either bounded from above by a constant or grows as a logarithm, or linearly. For other types of trees and problems (decision trees solving the problem of recognition nondeterministically and decision trees solving the membership problem deterministically and nondeterministically), with the growth of n, the minimum depth of the decision trees is either bounded from above by a constant or grows linearly. We study the joint behavior of the minimum depths of the considered four types of decision trees and describe five complexity classes of binary subword-closed languages.
... Every non-Kuratowski space satisfies operator equations that trivially disallow certain φ-numbers. Excluding φ-number 30, up to duality the only combinations allowed are: ED 11,13,16,22,25,28;OU 12,13,18,20,22,24,26,28; EO 13, 22, 28; partition 25; and none in discrete spaces. We verified by computer that all of their space-and-ψ-number subcombinations occur. ...
... The following papers cite the use of computers: [5,13,15,27,34,52,62,63,88,112,117]. Papers with at least one author from a computer science department include: [5,15,23,24,26,34,44,49,66,88,89,90,91,113,141]. Many papers besides those of Kuratowski [101], Zarycki [178], and GE have been authored or co-authored by graduate students [5,80,81,113,122,123,130,134,137,156,160] and undergraduates [13,23,27,32,33,40]. ...
... By Proposition 3(v) when a subset satisfies one or more equations in O the inequalities in O it satisfies are determined. When no equations in O are satisfied then either zero or two inequalities in O are.Note that the ψ-numbers without an equation in O are:1,7,9,14,16,[23][24][25]27.Since ffA = fiA ⇐⇒ fbA ⊆ fbiA it follows that ψA ∈ {7, 23} =⇒ fibA fbiA and ψA = 9 =⇒ fibA ⊆ fbiA. Hence, while subsets with ψ-number 9 satisfy exactly two inequalities in O, they satisfy none optionally.Dual results hold for ψ-numbers in {14, 23} and {16}, respectively. ...
The Kuratowski monoid $\mathcal{K}$ is generated under operator composition by closure and complement in a nonempty topological space. It satisfies $2\leq|\mathcal{K}|\leq14$. The Gaida-Eremenko (or GE) monoid $\mathcal{KF}$ extends $\mathcal{K}$ by adding the boundary operator. It satisfies $4\leq|\mathcal{KF}|\leq34$. We show that when $|\mathcal{K}|<14$ the GE monoid is determined by $\mathcal{K}$. When $|\mathcal{K}|=14$ if the interior of the boundary of every subset is clopen, then $|\mathcal{KF}|=28$. This defines a new type of topological space we call Kuratowski disconnected. Otherwise $|\mathcal{KF}|=34$. When applied to an arbitrary subset the GE monoid collapses in one of 70 possible ways. We use this result to settle two questions raised by Gardner and Jackson. Computer experimentation played a key role in our research.
... Some suitable examples include results in the classes of prefixand suffix-free languages by Han et al. [11,12]. Brzozowski et al. examined the classes of factor-and subword-free languages [3], ideal languages [4], and closed languages [5]. Particularly interesting are the classes of free languages since they are uniquely decodable codes arising often in practice as a result of Huffman coding [15] and other widespread coding schemes. ...
The cut of two languages is a subset of their concatenation given by the leftmost maximal substring match. We study the state complexity of the cut operation assuming that both operands belong to some, possibly different, subclasses of convex languages, namely, right, left, two-sided, and all-sided ideal, prefix-, suffix-, factor-, and subword-closed, and -free languages. For all considered pairs of classes, we get the exact state complexity of cut. We show that it is m whenever the first language is a right ideal, and it is m+n-1 or m+n-2 if the first language is prefix-closed or prefix-free. In the other cases, the state complexity of cut is between mn-2n-m+4 and mn-n+m, the latter being the known state complexity of cut on regular languages. All our witnesses are described over a fixed alphabet of size at most three, except for three cases when the witness languages are described over an alphabet of size m or m-1.
... In this paper, we study arbitrary binary languages (languages over the alphabet E = {0, 1}) that are subword-closed: if a word w 1 u 1 w 2 · · · w m u m w m+1 belongs to a language, then the word u 1 · · · u m belongs to this language [1,2,6]. ...
... We can prove in a similar way that |w 2 | ≤ 2m and |w 3 | ≤ 2m. Therefore w can be represented in the form (1). ...
In this paper, we study arbitrary subword-closed languages over the alphabet $\{0,1\}$ (binary subword-closed languages). For the set of words $L(n)$ of the length $n$ belonging to a binary subword-closed language $L$, we investigate the depth of decision trees solving the recognition and the membership problems deterministically and nondeterministically. In the case of recognition problem, for a given word from $L(n)$, we should recognize it using queries each of which, for some $i\in \{1,\ldots ,n\}$, returns the $i$th letter of the word. In the case of membership problem, for a given word over the alphabet $\{0,1\}$ of the length $n$, we should recognize if it belongs to the set $L(n)$ using the same queries. With the growth of $n$, the minimum depth of decision trees solving the problem of recognition deterministically is either bounded from above by a constant, or grows as a logarithm, or linearly. For other types of trees and problems (decision trees solving the problem of recognition nondeterministically, and decision trees solving the membership problem deterministically and nondeterministically), with the growth of $n$, the minimum depth of decision trees is either bounded from above by a constant or grows linearly. We study joint behavior of minimum depths of the considered four types of decision trees and describe five complexity classes of binary subword-closed languages.
... The operational state complexity in several subclasses of regular languages has been studied in the literature. Câmpeanu et al. [6] considered finite languages, Han and Salomaa [8,9] investigated prefix-free and suffix-free languages, and Brzozowski et al. [3,4] examined ideal and closed languages. The classes of co-finite, star-free, union-free, and unary languages have been studied as well [1,5,13,18]. ...
We investigate the class of languages recognized by permutation deterministic finite automata. Using automata constructions and some properties of permutation automata, we show that this class is closed under Boolean operations, reversal, and quotients, and it is not closed under concatenation, power, Kleene closure, positive closure, cut, shuffle, cyclic shift, and permutation. We prove that the state complexity of Boolean operations, Kleene closure, positive closure, and right quotient on permutation languages is the same as in the general case of regular languages. Next, we get the tight upper bounds on the state complexity of concatenation (\(m2^n-2^{n-1}-m+1\)), square (\(n2^{n-1}-2^{n-2}\)), reversal (\(n\atopwithdelims ()\lceil n/2\rceil \)), and left quotient (\(m\atopwithdelims ()\lceil m/2\rceil \); tight if \(m\le n\)). All our witnesses are unary or binary, and the binary alphabet is always optimal, except for Boolean operations in the case of \(\gcd (m,n)=1\). In the unary case, the state complexity of all considered operations is the same as for regular languages, except for quotients and cut. In case of quotients, it is \(\min \{m,n\}\), and in case of cut, it is either \(2m-1\) or \(2m-2\), depending on whether there exists an integer \(\ell \) with \(2\le \ell \le n\) such that \(m\bmod \ell \ne 0\).
... The concepts of absolute liveness [106] and suffix-closedness [18] are not new, but they are not leveraged in the context of control synthesis to the best knowledge of the author. In the literature, there are also many works studying efficient synthesis algorithms for specific fragements of LTL formulas, such as reach-stay-avoid game [82], mode-target game [8] and GR(1) [86]. ...
Correct-by-construction control synthesis methods refer to a collection of model-based techniques to algorithmically generate controllers/strategies that make the systems satisfy some formal specifications. Such techniques attract much attention as they provide formal guarantees on the correctness of cyber-physical systems, where corner cases may arise due to the interaction among different modules. The controllers synthesized through such methods, however, may still malfunction due to faults, such as physical component failures and unexpected operating conditions, which lead to a sudden change of the system model. In these cases, we want to guarantee that the performance of the faulty system degrades gracefully, and hence achieve fault tolerance. This thesis is about 1) incorporating fault detection and detectability analysis algorithms in correct-by-construction control synthesis, 2) formalizing the graceful degradation specification for fault tolerant systems with temporal logic, and 3) developing algorithms to synthesize correct-by-construction controllers that achieve such graceful degradation, with possible delay in the fault detection. In particular, two sets of approaches from the temporal logic planning domain, i.e., abstraction-based synthesis and optimization-based path planning, are considered. First, for abstraction-based approaches, we propose a recursive algorithm to reduce the fault tolerant controller synthesis problem into multiple small synthesis problems with simpler specifications. Such recursive reduction leverages the structure of the fault propagation and hence avoids the high complexity of solving the problem monolithically as one general temporal logic game. Furthermore, by exploring the structural properties in the specifications, we show that, even when the fault is detected with delay, the problem can be solved by a similar recursive algorithm without constructing the belief space. Secondly, optimization-based path planning is considered. The proposed approach leverages the recently developed temporal logic encodings and state-of-art mixed integer programming (MIP) solvers. The novelty of this work is to enhance the open-loop strategy obtained through solving the MIP so that it can react contingently to faults and disturbance. Finally, the control synthesis techniques developed for discrete state systems is shown to be applicable to continuous states systems. This is demonstrated by fuel cell thermal management application. Particularly, to apply the abstraction-based synthesis methods to complex systems such as the fuel cell thermal management system, structural properties (e.g., mixed monotonicity) of the system are explored and leveraged to ease abstraction computation, and techniques are developed to improve the scalability of synthesis process whenever the system has a large number of control actions.
... The results of these papers were summarized in the journal version [18]. Let us mention that the (deterministic) state complexity in all above mentioned classes were considered by Brzozowski et al. for basic operations [2,3,4] and byČevorová for the second power, the square operation [6,7]. ...
... Now we are going to examine the positive closure on DFAs accepting free, ideal, and closed languages. Since ε ∈ L implies L + = L * , the known results on Kleene star on closed languages [4,8] hold also for positive closure. Proof. ...
... To get the DFA for L + , it is enough to replace each transition (f, a, d) by (f, a, s · a) where s is the initial state and · is the transition function of A. The resulting DFA has n states and this upper bound is met by the language b * a n−2 if n ≥ 4, cf. [23,Lemmas 3,4]. (b) We may assume that the DFA A for L is non-returning and has a nonfinal sink state. ...
We study the descriptional complexity of the k-th power and positive closure operations on the classes of prefix-, suffix-, factor-, and subword-free, -closed, and -convex regular languages, and on the classes of right, left, two-sided, and all-sided ideal languages. We show that the upper bound kn on the nondeterministic complexity of the k-th power in the class of regular languages is tight for closed and convex classes, while in the remaining classes, the tight upper bound is \(k(n-1)+1\). Next we show that the upper bound n on the nondeterministic complexity of the positive closure operation in the class of regular languages is tight in all considered classes except for classes of factor-closed and subword-closed languages, where the complexity is one. All our worst-case examples are described over a unary or binary alphabet, except for witnesses for the k-th power on subword-closed and subword-convex languages which are described over a ternary alphabet. Moreover, whenever a binary alphabet is used for describing a worst-case example, it is optimal in the sense that the corresponding upper bounds cannot be met by a language over a unary alphabet. The most interesting result is the description of a binary factor-closed language meeting the upper bound kn for the k-th power. To get this result, we use a method which enables us to avoid tedious descriptions of fooling sets. We also provide some results concerning the deterministic state complexity of these two operations on the classes of free, ideal, and closed languages.
