State Complexity Bounds for the Commutative Closure of Group Languages
Stefan Hoffmann
备注：
12 pages paper, including 3 figures + 11 pages appendix; update: minor changes, a few sentences rearranged in the introduction and before definition 2, one footnote in appendix added for a reference; 2nd update: improved the bound and fixed some typos; 3rd update: added new section with intuitive explanations, removed the statements about jumping finite automata to the appendix
In this work we construct an automaton for the commutative closure of a given regular group language. The number of states of the resulting automaton is bounded by the number of states of the original automaton, raised to the power of the alphabet size, times the product of the order of the letters, viewed as permutations of the state set. This gives the asymptotic state bound $O((n\exp(\sqrt{n\ln n}))^{|Σ|})$, if the original regular language is accepted by an automaton with $n$ states. Depending on the automaton in question, we label points of $\mathbb N_0^{|Σ|}$ by subsets of states and introduce unary automata which decompose the thus labelled grid. Based on these constructions, we give a general regularity condition, which is fulfilled for group languages.
We investigate the state complexity of the upward and downward closure and interior operations on commutative regular languages. Then, we systematically study the state complexity of these operations and of the shuffle operation on commutative group languages and commutative aperiodic (or star-free) languages.
We show that the commutative closure combined with the iterated shuffle is a regularity-preserving operation on group languages. In particular, for commutative group languages, the iterated shuffle is a regularity-preserving operation. We also give bounds for the size of minimal recognizing automata. Then, we use these results to deduce that the commutative closure of any shuffle expression over group languages, i.e., expressions involving shuffle, iterated shuffle, concatenation, Kleene star and union in any order, starting with the group languages, always yields a regular language.
We identify a subclass of the regular commutative languages that is closed under the iterated shuffle, or shuffle closure. In particular, it is regularity-preserving on this subclass. This subclass contains the commutative group languages and, for every alphabet $Σ$, the class $\textbf{Com}^+(Σ^*)$ given by the ordered variety $\textbf{Com}^+$. Then, we state a simple characterization when the iterated shuffle on finite commutative languages gives a regular language again and state partial results for aperiodic commutative languages. We also show that the aperiodic, or star-free, commutative l…
The downward and upward closures of a regular language $L$ are obtained by collecting all the subwords and superwords of its elements, respectively. The downward and upward interiors of $L$ are obtained dually by collecting words having all their subwords and superwords in $L$, respectively. We provide lower and upper bounds on the size of the smallest automata recognizing these closures and interiors. We also consider the computational complexity of decision problems for closures of regular languages.
We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing sub- or superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of $2^{\mathcal{O}(n)}$ on the size of nondeterministic finite automata (NFAs) representing the subword closure of a CFG of size $n$. (2) We present a family of CFGs for which the minimal deterministic finite automata representing their subword closure matches the upper-bound of $2^{2^{\mathcal{O}(n)}}$ following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing su…
The downward closure of a language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of every language is regular. Moreover, recent results show that downward closures are computable for quite powerful system models. One advantage of abstracting a language by its downward closure is that then equivalence and inclusion become decidable. In this work, we study the complexity of these two problems. More precisely, we consider the following decision problems: Given languages $K$ and $L$ from classes $\mathcal{C}$ and $\mathcal{D}$, r…
We consider commutative regular and context-free grammars, or, in other words, Parikh images of regular and context-free languages. By using linear algebra and a branching analog of the classic Euler theorem, we show that, under an assumption that the terminal alphabet is fixed, the membership problem for regular grammars (given v in binary and a regular commutative grammar G, does G generate v?) is P, and that the equivalence problem for context free grammars (do G_1 and G_2 generate the same language?) is in $\mathrm{Π_2^P}$.
The paper is about a class of languages that extends context-free languages (CFL) and is stable under shuffle. Specifically, we investigate the class of partially-commutative context-free languages (PCCFL), where non-terminal symbols are commutative according to a binary independence relation, very much like in trace theory. The class has been recently proposed as a robust class subsuming CFL and commutative CFL. This paper surveys properties of PCCFL. We identify a natural corresponding automaton model: stateless multi-pushdown automata. We show stability of the class under natural operations…
We consider Parikh images of languages accepted by non-deterministic finite automata and context-free grammars; in other words, we treat the languages in a commutative way --- we do not care about the order of letters in the accepted word, but rather how many times each one of them appears. In most cases we assume that the alphabet is of fixed size. We show tight complexity bounds for problems like membership, equivalence, and disjointness. In particular, we show polynomial algorithms for membership and disjointness for Parikh images of non-deterministic finite automata over fixed alphabet, an…
We introduce a subclass of the commutative regular languages that is characterized by the property that the state set of the minimal deterministic automaton can be written as a certain Cartesian product. This class behaves much better with respect to the state complexity of the shuffle, for which we find the bound~$2nm$ if the input languages have state complexities $n$ and $m$, and the upward and downward closure and interior operations, for which we find the bound~$n$. In general, only the bounds $(2nm)^{|Σ|}$ and $n^{|Σ|}$ are known for these operations in the commutative case. We prove dif…
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 the same way, where by subword we mean subsequence. We study the quotient complexity (usually called state complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the prefix-, suffix-, factor-, and subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star and reversal in each of the four classes of closed languages.…
We show that the language equivalence problem for regular and context-free commutative grammars is coNEXP-complete. In addition, our lower bound immediately yields further coNEXP-completeness results for equivalence problems for communication-free Petri nets and reversal-bounded counter automata. Moreover, we improve both lower and upper bounds for language equivalence for exponent-sensitive commutative grammars.
Here we study the computational complexity of the constrained synchronization problem for the class of regular commutative constraint languages. Utilizing a vector representation of regular commutative constraint languages, we give a full classification of the computational complexity of the constraint synchronization problem. Depending on the constraint language, our problem becomes PSPACE-complete, NP-complete or polynomial time solvable. In addition, we derive a polynomial time decision procedure for the complexity of the constraint synchronization problem, given some constraint automaton a…
Given a regular language $L$, we study the language of words $\mathsf{D}(L)$, that distinguish between pairs of different left-quotients of $L$. We characterize this distinguishability operation, show that its iteration has always a fixed point, and we generalize this result to operations derived from closure operators and Boolean operators. We give an upper bound for the state complexity of the distinguishability operation, and prove its tightness. We show that the set of minimal words that can be used to distinguish between different left-quotients of a language $L$ has at most $n-1$ element…
The projected language of a general deterministic automaton with $n$ states is recognizable by a deterministic automaton with $2^{n-1} + 2^{n-m} - 1$ states, where $m$ denotes the number of states incident to unobservable non-loop transitions, and this bound is best possible. Here, we derive the tight bound $2^{n - \lceil \frac{m}{2} \rceil} - 1$ for permutation automata. For a state-partition automaton with $n$ states (also called automata with the observer property) the projected language is recognizable with $n$ states. Up to now, these, and finite languages projected onto unary languages, …
We study the computational and descriptional complexity of the following transformation: Given a one-counter automaton (OCA) A, construct a nondeterministic finite automaton (NFA) B that recognizes an abstraction of the language L(A): its (1) downward closure, (2) upward closure, or (3) Parikh image. For the Parikh image over a fixed alphabet and for the upward and downward closures, we find polynomial-time algorithms that compute such an NFA. For the Parikh image with the alphabet as part of the input, we find a quasi-polynomial time algorithm and prove a completeness result: we construct a s…
Generalized pseudostandard words have been introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words. We present two new results concerning these words. The first one is a necessary and sufficient condition for their periodicity. The second result is a counterexample to Conjecture 43 from the paper: A. B. Masse, G.Paquin, H. Tremblay, and L. Vuillon, On Generalized Pseudostandard Words over Binary Alphabet (Journal of Int. Seq…
We study the size and the complexity of computing finite state automata (FSA) representing and approximating the downward and the upward closure of Petri net languages with coverability as the acceptance condition. We show how to construct an FSA recognizing the upward closure of a Petri net language in doubly-exponential time, and therefore the size is at most doubly exponential. For downward closures, we prove that the size of the minimal automata can be non-primitive recursive. In the case of BPP nets, a well-known subclass of Petri nets, we show that an FSA accepting the downward/upward cl…
