Quotient Complexity of Bifix-, Factor-, and Subword-Free Regular Language

A language $L$ is prefix-free if whenever words $u$ and $v$ are in $L$ and $u$ is a prefix of $v$, then $u=v$. Suffix-, factor-, and subword-free languages are defined similarly, where by ``subword" we mean ``subsequence", and a language is bifix-free if it is both prefix- and suffix-free. These languages have important applications in coding theory. The quotient complexity of an operation on regular languages is defined as the number of left quotients of the result of the operation as a function of the numbers of left quotients of the operands. The quotient complexity of a regular language is the same as its state complexity, which is the number of states in the complete minimal deterministic finite automaton accepting the language. The state/quotient complexity of operations in the classes of prefix- and suffix-free languages has been studied before. Here, we study the complexity of operations in the classes of bifix-, factor-, and subword-free languages. We find tight upper bounds on the quotient complexity of intersection, union, difference, symmetric difference, concatenation, star, and reversal in these three classes of languages.Natural Sciences and Engineering Research Council of Canada [OGP0000871]Slovak Research and Development Agency [APVV-0035-10]Algorithms, Automata, and Discrete Data Structures VEGA, [2/0183/11

Syntactic Complexity of Prefix-, Suffix-, Bifix-, and Factor-Free Regular Languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity $n$ of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that $n^{n-2}$ is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be $(n-1)^{n-2}+(n-2)$, $(n-1)^{n-3} + (n-2)^{n-3} + (n-3)2^{n-3}$, and $(n-1)^{n-3} + (n-3)2^{n-3} + 1$, respectively, and exhibit languages with these syntactic complexities.Comment: 28 pages, 6 figures, 3 tables. An earlier version of this paper was presented in: M. Holzer, M. Kutrib, G. Pighizzini, eds., 13th Int. Workshop on Descriptional Complexity of Formal Systems, DCFS 2011, Vol. 6808 of LNCS, Springer, 2011, pp. 93-106. The current version contains improved bounds for suffix-free languages, new results about factor-free languages, and new results about reversa

Quotient Complexity Of Star-Free Languages

Electronic version of an article published as International Journal of Foundations of Computer Science, 23(06), 2012, 1261–1276. http://dx.doi.org/10.1142/S0129054112400515 © World Scientific Publishing Company http://www.worldscientific.com/The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quotient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient complexities of the operands. The class of star free languages is the smallest class containing the finite languages and closed under boolean operations and concatenation. We prove that the tight bounds on the quotient complexities of union, intersection, difference, symmetric difference, concatenation and star for star-free languages are the same as those for regular languages, with some small exceptions, whereas 2(n) - 1 is a lower bound for reversal.Natural Sciences and Engineering Research Council of Canada [OGP0000871

Quotient complexity of bifix-, factor-, and subword-free regular languages

A language L is prefix-free if whenever words u and v are in L and u is a prefix of v, then u = v. Suffix-, factor-, and subword-free languages are defined similarly, where by "subword" we mean "subsequence", and a language is bifix-free if it is both prefix- and suffix-free. These languages have important applications in coding theory. The quotient complexity of an operation on regular languages is defined as the number of left quotients of the result of the operation as a function of the numbers of left quotients of the operands. The quotient complexity of a regular language is the same as its state complexity, which is the number of states in the complete minimal deterministic finite automaton accepting the language. The state/quotient complexity of operations in the classes of prefix- and suffix-free languages has been studied before. Here, we study the complexity of operations in the classes of bifix-, factor-, and subword-free languages. We find tight upper bounds on the quotient complexity of intersection, union, difference, symmetric difference, concatenation, star, and reversal in these three classes of languages

Quotient Complexity of Ideal Languages

The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.tcs.2012.10.055 © 2013. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A language L over an alphabet Σ is a right (left) ideal if it satisfies L=LΣ∗ (L=Σ∗L). It is a two-sided ideal if L=Σ∗LΣ∗, and an all-sided ideal if L=Σ∗L, the shuffle of Σ∗ with L. Ideal languages are not only of interest from the theoretical point of view, but also have applications to pattern matching. We study the state complexity of common operations in the class of regular ideal languages, but prefer to use the equivalent term “quotient complexity”, which is the number of distinct left quotients of a language. We find tight upper bounds on the complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of the minimal generator, and also on the complexity of the minimal generator in terms of the complexity of the language. Moreover, tight upper bounds on the complexity of union, intersection, set difference, symmetric difference, concatenation, star, and reversal of ideal languages are derived.Natural Sciences and Engineering Research Council of Canada grant [OGP0000871]VEGA grant 2/0111/0

Complexity of Suffix-Free Regular Languages

The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.jcss.2017.05.011 © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We study various complexity properties of suffix-free regular languages. A sequence (Lk,Lk+1,…) of regular languages in some class, where n is the quotient complexity of Ln, is most complex if its languages Ln meet the complexity upper bounds for all basic measures. It is known that there exist such most complex sequences in several classes of regular languages. In contrast to this, we prove that there does not exist a most complex sequence in the class of suffix-free regular languages. However, we do exhibit two such sequences that together meet upper bounds for all basic measures.Natural Sciences and Engineering Research Council of Canada (NSERC) grant No. OGP000087National Science Centre, Poland project number 2014/15/B/ST6/0061

Syntactic Complexities of Nine Subclasses of Regular Languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of suffix-, bifix-, and factor-free regular languages, star-free languages including three subclasses, and R- and J-trivial regular languages. We found upper bounds on the syntactic complexities of these classes of languages. For R- and J-trivial regular languages, the upper bounds are n! and ⌊e(n-1)!⌋, respectively, and they are tight for n >= 1. Let C^n_k be the binomial coefficient ``n choose k''. For monotonic languages, the tight upper bound is C^{2n-1}_n. We also found tight upper bounds for partially monotonic and nearly monotonic languages. For the other classes of languages, we found tight upper bounds for languages with small state complexities, and we exhibited languages with maximal known syntactic complexities. We conjecture these lower bounds to be tight upper bounds for these languages. We also observed that, for some subclasses C of regular languages, the upper bound on state complexity of the reversal operation on languages in C can be met by languages in C with maximal syntactic complexity. For R- and J-trivial regular languages, we also determined tight upper bounds on the state complexity of the reversal operation

Algebraic Approaches to State Complexity of Regular Operations

The state complexity of operations on regular languages is an active area of research in theoretical computer science. Through connections with algebra, particularly the theory of semigroups and monoids, many problems in this area can be simplified or completely reduced to combinatorial problems. We describe various algebraic techniques for attacking state complexity problems. We present a general method for constructing witness languages for operations -- languages that attain the worst-case state complexity when used as the argument(s) of the operation. Our construction is based on full transformation monoids, which contain all functions from a finite set into itself. When a witness for an operation is known, determining the state complexity essentially becomes a counting problem. These counting problems, however, are not necessarily easy, and the witness languages produced by this method are not ideal in the sense that they have extremely large alphabets. We thus investigate some commonly used operations in detail, and look for algebraic techniques to simplify the combinatorial side of state complexity problems and to simplify the search for small-alphabet witnesses. For boolean operations (e.g., union, intersection, difference) we show that these combinatorial problems can be solved easily in special cases by studying the subgroup of permutations in the syntactic monoid of a witness candidate. If the subgroup of permutations is known to have some strong transitivity property, such as primitivity or 2-transitivity, we can draw conclusions about the worst-case state complexity when this language is used in a boolean operation. For the operations of concatenation and Kleene star (an iterated version of concatenation), we describe a “construction set” method to simplify state complexity lower-bound proofs, and determine some algebraic conditions under which this method can be applied. For the reversal operation, we show that the state complexity of the reverse of a language is closely related to the syntactic monoid of the language, and use this fact to investigate a generalized version of the reversal state complexity problem. After describing our techniques, we demonstrate them by applying them to some classical state complexity problems. We obtain complex generalizations of the classical results that would be difficult to prove without the machinery we develop