Operations on Automata with All States Final

We study the complexity of basic regular operations on languages represented by incomplete deterministic or nondeterministic automata, in which all states are final. Such languages are known to be prefix-closed. We get tight bounds on both incomplete and nondeterministic state complexity of complement, intersection, union, concatenation, star, and reversal on prefix-closed languages.Comment: In Proceedings AFL 2014, arXiv:1405.527

Complexity in Prefix-Free Regular Languages

We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next provide the tight bounds on state complexity of symmetric difference, and deterministic and nondeterministic state complexity of difference and cyclic shift of prefix-free languages.Comment: In Proceedings DCFS 2010, arXiv:1008.127

The Magic Number Problem for Subregular Language Families

We investigate the magic number problem, that is, the question whether there exists a minimal n-state nondeterministic finite automaton (NFA) whose equivalent minimal deterministic finite automaton (DFA) has alpha states, for all n and alpha satisfying n less or equal to alpha less or equal to exp(2,n). A number alpha not satisfying this condition is called a magic number (for n). It was shown in [11] that no magic numbers exist for general regular languages, while in [5] trivial and non-trivial magic numbers for unary regular languages were identified. We obtain similar results for automata accepting subregular languages like, for example, combinational languages, star-free, prefix-, suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free languages, showing that there are only trivial magic numbers, when they exist. For finite languages we obtain some partial results showing that certain numbers are non-magic.Comment: In Proceedings DCFS 2010, arXiv:1008.127

More Structural Characterizations of Some Subregular Language Families by Biautomata

We study structural restrictions on biautomata such as, e.g., acyclicity, permutation-freeness, strongly permutation-freeness, and orderability, to mention a few. We compare the obtained language families with those induced by deterministic finite automata with the same property. In some cases, it is shown that there is no difference in characterization between deterministic finite automata and biautomata as for the permutation-freeness, but there are also other cases, where it makes a big difference whether one considers deterministic finite automata or biautomata. This is, for instance, the case when comparing strongly permutation-freeness, which results in the family of definite language for deterministic finite automata, while biautomata induce the family of finite and co-finite languages. The obtained results nicely fall into the known landscape on classical language families.Comment: In Proceedings AFL 2014, arXiv:1405.527

Finitary languages

The class of omega-regular languages provides a robust specification language in verification. Every omega-regular condition can be decomposed into a safety part and a liveness part. The liveness part ensures that something good happens "eventually". Finitary liveness was proposed by Alur and Henzinger as a stronger formulation of liveness. It requires that there exists an unknown, fixed bound b such that something good happens within b transitions. In this work we consider automata with finitary acceptance conditions defined by finitary Buchi, parity and Streett languages. We study languages expressible by such automata: we give their topological complexity and present a regular-expression characterization. We compare the expressive power of finitary automata and give optimal algorithms for classical decisions questions. We show that the finitary languages are Sigma 2-complete; we present a complete picture of the expressive power of various classes of automata with finitary and infinitary acceptance conditions; we show that the languages defined by finitary parity automata exactly characterize the star-free fragment of omega B-regular languages; and we show that emptiness is NLOGSPACE-complete and universality as well as language inclusion are PSPACE-complete for finitary parity and Streett automata

Syntactic Complexity of Finite/Cofinite, Definite, and Reverse Definite Languages

We study the syntactic complexity of finite/cofinite, definite and reverse definite languages. The syntactic complexity of a class of languages is defined as the maximal size of syntactic semigroups of languages from the class, taken as a function of the state complexity n of the languages. We prove that (n-1)! is a tight upper bound for finite/cofinite languages and that it can be reached only if the alphabet size is greater than or equal to (n-1)!-(n-2)!. We prove that the bound is also (n-1)! for reverse definite languages, but the minimal alphabet size is (n-1)!-2(n-2)!. We show that \lfloor e\cdot (n-1)!\rfloor is a lower bound on the syntactic complexity of definite languages, and conjecture that this is also an upper bound, and that the alphabet size required to meet this bound is \floor{e \cdot (n-1)!} - \floor{e \cdot (n-2)!}. We prove the conjecture for n\le 4.Comment: 10 pages. An error concerning the size of the alphabet has been corrected in Theorem

Boundedness in languages of infinite words

We define a new class of languages of $\omega$-words, strictly extending $\omega$-regular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of $\omega$-regular expressions where two new variants of the Kleene star $L^*$ are added: $L^B$ and $L^S$. These new exponents are used to say that parts of the input word have bounded size, and that parts of the input can have arbitrarily large sizes, respectively. For instance, the expression $(a^Bb)^\omega$ represents the language of infinite words over the letters $a,b$ where there is a common bound on the number of consecutive letters $a$. The expression $(a^Sb)^\omega$ represents a similar language, but this time the distance between consecutive $b$'s is required to tend toward the infinite. We develop a theory for these languages, with a focus on decidability and closure. We define an equivalent automaton model, extending B\"uchi automata. The main technical result is a complementation lemma that works for languages where only one type of exponent---either $L^B$ or $L^S$---is used. We use the closure and decidability results to obtain partial decidability results for the logic MSOLB, a logic obtained by extending monadic second-order logic with new quantifiers that speak about the size of sets