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

Nondeterministic State Complexity for Suffix-Free Regular Languages

We investigate the nondeterministic state complexity of basic operations for suffix-free regular languages. The nondeterministic state complexity of an operation is the number of states that are necessary and sufficient in the worst-case for a minimal nondeterministic finite-state automaton that accepts the language obtained from the operation. We consider basic operations (catenation, union, intersection, Kleene star, reversal and complementation) and establish matching upper and lower bounds for each operation. In the case of complementation the upper and lower bounds differ by an additive constant of two.Comment: In Proceedings DCFS 2010, arXiv:1008.127

On the descriptional complexity of iterative arrays

The descriptional complexity of iterative arrays (lAs) is studied. Iterative arrays are a parallel computational model with a sequential processing of the input. It is shown that lAs when compared to deterministic finite automata or pushdown automata may provide savings in size which are not bounded by any recursive function, so-called non-recursive trade-offs. Additional non-recursive trade-offs are proven to exist between lAs working in linear time and lAs working in real time. Furthermore, the descriptional complexity of lAs is compared with cellular automata (CAs) and non-recursive trade-offs are proven between two restricted classes. Finally, it is shown that many decidability questions for lAs are undecidable and not semidecidable

Quotient Complexity of Regular Languages

The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formal-language terms as the number of distinct quotients of the language, and to call it "quotient complexity". The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular languages and f is a regular operation, for example, union or concatenation. Since quotients can be represented by derivatives, one can find a formula for the typical quotient of f(K,L) in terms of the quotients of K and L. To obtain an upper bound on the number of quotients of f(K,L) all one has to do is count how many such quotients are possible, and this makes automaton constructions unnecessary. The advantages of this point of view are illustrated by many examples. Moreover, new general observations are presented to help in the estimation of the upper bounds on quotient complexity of regular operations

History-Register Automata

Programs with dynamic allocation are able to create and use an unbounded number of fresh resources, such as references, objects, files, etc. We propose History-Register Automata (HRA), a new automata-theoretic formalism for modelling such programs. HRAs extend the expressiveness of previous approaches and bring us to the limits of decidability for reachability checks. The distinctive feature of our machines is their use of unbounded memory sets (histories) where input symbols can be selectively stored and compared with symbols to follow. In addition, stored symbols can be consumed or deleted by reset. We show that the combination of consumption and reset capabilities renders the automata powerful enough to imitate counter machines, and yields closure under all regular operations apart from complementation. We moreover examine weaker notions of HRAs which strike different balances between expressiveness and effectiveness.Comment: LMCS (improved version of FoSSaCS

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

Descriptional complexity of cellular automata and decidability questions

We study the descriptional complexity of cellular automata (CA), a parallel model of computation. We show that between one of the simplest cellular models, the realtime-OCA. and "classical" models like deterministic finite automata (DFA) or pushdown automata (PDA), there will be savings concerning the size of description not bounded by any recursive function, a so-called nonrecursive trade-off. Furthermore, nonrecursive trade-offs are shown between some restricted classes of cellular automata. The set of valid computations of a Turing machine can be recognized by a realtime-OCA. This implies that many decidability questions are not even semi decidable for cellular automata. There is no pumping lemma and no minimization algorithm for cellular automata

Operational State Complexity of Deterministic Unranked Tree Automata

We consider the state complexity of basic operations on tree languages recognized by deterministic unranked tree automata. For the operations of union and intersection the upper and lower bounds of both weakly and strongly deterministic tree automata are obtained. For tree concatenation we establish a tight upper bound that is of a different order than the known state complexity of concatenation of regular string languages. We show that (n+1) ( (m+1)2^n-2^(n-1) )-1 vertical states are sufficient, and necessary in the worst case, to recognize the concatenation of tree languages recognized by (strongly or weakly) deterministic automata with, respectively, m and n vertical states.Comment: In Proceedings DCFS 2010, arXiv:1008.127