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 Measuring Non-Recursive Trade-Offs

We investigate the phenomenon of non-recursive trade-offs between descriptional systems in an abstract fashion. We aim at categorizing non-recursive trade-offs by bounds on their growth rate, and show how to deduce such bounds in general. We also identify criteria which, in the spirit of abstract language theory, allow us to deduce non-recursive tradeoffs from effective closure properties of language families on the one hand, and differences in the decidability status of basic decision problems on the other. We develop a qualitative classification of non-recursive trade-offs in order to obtain a better understanding of this very fundamental behaviour of descriptional systems

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

Pushdown automata and constant height: Decidability and bounds: Extended abstract

It cannot be decided whether a pushdown automaton accepts using constant pushdown height, with respect to the input length, or not. Furthermore, in the case of acceptance in constant height, the height cannot be bounded by any recursive function in the size of the description of the machine. In contrast, in the restricted case of pushdown automata over a one-letter input alphabet, i.e., unary pushdown automata, the above property becomes decidable. Moreover, if the height is bounded by a constant in the input length, then it is at most exponential with respect to the size of the description of the pushdown automaton. This bound cannot be reduced. Finally, if a unary pushdown automaton uses nonconstant height to accept, then the height should grow at least as the logarithm of the input length. This bound is optimal

Descriptional Complexity of Finite Automata -- Selected Highlights

The state complexity, respectively, nondeterministic state complexity of a regular language $L$ is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for $L$. Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable

Multi-Head Finite Automata: Characterizations, Concepts and Open Problems

Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg, 1966). Since that time, a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices has been developed. Although multi-head finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even non-semi-decidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. These strong negative results trigger the study of subclasses and alternative characterizations of multi-head finite automata for a better understanding of the nature of non-recursive trade-offs and, thus, the borderline between decidable and undecidable problems. In the present paper, we tour a fragment of this literature

Converting Nondeterministic Automata and Context-Free Grammars into Parikh Equivalent Deterministic Automata

We investigate the conversion of nondeterministic finite automata and context-free grammars into Parikh equivalent deterministic finite automata, from a descriptional complexity point of view. We prove that for each nondeterministic automaton with n states there exists a Parikh equivalent deterministic automaton with eO( 1an\ub7ln n) states. Furthermore, this cost is tight. In contrast, if all the strings accepted by the given automaton contain at least two different letters, then a Parikh equivalent deterministic automaton with a polynomial number of states can be found. Concerning context-free grammars, we prove that for each grammar in Chomsky normal form with n variables there exists a Parikh equivalent deterministic automaton with 2O(n2) states. Even this bound is tight