We characterize hyperbolic groups in terms of quasigeodesics in the Cayley graph forming regular languages. We also obtain a quantitative characterization of hyperbolicity of geodesic metric spaces by the non-existence of certain local $(3,0)$-quasigeodesic loops. As an application, we make progress towards a question of Shapiro regarding groups admitting a uniquely geodesic Cayley graph.

This paper introduces a class of automata and associated languages, suitable to model a computational paradigm of fuzzy systems, in which both vagueness and simultaneity are taken as first-class citizens. This requires a weighted semantics for transitions and a precise notion of a synchronous product to enforce the simultaneous occurrence of actions. The usual relationships between automata and languages are revisited in this setting, including a specific Kleene theorem.

This paper presents an efficient approach for planning collision-free and dynamically feasible trajectories that enable a mobile robot to carry out tasks specified as regular languages over workspace regions. A sampling-based tree search is conducted over the feasible motions and over an abstraction obtained by combining the automaton representing the regular language with a workspace decomposition. The abstraction is used to partition the motion tree into equivalence classes and estimate the feasibility of reaching accepting automaton states from these equivalence classes. The partition is continually refined to discover new ways to expand the search. Comparisons to related work show significant speedups.

We present a new incremental algorithm for minimising deterministic finite automata. Itruns in quadratic time for any practical application and may be halted at any point,returning a partially minimised automaton. Hence, the algorithm may be applied to a givenautomaton at the same time as it is processing a string for acceptance. We also includesome experimental comparative results.

LetLϕ,λ = {ω ∈ Σ∗| ϕ(ω) > λ} be thelanguage recognized by a formal seriesϕ:Σ∗ → ℝ with isolated cut pointλ. We provide new conditions that guarantee the regularity of thelanguage Lϕ,λ in the case thatϕ is rational or ϕ is a Hadamard quotient of rationalseries. Moreover the decidability property of such conditions is investigated.

Two deterministic finite automata are almost equivalent if they disagree in acceptanceonly for finitely many inputs. An automaton A is hyper-minimized if noautomaton with fewer states is almost equivalent to A. A regular languageL is canonical if the minimal automaton accepting L ishyper-minimized. The asymptotic state complexitys∗(L) of a regular languageL is the number of states of a hyper-minimized automaton for a languagefinitely different from L. In this paper we show that: (1) the class ofcanonical regular languages is not closed under: intersection, union, concatenation,Kleene closure, difference, symmetric difference, reversal, homomorphism, and inversehomomorphism; (2) for any regular languages L1 andL2 the asymptotic state complexity of their sumL1 ∪ L2, intersectionL1 ∩ L2, differenceL1 − L2, and symmetricdifference L1 ⊕ L2 can be boundedbys∗(L1)·s∗(L2).This bound is tight in binary case and in unary case can be met in infinitely many cases.(3) For any regular language L the asymptotic state complexity of itsreversal LR can be bounded by2s∗(L). This bound is tightin binary case. (4) The asymptotic state complexity of Kleene closure and concatenationcannot be bounded. Namely, for every k ≥ 3, there exist languagesK, L, and M such thats∗(K) = s∗(L) = s∗(M) = 1ands∗(K∗) = s∗(L·M) = k.These are answers to open problems formulated by Badr et al.[RAIRO-Theor. Inf. Appl. 43 (2009) 69–94].

We study the relation between the standard two-way automata andmore powerful devices, namely, two-way finite automata equippedwith some $\ell$ additional “pebbles” that are movable alongthe input tape, but their use is restricted (nested) ina stack-like fashion. Similarly as in the case of the classicaltwo-way machines, it is not known whether there existsa polynomial trade-off, in the number of states, between thenondeterministic and deterministic two-way automata with $\ell$ nested pebbles. However, we show that these two machine modelsare not independent: if there exists a polynomial trade-off forthe classical two-way automata, then, for each $\ell$ ≥ 0,there must also exist a polynomial trade-off for the two-wayautomata with $\ell$ nested pebbles. Thus, we have an upwardcollapse (or a downward separation) from the classical two-wayautomata to more powerful pebble automata, still staying withinthe class of regular languages. The same upward collapse holdsfor complementation of nondeterministic two-way machines. These results are obtained by showing that each pebble machinecan be, by using suitable inputs, simulated by a classicaltwo-way automaton (and vice versa), with only a linear number ofstates, despite the existing exponential blow-up between theclassical and pebble two-way machines.

We define H- and EH-expressions as extensions of regular expressions by adding homomorphic and iterated homomorphic replacement as new operations, resp. The definition is analogous to the extension given by Gruska in order to characterize context-free languages. We compare the families of languages obtained by these extensions with the families of regular, linear context-free, context-free, and EDT0Llanguages. Moreover, relations to language families based on patterns, multi-patterns, pattern expressions, H-systems and uniform substitutions are also investigated. Furthermore, we present their closure properties with respect to TRIO operations and discuss the decidability status and complexity of fixed and general membership, emptiness, and the equivalence problem.

The paper treats the question whether there always exists a minimal nondeterministic finite automaton of n states whose equivalent minimal deterministic finite automaton has α states for any integers n and α with n ≤ α ≤ 2n .Partial answers to this question were given by Iwama, Kambayashi, and Takaki (2000) and by Iwama, Matsuura, and Paterson (2003). In the present paper, the question is completely solved by presenting appropriate automata for all values of n and α. However, in order to give an explicit construction of the automata, we increase the input alphabet to exponential sizes. Then we prove that 2n letters would be sufficient but we describe the related automata only implicitly. In the last section, we investigate the above question for automata over binary and unary alphabets.

Resolving an open problem of Ravikumar and Quan, we show that equivalence of prefix grammars is complete in PSPACE. We also showthat membership for these grammars is complete in P (it was known that this problem is in P) and characterize thecomplexity of equivalence and inclusion for monotonic grammars. For grammars with several premises we show that membershipis complete in EXPTIME and hard for PSPACE for monotonicgrammars.