In this paper we give the cost, in terms of states, of some basic operations (union, intersection, concatenation, and Kleene star) on regular languages in the unary case (where the alphabet contains only one symbol). These costs are given by explicitly determining the number of states in the noncyclic and cyclic parts of the resulting automata. Furthermore, we prove that our bounds are optimal. We also present an interesting connection to Jacobsthal's function from number theory.
Limited automata are one-tape Turing machines which are allowed to rewrite each tape cell only in the first d visits, for a given constant d. For each d ≥ 2, these devices characterize the class of context-free languages. We investigate the equivalence between 2-limited automata and pushdown automata, comparing the relative sizes of their descriptions. We prove exponential upper and lower bounds for the sizes of pushdown automata simulating 2-limited automata. In the case of the conversion of deterministic 2-limited automata into deterministic pushdown automata the upper bound is double exponential and we conjecture that it cannot be reduced. On the other hand, from pushdown automata we can obtain equivalent 2-limited automata of polynomial size, also preserving determinism. From our results, it follows that the class of languages accepted by deterministic 2limited automata coincides with the class of deterministic context-free languages. grammars of type k (k = 1, 2, 3) is a restriction of the form used for grammars of type k − 1. From the point of view of language acceptors, each class of the hierarchy is characterized by some kind of device. However, while linear bounded automata, used to characterize context-sensitive languages (type 1) and finite state automata, used to characterize regular languages (type 3), can be seen as restrictions of (one-tape) Turing machines, which characterize type 0 languages, for context-free languages (type 2) the characterization in terms of pushdown automata is usually presented. Devices of this kind are very useful to investigate and manipulate context-free languages. They also emphasize the main difference between regular and context-free languages, namely the possibility of representing recursive structures which, in terms of accepting devices, corresponds to increase the power of finite automata by adding a pushdown store. However, in a hierarchical view, pushdown automata do not appear as a special case of linear bounded automata.Almost half a century ago, Hibbard discovered a different characterization of context-free languages, which uses a restricted version of Turing machines, called scan limited automata or, simply, limited automata [5]. For each integer d ≥ 0, a d-limited automaton is a one-way nondeterministic Turing machine which is allowed to rewrite the content of each tape cell only in the first d visits. He proved that, for each d ≥ 2, the class of languages accepted by d-limited automata coincides with the class of contextfree languages. Furthermore, since restricting these devices to use only the part of the tape containing the input string, without any available extra storage, does not reduce their computational power, these models can be seen as a restriction of linear bounded automata and, clearly, they are extensions of finite state automata. Hence, together with the above mentioned models related to type 1 and type 3 languages, this gives a hierarchy of classes of Turing machines corresponding to the Chomsky hierarchy.In this paper we investigate the ca...
It is well known that a context-free language defined over a one-letter alphabet is regular. This implies that unary context-free grammars and unary pushdown automata can be transformed into equivalent finite automata. In this paper, we study these transformations from a descriptional complexity point of view. In particular, we give optimal upper bounds for the number of states of nondeterministic and deterministic finite automata equivalent to unary context-free grammars in Chomsky normal form. These bounds are functions of the number of variables of the given grammars. We also give upper bounds for the number of states of finite automata simulating unary pushdown automata. As a main consequence, we are able to prove a log log n lower bound for the workspace used by one-way auxiliary pushdown automata in order to accept nonregular unary languages. The notion of space we consider is the so-called weak space concept. # 2002 Elsevier Science (USA)
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