Unambiguous conjunctive grammars with 1 nonterminal symbol are shown to be strictly weaker than the grammars with 2 nonterminal symbols, which are in turn strictly weaker that the grammars with 3 or more nonterminal symbols. This hierarchy is established by considering grammars over a one-symbol alphabet, for which it is shown that 1-nonterminal grammars describe only regular languages, 2-nonterminal grammars describe some non-regular languages, but all of them are in a certain sense sparse, whereas 3-nonterminal grammars may describe some non-regular languages of non-zero density. It is also proved that one can test a 2-nonterminal grammar for equivalence with a regular language, whereas the equivalence between a pair of 2-nonterminal grammars is undecidable.
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We investigate and compare the descriptional power of unary probabilistic and nondeterministic automata (pfa's and nfa's, respectively). We show the existence of a family of languages hard for pfa’s in the following sense: For any positive integer d, there exists a unary d-cyclic language such that any pfa accepting it requires d states, as the smallest deterministic automaton. On the other hand, we prove that there exist infinitely many languages having pfa’s which from one side do not match a known optimal state lower bound and, on the other side, they are smaller than nfa’s which, in turn, are smaller than deterministic automata.
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We study the stochastic events induced by MM-qfa's working on unary alphabets. We give two algorithms for unary MM-qfa's: the first computes the dimension of the ergodic and transient components of the non halting subspace, while the second tests whether the induced event is d-periodic. These algorithms run in polynomial time whenever the MM-qfa given in input has complex amplitudes with rational components. We also characterize the recognition power of unary MM-qfa's, by proving that any unary regular language can be accepted by a MM-qfa with constant cut point and isolation. Yet, the amount of states of the resulting MM-qfa is linear in the size of the corresponding minimal dfa. We also single out families of unary regular languages for which the size of the accepting MM-qfa's can be exponentially decreased.
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