Deterministic tree-walking-storage automata

We study deterministic tree-walking-storage automata, which are finite-state devices equipped with a tree-like storage. These automata are generalized stack automata, where the linear stack storage is replaced by a non-linear tree-like stack. Therefore, tree-walking-storage automata have the ability to explore the interior of the tree storage without altering the contents, with the possible moves of the tree pointer corresponding to those of tree-walking automata. In addition, a tree-walking-storage automaton can append (push) non-existent descendants to a tree node and remove (pop) leaves from the tree. Here we are particularly considering the capacities of deterministic tree-walking-storage automata working in real time. It is shown that even the non-erasing variant can accept rather complicated unary languages as, for example, the language of words whose lengths are powers of two, or the language of words whose lengths are double Fibonacci numbers. Comparing the computational capacities with automata from the classical automata hierarchy, we derive that the family of languages accepted by real-time deterministic (non-erasing) tree-walking-storage automata is located between the regular and the deterministic context-sensitive languages. Moreover, the families are incomparable with the families of context-free and growing context-sensitive languages. It turns out that the devices under consideration accept unary languages in non-erasing mode that cannot be accepted by any classical stack automaton, even in erasing mode and arbitrary time. Basic closure properties of the induced families of languages are shown. In particular, we consider Boolean operations and AFL operations. It turns out that the two families in question have the same properties and, in particular, share all but one of these closure properties with the important family of deterministic context-free languages. Then, we consider the computational capacity of the counterpart to counter- and stack-counter automata, where the set of stack symbols is a singleton. Finally, we explore several decidability problems and show, that even for devices with a single tree symbol, the problems are all non-semidecidable by reductions of non-semidecidable problems of Turing machines.

Deterministic one-way Turing machines with sublinear space bounds are systematically studied. We distinguish among the notions of strong, weak, and restricted space bounds. The latter is motivated by the study of P automata. The space available on the work tape depends on the number of input symbols read so far, instead of the entire input. The class of functions space constructible by such machines is investigated, and it is shown that every function f that is space constructible by a deterministic two-way Turing machine, is space constructible by a strongly f space-bounded deterministic one-way Turing machine as well. We prove that the restricted mode coincides with the strong mode for space constructible functions. The known infinite, dense, and strict hierarchy of strong space complexity classes is derived also for the weak mode by Kolmogorov complexity arguments. Finally, closure properties under AFL operations, Boolean operations and reversal are shown.

Two-dimensional Boolean RAM (random access machine with Boolean operations and shifts) can speed the results and lower the costs that appear in performing two-dimensional computations such as matrix manipulation. This paper presents an implementation of a fast algorithm for matrix multiplication; the product of (m,m)-matrices A and B with 0 ≤ aij, bij ≤ k−1, for all 1 ≤ i,j ≤ m, is computed in 0(m·log2k (log2m + log2k)) steps in linear storage.

We introduce and investigate tree-walking-storage automata, which are finite-state devices equipped with a tree-like storage. The automata are generalized stack automata, where the linear stack storage is replaced by a non-linear tree-like stack. Therefore, tree-walking-storage automata have the ability to explore the interior of the tree storage without altering the contents, where the possible moves of the tree pointer correspond to those of tree walking automata. In addition, a tree-walking-storage automaton can append (push) non-existent descendants to a tree node and remove (pop) leaves from the tree. As for classical stack automata, we also consider non-erasing and checking variants. As first steps to investigate these models we consider the computational capacities of deterministic one-way variants. In particular, a main focus is on the comparisons of the different variants of tree-walking-storage automata as well as on the comparisons with classical stack automata, and we can draw a complete picture.

Finite turns and the regular closure of linear context-free languages

In this paper, we investigate the possibility of using gain material to overcome losses in periodic semiconductor layered metamaterial. A generic approach, based on the modified transfer matrix method and perturbation theory, was devised here to take into account a gain saturation effect. It is demonstrated that the introduction of nonlinear saturation of the linear gain leads to nonreciprocal response of the system. Taking advantage of the conservation relation, we determine the relation between the gain coefficient and geometrical parameters of the layers to get full compensation of the losses in the nonlinear stack. It is obtained that depending on the parameters of the layers and angle of electromagnetic wave incidence, the full gain–loss balance in the nonlinear system can be maintained taking the gain coefficient above or below the threshold determined for the lossless linear stack. It is shown that the input intensity strongly affects the reflection and transmission coefficients for the nonlinear periodic stack and can be used as the additional mechanism to mitigate the losses.

Uniformly erasable AFL

While the closure of a language family [Formula: see text] under certain language operations is the least family of languages which contains all members of [Formula: see text] and is closed under all of the operations, a kernel of [Formula: see text] is a maximal family of languages which is a sub-family of [Formula: see text] and is closed under all of the operations. Here we investigate properties of kernels of general language families and operations defined thereon as well as kernels of (deterministic) (linear) context-free languages with a focus on Boolean operations. While the closures of language families are unique, this uniqueness is not obvious for kernels. We consider properties of language families and of operations that yield unique and non-unique, i.e. a set, of kernels. For the latter case, the question whether the union of all kernels coincides with the language family, or whether there are languages that do not belong to any kernel is addressed. Additionally, languages that are mandatory for each (Boolean) kernel and languages that are optional for (Boolean) kernels are studied. That is, we consider the intersection of all Boolean kernels as well as their union. The expressive capacities of these families are addressed leading to a hierarchical structure. Further closure properties are considered. Furthermore, we study descriptional complexity aspects of these families, where languages are represented by context-free grammars with proofs attached. It turns out that the size trade-offs between all families in question and deterministic context-free languages are non-recursive. That is, one can choose an arbitrarily large recursive function [Formula: see text], but the gain in economy of description eventually exceeds [Formula: see text] when changing from the latter system to the former.

The chop of languages

The dot-depth hierarchy of star-free languages is infinite

Context-dependent nondeterminism for pushdown automata

Pushdown automata using a limited and unlimited amount of nondeterminism are investigated. Moreover, nondeterministic steps are allowed only within certain contexts, i.e., in configurations that meet particular conditions. The relationships of the accepted language families with closures of the deterministic context-free languages ($\textrm{DCFL}$) under regular operations are studied. For example, automata with unbounded nondeterminism that have to empty their pushdown store up to the initial symbol in order to make a guess are characterized by the regular closure of $\textrm{DCFL}$. Automata that additionally have to reenter the initial state are (almost) characterized by the Kleene star closure of the union closure of the prefix-free deterministic context-free languages. Pushdown automata with bounded nondeterminism are characterized by the union closure of $\textrm{DCFL}$ in any of the considered contexts. Proper inclusions between all language classes discussed are shown. Finally, closure properties of these families under AFL operations are investigated.

A generalization of pushdown automata towards regulated nondeterminism is studied. The nondeterminism is governed in such a way that the decision, whether or not a nondeterministic rule is applied, depends on the whole content of the stack. More precisely, the content of the stack is considered as a word over the stack alphabet, and the pushdown automaton is allowed to act nondeterministically, if this word belongs to some given set R of control words. Otherwise its behavior is deterministic. The computational capacity of such R-PDAs depends on the complexity of R. It turns out that non-context-free languages are accepted even if R is a linear, deterministic context-free language. On the other hand, regular control sets R do not increase the computational capacity of nondeterministic pushdown automata. This raises the natural question for the relations between the structure and complexity of regular sets R on one hand and the computational capacity of the corresponding R-PDA on the other hand. Clearly, if R is empty, the deterministic context-free languages are characterized. For R = {a, b}* one obtains all context-free languages. Furthermore, if R is finite, then the regular closure of the deterministic context-free languages is described. We investigate these questions, and discuss closure properties of the language classes in question under AFL operations.

Closures of linear context-free languages under Boolean operations are investigated. The intersection closure and the complementation closure are incomparable. By closing these closures under further Boolean operations we obtain several new language families. The hierarchy obtained by such closures of closures is proper up to level four, where it collapses to the Boolean closure which, in turn, is incomparable with several closures of the family of context-free languages. The Boolean closure of the linear context-free languages is properly contained in the Boolean closure of the context-free languages. A characterization of a class of non-unary languages that cannot be expressed as a Boolean formula over the linear context-free languages is presented.

Closures of linear context-free languages under Boolean operations are investigated. The intersection closure and the complementation closure are incomparable. By closing these closures under further Boolean operations we obtain several new language families. The hierarchy obtained by such closures of closures is proper up to a certain level, where it collapses to the Boolean closure which, in turn, is incomparable with several closures of the family of context-free languages. The Boolean closure of the linear context-free languages is properly contained in the Boolean closure of the context-free languages. A characterization of a class of non-unary languages that cannot be expressed as a Boolean formula over the linear context-free languages is presented.