Three hierarchies of transducers

Composition of top-down tree transducers yields a proper hierarchy of transductions and of output languages. The same is true for ETOL systems (viewed as transducers) and for two-way generalized sequential machines

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

Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in d=1 lead to distinct classes in d >= 2. We explore the closure properties and computational complexity of these classes, including undecidability and L-, NL- and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d >= 2 has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t^{-d} unless it maps every initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy

The copying power of one-state tree transducers

One-state deterministic top-down tree transducers (or, tree homomorphisms) cannot handle "prime copying," i.e., their class of output (string) languages is not closed under the operation L → {$(w$)f(n) w ε L, f(n) ≥ 1}, where f is any integer function whose range contains numbers with arbitrarily large prime factors (such as a polynomial). The exact amount of nonclosure under these copying operations is established for several classes of input (tree) languages. These results are relevant to the extended definable (or, restricted parallel level) languages, to the syntax-directed translation of context-free languages, and to the tree transducer hierarchy.\ud \u

Dissecting Power of a Finite Intersection of Context Free Languages

Let $\exp^{k,\alpha}$ denote a tetration function defined as follows: $\exp^{1,\alpha}=2^{\alpha}$ and $\exp^{k+1,\alpha}=2^{\exp^{k,\alpha}}$, where $k,\alpha$ are positive integers. Let $\Delta_n$ denote an alphabet with $n$ letters. If $L\subseteq\Delta_n^*$ is an infinite language such that for each $u\in L$ there is $v\in L$ with $\vert u\vert<\vert v\vert\leq \exp^{k,\alpha}\vert u\vert$ then we call $L$ a language with the \emph{growth bounded by} $(k,\alpha)$-tetration. Given two infinite languages $L_1,L_2\in \Delta_n^*$, we say that $L_1$ \emph{dissects} $L_2$ if $\vert L_1\cap L_2\vert=\infty$ and $\vert(\Delta_n^*\setminus L_1)\cap L_2\vert=\infty$. Given a context free language $L$, let $\kappa(L)$ denote the size of the smallest context free grammar $G$ that generates $L$. We define the size of a grammar to be the total number of symbols on the right sides of all production rules. Given positive integers $n,k$ with $k\geq 2$, we show that there are context free languages $L_1,L_2,\dots, L_{3k-3}\subseteq \Delta^*_n$ with $\kappa(L_i)\leq 40 k$ such that if $\alpha$ is a positive integer and $L\subseteq\Delta_n^*$ is an infinite language with the growth bounded by $(k,\alpha)$-tetration then there is a regular language $M$ such that $M\cap\left(\bigcap_{i=1}^{3k-3}L_i\right)$ dissects $L$ and the minimal deterministic finite automaton accepting $M$ has at most $k+\alpha+3$ states

Tree transducers, L systems, and two-way machines

A relationship between parallel rewriting systems and two-way machines is investigated. Restrictions on the “copying power” of these devices endow them with rich structuring and give insight into the issues of determinism, parallelism, and copying. Among the parallel rewriting systems considered are the top-down tree transducer; the generalized syntax-directed translation scheme and the ETOL system, and among the two-way machines are the tree-walking automaton, the two-way finite-state transducer, and (generalizations of) the one-way checking stack automaton. The. relationship of these devices to macro grammars is also considered. An effort is made .to provide a systematic survey of a number of existing results

The IO- and OI-hierarchies

AbstractAn analysis of recursive procedures in ALGOL 68 with finite modes shows, that a denotational semantics of this language can be described on the level of program schemes using a typed λ-calculus with fixed-point operators. In the first part of this paper, we derive classical schematological theorems for the resulting class of level-n schemes. In part two, we investigate the language families obtained by call-by-value and call-by-name interpretation of level-n schemes over the algebra of formal languages. It is proved, that differentiating according to the functional level of recursion leads to two infinite hierarchies of recursive languages, the IO- and OI-hierarchies, which can be characterized as canonical extensions of the regular, context-free, and IO- and OI-macro languages, respectively. Sufficient conditions are derived to establish strictness of IO-like hierarchies. Finally we derive, that recursion on higher types induces an infinite hierarchy of control structures by proving that level-n schemes are strictly less powerful than level-n+1 schemes