A note on a decomposition theorem for simple deterministic languages

A procedure to resolve simple deterministic languages into the concatenation of other simple deterministic languages is presented

Top-down tree transducers with regular look-ahead

Top-down tree transducers with regular look-ahead are introduced. It is shown how these can be decomposed and composed, and how this leads to closure properties of surface sets and tree transformation languages. Particular attention is paid to deterministic tree transducers

An Algorithmic Metatheorem for Directed Treewidth

The notion of directed treewidth was introduced by Johnson, Robertson, Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as a first step towards an algorithmic metatheory for digraphs. They showed that some NP-complete properties such as Hamiltonicity can be decided in polynomial time on digraphs of constant directed treewidth. Nevertheless, despite more than one decade of intensive research, the list of hard combinatorial problems that are known to be solvable in polynomial time when restricted to digraphs of constant directed treewidth has remained scarce. In this work we enrich this list by providing for the first time an algorithmic metatheorem connecting the monadic second order logic of graphs to directed treewidth. We show that most of the known positive algorithmic results for digraphs of constant directed treewidth can be reformulated in terms of our metatheorem. Additionally, we show how to use our metatheorem to provide polynomial time algorithms for two classes of combinatorial problems that have not yet been studied in the context of directed width measures. More precisely, for each fixed $k,w \in \mathbb{N}$, we show how to count in polynomial time on digraphs of directed treewidth $w$, the number of minimum spanning strong subgraphs that are the union of $k$ directed paths, and the number of maximal subgraphs that are the union of $k$ directed paths and satisfy a given minor closed property. To prove our metatheorem we devise two technical tools which we believe to be of independent interest. First, we introduce the notion of tree-zig-zag number of a digraph, a new directed width measure that is at most a constant times directed treewidth. Second, we introduce the notion of $z$-saturated tree slice language, a new formalism for the specification and manipulation of infinite sets of digraphs.Comment: 41 pages, 6 figures, Accepted to Discrete Applied Mathematic

Macro tree transducers

Macro tree transducers are a combination of top-down tree transducers and macro grammars. They serve as a model for syntax-directed semantics in which context information can be handled. In this paper the formal model of macro tree transducers is studied by investigating typical automata theoretical topics like composition, decomposition, domains, and ranges of the induced translation classes. The extension with regular look-ahead is considered

Bottom-up and top-down tree transformations - a comparison

The top-down and bottom-up tree transducer are incomparable with respect to their transformation power. The difference between them is mainly caused by the different order in which they use the facilities of copying and nondeterminism. One can however define certain simple tree transformations, independent of the top-down/bottom-up distinction, such that each tree transformation, top-down or bottom-up, can be decomposed into a number of these simple transformations. This decomposition result is used to give simple proofs of composition results concerning bottom-up tree transformations.\ud \ud A new tree transformation model is introduced which generalizes both the top-down and the bottom-up tree transducer

Graph Spectral Properties of Deterministic Finite Automata

We prove that a minimal automaton has a minimal adjacency matrix rank and a minimal adjacency matrix nullity using equitable partition (from graph spectra theory) and Nerode partition (from automata theory). This result naturally introduces the notion of matrix rank into a regular language L, the minimal adjacency matrix rank of a deterministic automaton that recognises L. We then define and focus on rank-one languages: the class of languages for which the rank of minimal automaton is one. We also define the expanded canonical automaton of a rank-one language.Comment: This paper has been accepted at the following conference: 18th International Conference on Developments in Language Theory (DLT 2014), August 26 - 29, 2014, Ekaterinburg, Russi

Multiple Context-Free Tree Grammars: Lexicalization and Characterization

Multiple (simple) context-free tree grammars are investigated, where "simple" means "linear and nondeleting". Every multiple context-free tree grammar that is finitely ambiguous can be lexicalized; i.e., it can be transformed into an equivalent one (generating the same tree language) in which each rule of the grammar contains a lexical symbol. Due to this transformation, the rank of the nonterminals increases at most by 1, and the multiplicity (or fan-out) of the grammar increases at most by the maximal rank of the lexical symbols; in particular, the multiplicity does not increase when all lexical symbols have rank 0. Multiple context-free tree grammars have the same tree generating power as multi-component tree adjoining grammars (provided the latter can use a root-marker). Moreover, every multi-component tree adjoining grammar that is finitely ambiguous can be lexicalized. Multiple context-free tree grammars have the same string generating power as multiple context-free (string) grammars and polynomial time parsing algorithms. A tree language can be generated by a multiple context-free tree grammar if and only if it is the image of a regular tree language under a deterministic finite-copying macro tree transducer. Multiple context-free tree grammars can be used as a synchronous translation device.Comment: 78 pages, 13 figure