Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure

String languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first-order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the transitive closure. More precisely, first-order logic with k-ary deterministic transitive closure has the same power as deterministic automata walking on their input with k heads, additionally using a finite set of nested pebbles. This result is valid for strings, ordered trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toruses, and rectangular mazes. For nondeterministic automata, the logic is restricted to positive occurrences of transitive closure. The special case of k=1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary deterministic transitive closure. This refines our earlier result that placed these automata between first-order and monadic second-order logic on trees.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur

Advances and applications of automata on words and trees : abstracts collection

From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

The Parametric Ordinal-Recursive Complexity of Post Embedding Problems

Post Embedding Problems are a family of decision problems based on the interaction of a rational relation with the subword embedding ordering, and are used in the literature to prove non multiply-recursive complexity lower bounds. We refine the construction of Chambart and Schnoebelen (LICS 2008) and prove parametric lower bounds depending on the size of the alphabet.Comment: 16 + vii page

Generalizing input-driven languages: theoretical and practical benefits

Regular languages (RL) are the simplest family in Chomsky's hierarchy. Thanks to their simplicity they enjoy various nice algebraic and logic properties that have been successfully exploited in many application fields. Practically all of their related problems are decidable, so that they support automatic verification algorithms. Also, they can be recognized in real-time. Context-free languages (CFL) are another major family well-suited to formalize programming, natural, and many other classes of languages; their increased generative power w.r.t. RL, however, causes the loss of several closure properties and of the decidability of important problems; furthermore they need complex parsing algorithms. Thus, various subclasses thereof have been defined with different goals, spanning from efficient, deterministic parsing to closure properties, logic characterization and automatic verification techniques. Among CFL subclasses, so-called structured ones, i.e., those where the typical tree-structure is visible in the sentences, exhibit many of the algebraic and logic properties of RL, whereas deterministic CFL have been thoroughly exploited in compiler construction and other application fields. After surveying and comparing the main properties of those various language families, we go back to operator precedence languages (OPL), an old family through which R. Floyd pioneered deterministic parsing, and we show that they offer unexpected properties in two fields so far investigated in totally independent ways: they enable parsing parallelization in a more effective way than traditional sequential parsers, and exhibit the same algebraic and logic properties so far obtained only for less expressive language families

Hypertableau Reasoning for Description Logics

We present a novel reasoning calculus for the description logic SHOIQ^+---a knowledge representation formalism with applications in areas such as the Semantic Web. Unnecessary nondeterminism and the construction of large models are two primary sources of inefficiency in the tableau-based reasoning calculi used in state-of-the-art reasoners. In order to reduce nondeterminism, we base our calculus on hypertableau and hyperresolution calculi, which we extend with a blocking condition to ensure termination. In order to reduce the size of the constructed models, we introduce anywhere pairwise blocking. We also present an improved nominal introduction rule that ensures termination in the presence of nominals, inverse roles, and number restrictions---a combination of DL constructs that has proven notoriously difficult to handle. Our implementation shows significant performance improvements over state-of-the-art reasoners on several well-known ontologies

Cyclic Operator Precedence Grammars for Parallel Parsing

Operator precedence languages (OPL) enjoy the local parsability property, which essentially means that a code fragment enclosed within a pair of markers -- playing the role of parentheses -- can be compiled with no knowledge of its external context. Such a property has been exploited to build parallel compilers for languages formalized as OPLs. It has been observed, however, that when the syntax trees of the sentences have a linear substructure, its parsing must necessarily proceed sequentially making it impossible to split such a subtree into chunks to be processed in parallel. Such an inconvenience is due to the fact that so far much literature on OPLs has assumed the hypothesis that equality precedence relation cannot be cyclic. This hypothesis was motivated by the need to keep the mathematical notation as simple as possible. We present an enriched version of operator precedence grammars, called cyclic, that allows to use a simplified version of regular expressions in the right hand sides of grammar's rules; for this class of operator precedence grammars the acyclicity hypothesis of the equality precedence relation is no more needed to guarantee the algebraic properties of the generated languages. The expressive power of the cyclic grammars is now fully equivalent to that of other formalisms defining OPLs such as operator precedence automata, monadic second order logic and operator precedence expressions. As a result cyclic operator precedence grammars now produce also unranked syntax trees and sentences with flat unbounded substructures that can be naturally partitioned into chunks suitable for parallel parsing.Comment: 23 pages, 8 figures. arXiv admin note: text overlap with arXiv:2006.0123