Query Order and the Polynomial Hierarchy

Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] initiated the field of query order, which studies the ways in which computational power is affected by the order in which information sources are accessed. The present paper studies, for the first time, query order as it applies to the levels of the polynomial hierarchy. We prove that the levels of the polynomial hierarchy are order-oblivious. Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that all leaf language classes - and thus essentially all standard complexity classes - inherit all order-obliviousness results that hold for P.Comment: 14 page

On Second-Order Monadic Monoidal and Groupoidal Quantifiers

We study logics defined in terms of second-order monadic monoidal and groupoidal quantifiers. These are generalized quantifiers defined by monoid and groupoid word-problems, equivalently, by regular and context-free languages. We give a computational classification of the expressive power of these logics over strings with varying built-in predicates. In particular, we show that ATIME(n) can be logically characterized in terms of second-order monadic monoidal quantifiers

Leaf languages and string compression

AbstractTight connections between leaf languages and strings compressed by straight-line programs (SLPs) are established. It is shown that the compressed membership problem for a language L is complete for the leaf language class defined by L via logspace machines. A more difficult variant of the compressed membership problem for L is shown to be complete for the leaf language class defined by L via polynomial time machines. As a corollary, it is shown that there exists a fixed linear visibly pushdown language for which the compressed membership problem is PSPACE-complete. For XML languages, it is shown that the compressed membership problem is coNP-complete.Furthermore it is shown that the embedding problem for SLP-compressed strings is hard for PP (probabilistic polynomial time)

Evolución de la competencia comunicativa matemática en un contexto de master de formación de profesores de matemática: la evolución de Ester

En un contexto de master de formación de profesores de secundaria se encuentra Ester, una profesional que decide estudiar un Master de Formación del Profesorado de Secundaria en la especialidad de Matemática para ser profesor de matemática. En la investigación analizamos los textos producidos por los estudiantes de este Master y en este artículo queremos destacar las reflexiones de Ester en cuanto a lo que hemos definido como competencia comunicativa matemática en la formación de profesores. Por ello explicaremos el contexto de la investigación, cuáles fueron las tareas analizadas, específicamente, aquellas que se pensaron como desarrolladoras de reflexiones vinculadas a la comunicación de la matemática. Se concluye que debido a los procesos de formación vividos en el master de formación de profesores las reflexiones de Ester sufrieron una evolución progresiva en cuanto a competencia comunicativa matemática

The dot-depth and the polynomial hierarchy correspond on the delta levels

It is well-known that the Sigma_k- and Pi_k-levels of the dot-depth hierarchy and the polynomial hierarchy correspond via leaf languages. In this paper this correspondence will be extended to the Delta_k-levels of these hierarchies: Leaf^P(Delta_k^L) = Delta_k^p

On the Complexity of Universality for Partially Ordered NFAs

International audiencePartially ordered nondeterminsitic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, i.e., for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterise the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSpace-complete. Our results also prove the complexity of the inclusion and equivalence problems, since universality provides the lower bound, while the upper bound is mostly known or proved in the paper

On the Complexity of Universality for Partially Ordered NFAs

Partially ordered nondeterminsitic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, i.e., for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterise the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSPACE-complete. Our results also prove the complexity of the inclusion and equivalence problems, since universality provides the lower bound, while the upper bound is mostly known or proved in the paper

Partially Ordered Automata and Piecewise Testability

Partially ordered automata are automata where the transition relation induces a partial order on states. The expressive power of partially ordered automata is closely related to the expressivity of fragments of first-order logic on finite words or, equivalently, to the language classes of the levels of the Straubing-Th\'erien hierarchy. Several fragments (levels) have been intensively investigated under various names. For instance, the fragment of first-order formulae with a single existential block of quantifiers in prenex normal form is known as piecewise testable languages or $J$-trivial languages. These languages are characterized by confluent partially ordered DFAs or by complete, confluent, and self-loop-deterministic partially ordered NFAs (ptNFAs for short). In this paper, we study the complexity of basic questions for several types of partially ordered automata on finite words; namely, the questions of inclusion, equivalence, and ($k$-)piecewise testability. The lower-bound complexity boils down to the complexity of universality. The universality problem asks whether a system recognizes all words over its alphabet. For ptNFAs, the complexity of universality decreases if the alphabet is fixed, but it is open if the alphabet may grow with the number of states. We show that deciding universality for general ptNFAs is as hard as for general NFAs. Our proof is a novel and nontrivial extension of our recent construction for self-loop-deterministic partially ordered NFAs, a model strictly more expressive than ptNFAs. We provide a comprehensive picture of the complexities of the problems of inclusion, equivalence, and ($k$-)piecewise testability for the considered types of automata