Sublinearly space bounded iterative arrays

Iterative arrays (IAs) are a, parallel computational model with a sequential processing of the input. They are one-dimensional arrays of interacting identical deterministic finite automata. In this note, realtime-lAs with sublinear space bounds are used to accept formal languages. The existence of a proper hierarchy of space complexity classes between logarithmic anel linear space bounds is proved. Furthermore, an optimal spacc lower bound for non-regular language recognition is shown. Key words: Iterative arrays, cellular automata, space bounded computations, decidability questions, formal languages, theory of computatio

Unambiguous Turn Position and Rational Trace Languages

We show the existence of rational trace languages defined over direct products of free monoids that have inherent ambiguity of the order of log n and n 1/2 . This result is obtained by studying the relationship between trace languages and linear context-free grammars that satisfy a special unambiguity condition on the position of the last step of derivation

On the Computational Complexity of MapReduce

In this paper we study MapReduce computations from a complexity-theoretic perspective. First, we formulate a uniform version of the MRC model of Karloff et al. (2010). We then show that the class of regular languages, and moreover all of sublogarithmic space, lies in constant round MRC. This result also applies to the MPC model of Andoni et al. (2014). In addition, we prove that, conditioned on a variant of the Exponential Time Hypothesis, there are strict hierarchies within MRC so that increasing the number of rounds or the amount of time per processor increases the power of MRC. To the best of our knowledge we are the first to approach the MapReduce model with complexity-theoretic techniques, and our work lays the foundation for further analysis relating MapReduce to established complexity classes

Sublogarithmic bounds on space and reversals

The complexity measure under consideration is SPACE x REVERSALS for Turing machines that are able to branch both existentially and universally. We show that, for any function h(n) between log log n and log n, Pi(1) SPACE x REVERSALS(h(n)) is separated from Sigma(1)SPACE x REVERSALS(h(n)) as well as from co Sigma(1)SPACE x REVERSALS(h(n)), for middle, accept, and weak modes of this complexity measure. This also separates determinism from the higher levels of the alternating hierarchy. For "well-behaved" functions h(n) between log log n and log n, almost all of the above separations can be obtained by using unary witness languages. In addition, the construction of separating languages contributes to the research on minimal resource requirements for computational devices capable of recognizing nonregular languages. For any (arbitrarily slow growing) unbounded monotone recursive function f(n), a nonregular unary language is presented that can be accepted by a middle Pi(1) alternating Turing machine in s(n) space and i(n) input head reversals, with s(n) . i(n) is an element of O(log log n . f(n)). Thus, there is no exponential gap for the optimal lower bound on the product s(n) . i(n) between unary and general nonregular language acceptance-in sharp contrast with the one-way case

On Languages Generated by Signed Grammars

We consider languages defined by signed grammars which are similar to context-free grammars except productions with signs associated to them are allowed. As a consequence, the words generated also have signs. We use the structure of the formal series of yields of all derivation trees over such a grammar as a method of specifying a formal language and study properties of the resulting family of languages.Comment: In Proceedings NCMA 2023, arXiv:2309.0733

Hamiltonian simulation with nearly optimal dependence on all parameters

We present an algorithm for sparse Hamiltonian simulation whose complexity is optimal (up to log factors) as a function of all parameters of interest. Previous algorithms had optimal or near-optimal scaling in some parameters at the cost of poor scaling in others. Hamiltonian simulation via a quantum walk has optimal dependence on the sparsity at the expense of poor scaling in the allowed error. In contrast, an approach based on fractional-query simulation provides optimal scaling in the error at the expense of poor scaling in the sparsity. Here we combine the two approaches, achieving the best features of both. By implementing a linear combination of quantum walk steps with coefficients given by Bessel functions, our algorithm's complexity (as measured by the number of queries and 2-qubit gates) is logarithmic in the inverse error, and nearly linear in the product $\tau$ of the evolution time, the sparsity, and the magnitude of the largest entry of the Hamiltonian. Our dependence on the error is optimal, and we prove a new lower bound showing that no algorithm can have sublinear dependence on $\tau$.Comment: 21 pages, corrects minor error in Lemma 7 in FOCS versio

Entanglement entropy in homogeneus, fermionic chains: some results and some conjectures

El objetivo de esta tesis es el estudio de la entropía de entrelazamiento de Rényi en los estados estacionarios de cadenas de fermiones sin spin descritas por un Hamiltoniano cuadrático general con invariancia translacional y posibles acoplos a larga distancia.Nuestra investigación se basa en la relación que existe entre la matriz densidad de los estados estacionarios y la correspondiente matriz de correlaciones entre dos puntos. Esta propiedad reduce la complejidad de calcular numéricamente la entropía de entrelazamiento y permite expresar esta magnitud en términos del determinante del resolvente de la matriz de correlaciones.Dado que la cadena es invariante translacional, la matriz de correlaciones es una matriz block Toeplitz. En vista de este hecho, la filosofía que seguimos en esta tesis es la de aprovecharnos de las propiedades asintóticas de este tipo de determinantes para investigar la entropía de entrelazamiento de Rényi en el límite termodinámico. Un aspecto interesante es que los resultados conocidos sobre el comportamiento asintótico de los determinantes block Toeplitz no son válidos para algunas de las matrices de correlaciones que consideraremos. Intentando llenar esta laguna, obtenemos algunos resultados originales sobre el comportamiento asintótico de los determinantes de matrices de Toeplitz y block Toeplitz.Estos nuevos resultados combinados con los ya previamente conocidos nos permiten obtener analíticamente el término dominante en la expansión de la entropía de entrelazamiento, tanto para un único intervalo de puntos o sites contiguos de la cadena como para subsistemas formados por varios intervalos disjuntos. En particular, descubrimos que los acoplos de largo alcance dan lugar a nuevas propiedades del comportamiento asintótico de la entropía tales como la aparición de un término logarítmico no universal fuera de los puntos críticos cuando los términos de pairing decaen siguiendo una ley de potencias o un crecimiento sublogarítmico cuando dichos acoplos decaen logarítmicamente. El estudio de la entropía de entrelazamiento a través de los determinantes block Toeplitz también nos ha llevado a descubrir una nueva simetría de la entropía de entrelazamiento bajo transformaciones de Möbius que pueden verse como transformaciones de los acoplos de la teoría. En particular, encontramos que para teorías críticasesta simetría presenta un intrigante paralelismo con las transformaciones conformes en el espacio-tiempo. <br /