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

Algorithms and lower bounds in finite automata size complexity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 97-99).In this thesis we investigate the relative succinctness of several types of finite automata, focusing mainly on the following four basic models: one-way deterministic (1)FAs), one-way nondeterministic (1NFAs), two-way deterministic (2DFAS), and two-way nondeterministic (2NFAS). First, we establish the exact values of the trade-offs for all conversions from two-way to one-way automata. Specifically, we prove that the functions ... return the exact values of the trade-offs from 2DFAS to 1DFAS, from 2NFAS to 1DFAs, and from 2DFAs or 2NFAS to 1NFAs, respectively. Second, we examine the question whether the trade-offs from NFAs or 2NFAS to 2DiFAs are polynomial or not. We prove two theorems for liveness, the complete problem for the conversion from 1NFAS to 2DFAS. We first focus on moles, a restricted class of 2NFAs that includes the polynomially large 1NFAS which solve liveness. We prove that, in contrast, 2DFA moles cannot solve liveness, irrespective of size.(cont.) We then focus on sweeping 2NFAS, which can change the direction of their input head only on the end-markers. We prove that all sweeping 2NFAs solving the complement of liveness are of exponential size. A simple modification of this argument also proves that the trade-off from 2DFAS to sweeping 2NFAS is exponential. Finally, we examine conversions between two-way automata with more than one head-like devices (e.g., heads, linearly bounded counters, pebbles). We prove that, if the automata of some type A have enough resources to (i) solve problems that no automaton of some other type B can solve, and (ii) simulate any unary 2DFA that has additional access to a linearly-bounded counter, then the trade-off from automata of type A to automata of type B admits no recursive upper bound.by Christos Kapoutsis.Ph.D

The complexity of membership problems for circuits over sets of integers

AbstractWe investigate the complexity of membership problems for {∪,∩,-,+,×}-circuits computing sets of integers. These problems are a natural modification of the membership problems for circuits computing sets of natural numbers studied by McKenzie and Wagner [The complexity of membership problems for circuits over sets of natural numbers, Lecture Notes in Computer Science, Vol. 2607, 2003, pp. 571–582]. We show that there are several membership problems for which the complexity in the case of integers differs significantly from the case of the natural numbers: testing membership in the subset of integers produced at the output of a {∪,+,×}-circuit is NEXPTIME-complete, whereas it is PSPACE-complete for the natural numbers. As another result, evaluating {-,+}-circuits is shown to be P-complete for the integers and PSPACE-complete for the natural numbers. The latter result extends McKenzie and Wagner's work in nontrivial ways. Furthermore, evaluating {×}-circuits is shown to be NL∧⊕L-complete, and several other cases are resolved

Поведение конечных автоматов в лабиринтах

The paper is devoted to the study of problems on the behavior of finite automata in mazes. For any n, a maze is constructed that can be bypassed with 2n stones but you can’t get around with n stones. The range of tasks is extensive and touches upon key aspects of theoretical Computer Science. Of course, the solution of such problems does not mean the automatic solution of complex problems of complexity theory, however, the consideration of these issues can have a positive impact on the understanding of the essence of theoretical Computer Science. It is hoped that the behavior of automata in mazes is a good model for non-trivial information theoretic problems, and the development of methods and approaches to the study of robot behavior will give more serious results in the future. Problems related to automaton analysis of geometric media have a rather rich history of study. The first work that gave rise to this kind of problems, it is necessary to recognize the work of Shannon [24]. It deals with a model of a mouse in the form of an automaton, which must find a specific target in the maze. Another early work, one way or another affecting our problems, is the work of Fisher [9] on computing systems with external memory in the form of a discrete plane. A serious impetus to the study of the behavior of automata in mazes was the work of Depp [7, 8], in which the following model is proposed: there is a certain configuration of cells from mathbbZ^2 (chess maze), in which finite automata, surveying some neighborhood of the cell in which they are, can move to an adjacent cell in one of four directions. The main question posed in such a model is whether there is an automaton that bypasses all such mazes. In [20], Muller constructed a flat trap for a given automaton (a maze that does not completely bypass) in the form of a 3-graph. Budach [5] constructed a chess trap for any given finite automaton. Note that Budach’s solution was quite complex (the first versions contained 175 pages). More visual solutions to this question are presented here [29, 31, 33, 34]. Antelman [2] estimated the complexity of such a trap by the number of cells, and in [1] Antelman, Budach, and Rollick made a finite trap for any finite automaton system. In the formulation with a chess maze and one automaton, there are a number of results related to the problems of traversability of labyrinths with different numbers of holes, with bundles of labyrinths by the number of States of the automaton, and other issues. An overview of such problems can be found for example here [35]. The impossibility of traversing all flat chess labyrinths with one automaton raised the question of studying the possible amplifications of the automaton model, which will solve the problem of traversal. The main way of strengthening can be the consideration of a collective of automata, instead of one automaton, interacting with each other. A special and widely used case is the consideration of a system of one full-fledged automaton and a certain number of automata of stones, which have no internal state and can move only together with the main automaton. Interaction between machines is a key feature of this gain, it is allowed to have a collective (or one machine with stones) external memory, thereby significantly diversifies its behavior. If you get rid of the interaction of automata, the resulting  independent system will be little better than a single machine. Next, we discuss the known results associated with the collective automata.Работа посвящена исследованию задач о поведении конечных автоматов в лабиринтах. Для любого n строится лабиринт, который можно обойти с помощью 2n камней но нельзя обойти с помощью n камней. Спектр задач обхода обширен и затрагивает ключевые аспекты теоретической Computer Science. Конечно, решение таких задач не означает автоматическое решение сложных проблем теории сложности, тем не менее рассмотрение данных вопросов может положительно сказаться на понимании сути теоретической Computer Science. Есть надежда, что поведение автоматов в лабиринтах является хорошей моделью для нетривиальных теоретико-информационных задач, и отработка методов и подходов к исследованию поведения роботов даст более серьезные результаты с будущем. Задачи связанные c автоматным анализом геометрических сред имеют довольно богатую историю изучения. Первой работой, давшей начало подобного рода задачам, стоит признать работу Шеннона [24]. В ней рассматривается модель мыши в виде автомата, которая должна найти определенную цель в лабиринте. Другая ранняя работа, так или иначе затрагивающая нашу проблематику, это работа Фишера [9] о вычислительных системах с внешней памятью в виде дискретной плоскости. Серьёзным толчком к исследование поведения автоматов в лабиринтах послужила работы Деппа [7, 8], в которых предложена следующая модель: имеется некоторая конфигурация клеток из Z^2 (шахматный лабиринт), в которой конечные автоматы, обозревая некоторую окрестность клетки, в которой они находятся, могут перемещаться в соседнюю клетку в одном из четырёх направлений. Основной вопрос, который ставится в подобной модели, существует ли автомат обходящий все подобные лабиринты. В [20] Мюллер построил для заданного автомата плоскую ловушку (лабиринт который обходится не полностью) в виде 3-графа. Будах [5] построил шахматную ловушку для любого заданного конечного автомата. Отметим, что решение Будаха было довольно сложным (первые варианты содержали 175 страниц). Более наглядные решения данного вопроса представлены здесь [29, 31, 33, 34]. Антельман [2] оценил сложность подобной ловушки по числу клеток, а в [1] Антельман, Будах и Роллик сделали конечную ловушку для любой конечной системы автоматов. В постановке с шахматным лабиринтом и одним автоматом есть ещё ряд результатов, связанных с проблемами обходимости лабиринтов с различными числом дыр, с расслоениями лабиринтов по количеству состояний автомата и другими вопросами. Обзор подобных проблем можно найти например здесь [35]. Невозможность обхода всех плоских шахматных лабиринтов одним автоматом выдвинула вопрос об изучении возможных усилений модели автомата, которая решит задачу обхода. Основным способом усиления может являться рассмотрение коллектива автоматов,вместо одного автомата, взаимодействующих между собой. Частным и широко используемым случаем является рассмотрение системы из одного полноценного автомата и некоторого количества автоматов камней, которые не имеют внутреннего состояние и могут передвигаться только совместно с главным автоматом. Взаимодействие между автоматами является ключевой особенностью данного усиления, оно позволяется иметь коллективу (или одному автомату с камнями) внешнюю память, тем самым существенно разнообразит его поведение. Если от взаимодействия автоматов избавиться, то полученная независимая система будет немногим лучше одного автомата. Далее обсудим известные результаты связанные с коллективом автоматов

Upper and lower bounds for first order expressibility

AbstractWe study first order expressibility as a measure of complexity. We introduce the new class Var&Sz[v(n),z(n)] of languages expressible by a uniform sequence of sentences with v(n) variables and size O[z(n)]. When v(n) is constant our uniformity condition is syntactical and thus the following characterizations of P and PSPACE come entirely from logic. NSPACE|log n|⊆⋃k=1,2,…Var&Sz|k, log(n)|⊆DSPACE|log2(n)|,P=⋃k=1,2,…Var&Sz|k, nk|,PSPACE=⋃k=1,2,…Var&Sz|k, 2nk|. The above means, for example, that the properties expressible with constantly many variables in polynomial size sentences are just the polynomial time recognizable properties. These results hold for languages with an ordering relation, e.g., for graphs the vertices are numbered. We introduce an “alternating pebbling game” to prove lower bounds on the number of variables and size needed to express properties without the ordering. We show, for example, that k variables are needed to express Clique(k), suggesting that this problem requires DTIME[nk]

Balancing Bounded Treewidth Circuits

Algorithmic tools for graphs of small treewidth are used to address questions in complexity theory. For both arithmetic and Boolean circuits, it is shown that any circuit of size $n^{O(1)}$ and treewidth $O(\log^i n)$ can be simulated by a circuit of width $O(\log^{i+1} n)$ and size $n^c$, where $c = O(1)$, if $i=0$, and $c=O(\log \log n)$ otherwise. For our main construction, we prove that multiplicatively disjoint arithmetic circuits of size $n^{O(1)}$ and treewidth $k$ can be simulated by bounded fan-in arithmetic formulas of depth $O(k^2\log n)$. From this we derive the analogous statement for syntactically multilinear arithmetic circuits, which strengthens a theorem of Mahajan and Rao. As another application, we derive that constant width arithmetic circuits of size $n^{O(1)}$ can be balanced to depth $O(\log n)$, provided certain restrictions are made on the use of iterated multiplication. Also from our main construction, we derive that Boolean bounded fan-in circuits of size $n^{O(1)}$ and treewidth $k$ can be simulated by bounded fan-in formulas of depth $O(k^2\log n)$. This strengthens in the non-uniform setting the known inclusion that $SC^0 \subseteq NC^1$. Finally, we apply our construction to show that {\sc reachability} for directed graphs of bounded treewidth is in $LogDCFL$

Tight bounds for undirected graph exploration with pebbles and multiple agents

We study the problem of deterministically exploring an undirected and initially unknown graph with $n$ vertices either by a single agent equipped with a set of pebbles, or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles $\Theta(\log n)$ bits of memory are necessary and sufficient to explore any graph with at most $n$ vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles, or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory $\Theta(\log \log n)$ pebbles are necessary and sufficient for exploration. We further prove that the same bound holds for the number of collaborating agents needed for exploration. For the upper bound, we devise an algorithm for a single agent with constant memory that explores any $n$-vertex graph using $\mathcal{O}(\log \log n)$ pebbles, even when $n$ is unknown. The algorithm terminates after polynomial time and returns to the starting vertex. Since an additional agent is at least as powerful as a pebble, this implies that $\mathcal{O}(\log \log n)$ agents with constant memory can explore any $n$-vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph of size $n$ is already $\Omega(\log \log n)$ when we allow each agent to have at most $\mathcal{O}( \log n ^{1-\varepsilon})$ bits of memory for any $\varepsilon>0$. This also implies that a single agent with sublogarithmic memory needs $\Theta(\log \log n)$ pebbles to explore any $n$-vertex graph

DNA Computing: Modelling in Formal Languages and Combinatorics on Words, and Complexity Estimation

DNA computing, an essential area of unconventional computing research, encodes problems using DNA molecules and solves them using biological processes. This thesis contributes to the theoretical research in DNA computing by modelling biological processes as computations and by studying formal language and combinatorics on words concepts motivated by DNA processes. It also contributes to the experimental research in DNA computing by a scaling comparison between DNA computing and other models of computation. First, for theoretical DNA computing research, we propose a new word operation inspired by a DNA wet lab protocol called cross-pairing polymerase chain reaction (XPCR). We define and study a word operation called word blending that models and generalizes an unexpected outcome of XPCR. The input words are uwx and ywv that share a non-empty overlap w, and the output is the word uwv. Closure properties of the Chomsky families of languages under this operation and its iterated version, the existence of a solution to equations involving this operation, and its state complexity are studied. To follow the XPCR experimental requirement closely, a new word operation called conjugate word blending is defined, where the subwords x and y are required to be identical. Closure properties of the Chomsky families of languages under this operation and the XPCR experiments that motivate and implement it are presented. Second, we generalize the sequence of Fibonacci words inspired by biological concepts on DNA. The sequence of Fibonacci words is an infinite sequence of words obtained from two initial letters f(1) = a and f(2)= b, by the recursive definition f(n+2) = f(n+1)*f(n), for all positive integers n, where * denotes word concatenation. After we propose a unified terminology for different types of Fibonacci words and corresponding results in the extensive literature on the topic, we define and explore involutive Fibonacci words motivated by ideas stemming from theoretical studies of DNA computing. The relationship between different involutive Fibonacci words and their borderedness and primitivity are studied. Third, we analyze the practicability of DNA computing experiments since DNA computing and other unconventional computing methods that solve computationally challenging problems often have the limitation that the space of potential solutions grows exponentially with their sizes. For such problems, DNA computing algorithms may achieve a linear time complexity with an exponential space complexity as a trade-off. Using the subset sum problem as the benchmark problem, we present a scaling comparison of the DNA computing (DNA-C) approach with the network biocomputing (NB-C) and the electronic computing (E-C) approaches, where the volume, computing time, and energy required, relative to the input size, are compared. Our analysis shows that E-C uses a tiny volume compared to that required by DNA-C and NB-C, at the cost of the E-C computing time being outperformed first by DNA-C and then by NB-C. In addition, NB-C appears to be more energy efficient than DNA-C for some input sets, and E-C is always an order of magnitude less energy efficient than DNA-C