M. Blum and W. J. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In Proceedings of the Eighteenth Annual Symposium on Foundations of Computer Science, pages 147--161, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....dout(v) Communicated by S. Khuller: submitted January 2002; revised June 2002 Work by S. Even supported by the Fund for the Promotion of Research at the Technion. 1 Introduction and Model The question of how best to traverse an unknown maze, given only local knowledge, has been long studied. [8, 3, 5, 7, 6, 1, 4, 2]. Without the ability to store some knowledge about the maze the searcher can easily become trapped in cycles. In this paper we consider the case of directed Eulerian mazes graphs, and show that a finite state automaton, with access to some local memory stored in the vertices, can find an ....

M. Blum and W.J. Sakoda, On the Capability of Finite Automata in 2 and 3 Dimensional Space. In Proceeding of the Eighteenth Annual Symposium on Foundations of Computer Science, 1977. pp. 147-161.

....[52] improves this by giving an algorithm with competitive ratio 4 8 2:61. For rectilinear streets, the algorithm achieves a competitive ratio of 2. There are many other related papers in the literature, particularly in the area of robotics (e.g. 57] and maze searching (e.g. [25, 24]) Rao, Kareti, Shi, and Iyengar [68] give a survey of work on robot navigation in unknown terrains. 3.3 Formal model We model the robot s environment as a finite connected undirected graph G = V; E) with distinguished start vertex s. Vertices represent accessible locations. Edges represent ....

Manuel Blum and W. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science. IEEE, 1977.

....them. The pebbles represent vertex names that a structured algorithm might record in its workspace. Walking represents replacing a vertex name by some adjacent vertex found in the input. Jumping represents copying a previously recorded vertex name. Rabin (see [20] Savitch [31] Blum and Sakoda [13], Blum and Kozen [12] Hemmerling [24] and others have considered similar models; see Hemmerling s monograph for an extensive bibliography (going back over a century) emphasizing results for labyrinths graphs embedded in two or three dimensional Euclidean space. The JAG is a structured ....

....the tradeoff between time and space for the problem of undirected graph traversal. The JAG variant we consider is more restricted than the model introduced by Cook and Rackoff, because the pebbles are not permitted to jump. This nonjumping model is closer to the one studied by Blum and Sakoda [13], Blum and Kozen [12] and Hemmerling [24] We will distinguish this nonjumping variant by referring to it as a WAG walking automaton for graphs . Several authors have considered traversal of undirected regular graphs by a WAG with an unlimited number of states but only the minimum number ....

M. Blum and W. J. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In 18th Annual Symposium on Foundations of Computer Science, pages 147--161, Providence, RI, Oct. 1977. IEEE.

....cells traversed, i.e. the number of food pellets eaten. The food trail must not have any gaps but there are no restrictions on the width of the trail (the problem is then essentially that of searching a finite region of the plane, which can be solved by a 2 d finite automaton using 4 pebbles [1]) In this implementation, a trail is defined by an anti ant , which lays a trail site by site by moving around on the grid. 3 Methods The technique used to evolve computer programs in this paper is known as Genetic Programming (GP) 5] GP is an the extension of the conventional genetic ....

Blum, M., Sakoda, W. J.: On the capability of finite automata in 2 and 3 dimensional space, in Proceedings of 18th IEEE Conference on Foundations of Computer Science, pp. 147--161, (1977).

....A complete survey of graph algorithms for searching unknown graphs is not intended here; we only discuss some graph problems that appear to be very similar to the robot navigation algorithms in unknown terrains. In terms of searching by an automata of the type of last section, Blum and Sakoda [8] posed the question of weather it is easier to search mazes than planar graphs (planar graphs are the graphs that can be embedded in plane such that no two edges intersect [30] Mazes and regular planar graphs (planar graphs where each node has the same number of neighbors) appear similar on ....

M. Blum and W. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In 17th Annual Symposium on Foundations of Computer Science, pages 147--161, October 1977.

....are interested in the generative power of such devices. In the last section, we provide a list of open problems in the area as stimulus for further research. 2 Preliminaries In the main part of this section, we will introduce the definitions and notations for arrays and sequential array grammars [2, 11, 15, 25, 30] and give some explanatory examples and well known results. First, we recall some basic notions from the theory of formal languages (for more details, the reader is referred to [28] For an alphabet V , by V we denote the free monoid generated by V under the operation of concatenation; jxj ....

....v : Z n Z n is defined by v (w) w v for all w 2 Z n , and for any array A 2 V n we define v (A) the corresponding n dimensional array translated by v, by ( v (A) w) A (w Gamma v) for all w 2 Z n . The vector (0; 0) 2 Z n is denoted by Omega n . Usually [2, 25, 30, 31], arrays are regarded as equivalence classes of arrays with respect to linear translations, i.e. only the relative positions of the symbols 6= # in the plane are taken into account: The equivalence class [A] of an array A 2 V n is defined by [A] fB 2 V n j B = v (A) for some v 2 Z n ....

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M. Blum and W. J. Sakoda, On the capability of finite automata in 2 and 3 dimensional space. In: Proc. 18th Ann. Symp. on Foundations in Computer Science (1977), pp. 147--161.

....them. The pebbles represent vertex names that a structured algorithm might record in its workspace. Walking represents replacing a vertex name by some adjacent vertex found in the input. Jumping represents copying a previously recorded vertex name. Rabin (see [24] Savitch [45] Blum and Sakoda [13], Blum and Kozen [12] Hemmerling [30] and others have considered similar models; see Hemmerling s monograph for an extensive bibliography (going back over a century) emphasizing results for labyrinths graphs embedded in two or three dimensional Euclidean space. The JAG is a structured ....

....complete freedom. Such models are surprisingly powerful; see Section 3. Nevertheless, in a companion paper [9] we prove a lower bound on a model with freely moving pebbles, but without the ability to jump one pebble to another. This nonjumping model is closer to the one studied by Blum and Sakoda [13], Blum and Kozen [12] and Hemmerling [30] We will distinguish this nonjumping variant by referring to it as a WAG walking automaton for graphs . Following the preliminary appearance of some of these results [10] Edmonds [26] proved a much stronger result for traversing undirected graphs ....

M. Blum and W. J. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In 18th Annual Symposium on Foundations of Computer Science, pages 147--161, Providence, RI, Oct. 1977. IEEE.

....and partially by DARPA grant DABT63 96 C 0018. Laboratory for Computer Science, MIT, 545 Technology Square, Cambridge, MA 02139. Email: salil math.mit.edu. Supported by a DOD NDSEG doctoral fellowship and partially by DARPA grant DABT63 96 C 0018. tions on the form of the environment (cf. [13, 12, 16, 23, 14, 27, 7, 4, 1]. In this paper, we consider a model that makes very limited assumptions about the environment, and give efficient algorithms to solve the mapping problem in this general setting. A natural way to model the problem is by a robot exploring a graph G = V;E) The case in which the graph has both ....

....algorithm. 6 Most early work on graph exploration assumed that the robot is a finite automaton. Rabin [24] first proposed the idea of providing the automaton with pebbles to help it explore. This led to a body of work examining the number of pebbles needed to explore various environments [29, 13, 12, 3, 25]. For a survey on automata exploring labyrinths, see [21] Deng and Papadimitriou [16] propose and study the problem of exploring an unknown directed graph having labeled vertices. Albers and Henzinger [1] give improved al 6 In light of our results and those of Bender and Slonim, we see that a ....

M. Blum and W. J. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In Proceedings of the Eighteenth Annual Symposium on Foundations of Computer Science, pages 147--161, 1977.

....scan the same symbol k times (each of them can read the same information once) in the array case for our purposes we require that the whole automaton may scan a certain position only once. Let us mention that two dimensional automata which cannot visit one point twice are also called worms in [2]. In passing, this excludes a head sensing ability: it is not possible for two heads to assume the same position at the same time. Moreover, in our model we include the possibility that an automaton head may split in a certain, local way, and that it may be totally removed if it is not necessary ....

M. Blum and W. J. Sakoda, On the capability of finite automata in 2 and 3 dimensional space. In: Proc. 18th Ann. Symp. on Foundations in Computer Science (1977), pp. 147--161.

....an unknown environment is a fundamental problem with applications ranging from robot navigation to searching the World Wide Web. As such, a large body of work has focused on finding efficient solutions to variants of the problem, with restrictive assumptions on the form of the environment (cf. [13, 12, 16, 22, 14, 26, 7, 4, 1]. In this paper, we consider a model that makes very limited assumptions about the environment, and give efficient algorithms to solve the mapping problem in this general setting. A natural way to model the problem is by a robot exploring a graph. The case where the graph has both undirected ....

....algorithm. 6 Most early work on graph exploration assumed that the robot is a finite automaton. Rabin [23] first proposed the idea of providing the automaton with pebbles to help it explore. This led to a body of work examining the number of pebbles needed to explore various environments [28, 13, 12, 3, 24]. Deng and Papadimitriou [16] propose and study the problem of exploring an unknown directed graph having labeled vertices. 5 Actually, the robot may be at vertices equivalent under automorphism, but we avoid this issue in the introduction. 6 In light of our results and those of Bender and ....

M. Blum and W. J. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In Proceedings of the Eighteenth Annual Symposium on Foundations of Computer Science, pages 147-- 161, 1977.

.... also many results on learning unknown graphs under various conditions (e.g. BRS93] DP90] RS87] RS93] Rabin proposed the idea of dropping pebbles to mark nodes [Rab67] This suggestion led to work exploring the searching capabilities of a finite automaton supplied with pebbles (e.g. BS77] BK78] Sav72] Cook and Rackoff generalized the idea of pebbles to jumping automata [CR80] However, most previous work has concentrated on learning undirected graphs or graphs with distinguishable nodes. The power behind the two robot model lies in the robots abilities to recognize each ....

M. Blum and W. J. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In 18th Annual Symposium on Foundations of Computer Science, pages 147--161. IEEE, 1977.

....a number of pebbles that can be used to mark nodes, the single robot s plight improves. Rabin first proposed the idea of dropping pebbles to mark nodes [Rab67] This suggestion led to a body of work exploring the searching capabilities of a finite automaton supplied with pebbles. Blum and Sakoda [BS77] consider the question of whether a finite set of finite automata can search a 2 or 3 dimensional obstructed grid. They prove that a single automaton with just four pebbles can completely search any 2 dimensional finite maze, and that a single automaton with seven pebbles can completely search ....

M. Blum and W. J. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In 18th Annual Symposium on Foundations of Computer Science, pages 147--161. IEEE, 1977.

....them. The pebbles represent vertex names that a structured algorithm might record in its workspace. Walking represents replacing a vertex name by some adjacent vertex found in the input. Jumping represents copying a previously recorded vertex name. Rabin (see [19] Savitch [33] Blum and Sakoda [9], Blum and Kozen [8] Hemmerling [21] and others have considered similar models; see Hemmerling s monograph for an extensive bibliography (going back over a century) emphasizing results for labyrinths graphs embedded in two or three dimensional Euclidean space. The JAG is a structured ....

....move with complete freedom. Such models are surprisingly powerful; see Section 3. Nevertheless, in Section 5 we prove a lower bound on a model with freely moving pebbles, but without the ability to jump one pebble to another. This nonjumping model is closer to the one studied by Blum and Sakoda [9], Blum and Kozen [8] and Hemmerling [21] We will distinguish this nonjumping variant by referring to it as a WAG walking automaton for graphs . More specifically, using a very different and more complex argument, we prove lower bounds on time that are nonlinear in m for a wide range of ....

M. Blum and W. J. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In 18th Annual Symposium on Foundations of Computer Science, pages 147--161, Providence, RI, Oct. 1977. IEEE.

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M. Blum and W. J. Sakoda. On the capability of finite automata in 2 and 3 dimensional space. In Proceedings of the Eighteenth Annual Symposium on Foundations of Computer Science, pages 147--161, 1977.

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