M. Blum and D. Kozen. On the power of a compass (or, why mazes are easier to search than graphs). In Proc. 19th Symp. Foundations of Comp. Sc. (FOCS), 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[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 Dexter Kozen. On the power of the compass (or, why mazes are easier to search than graphs. In Proceedings of the 19th Annual Symposium on Foundations of Computer Science. IEEE, 1978.

....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 model, but not a weak ....

....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 (one) of pebbles, a ....

M. Blum and D. C. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In 19th Annual Symposium on Foundations of Computer Science, pages 132--142, Ann Arbor, MI, Oct. 1978. IEEE.

....ways in which the pursuer can detect returning to the same junction along the face boundary. It can drop a marker in the environment, and detect it upon return (similar to the use of pebbles in on line exploration [1] Alternatively, it could use information from a poor odometer or compass (e.g. [2]) It is acceptable if the pursuer must continue past the junction before detection. It may be possible to detect the same junction by simply analyzing the history of the gap sensor and motion commands; however, this remains to be proven. In all cases we have seen so far, it is possible to avoid ....

M. Blum and D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In Proc. Annual Symposium on Foundations of Computer Science, pages 132-142, 1978.

....It is shown (implicitly) in [10] and (explicitly) in [14] that this automaton recognizes exactly the regular tree languages. Another, classical remedy against getting lost is to use pebbles. For instance, arbitrary mazes can be searched by maze walking nite automata with two pebbles, see [5]. The main aim of this paper is to investigate the power of tree walking automata with pebbles. Obviously, the unrestricted use of pebbles leads to a class of tree languages much larger than the regular tree languages, in fact to all tree languages in NSPACE(logn) Thus, we restrict the automaton ....

M. Blum, D. Kozen; On the power of the compass (or, why mazes are easier to search than graphs), Proc. 19th FOCS (Annual Symposium on Foundations of Computer Science), 1978, pp.132-142

....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 model, but not a weak ....

....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 than that proved in ....

M. Blum and D. C. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In 19th Annual Symposium on Foundations of Computer Science, pages 132--142, Ann Arbor, MI, Oct. 1978. IEEE.

.... by Erdmann s monograph on sensor design [Erd3] and the information invariants that Erdmann introduced to the robotics community in 1989 [Erd2] We also observe that rigorous examples of information invariants can be found in the theoretical literature from as far back as 1978 (see, for example, [BK, Koz]) This work was motivated by the theoretical attack on perceptual equivalence begun by [DJ] and by the experimental studies of [JR] Horswill [Hors] has developed a semantics for sensory systems that models and quantifies the kinds of assumptions a sensori computational program makes about its ....

Blum, M. and Kozen, D. On the power of the compass (or, why mazes are easier to search than graphs), Proc. 19 th Symp. Found. Computer Science, Ann Arbor, MI, pp. 132-42 (1978).

....for robot tasks. Our work takes as its inspiration the information invariants that Erdmann 2 introduced to the robotics community in 1989 [Erd89] although rigorous examples of information invariants can be found in the theoretical literature from as far back as 1978 (see, for example, [BK, Koz]) Part I of this book develops the basic concepts and tools behind information invariants in plain language. Therein, we develop a number of motivating examples. In part II, we provide a fairly detailed analysis. In particular, we admit more sophisticated models of sensors and computation. This ....

....appear secondary, if not artificial. In part I of this book, we describe two working robots Tommy and Lily, which may be viewed as online robots. We discuss their capabilities, and how they are programmed. We also consider formal models of online robots, foregrounding the situated automata of [BK]. The examples in part I link our work to the recent but intense interest in online paradigms for situated autonomous agents. In particular, we discuss what kind of data structures robots can build to represent the environment. We also discuss the externalization of state, and the distribution of ....

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Blum, M. and D. Kozen On the power of the compass (or, why mazes are easier to search than graphs), Proc. 19 th Symp. Found. Computer Science, Ann Arbor, MI, pp. 132-42 (1978).

....from the goal. This is essentially the approach taken by Erdmann [Erd94] As we shall see, thinking in terms of knowledge gives us a highlevel tool to clarify what is going on. We illustrate this point by applying our ideas to a maze searching example originally analyzed by Blum and Kozen [BK78]; see Section 4. To provide intuition, throughout this paper we will anchor the formal development in the following example. Although simple, the example embodies two important ingredients imprecise sensing, and the need to coordinate the actions of spatially distributed actuators. Example ....

....to the first question provide necessary insight for the design of agents capable of performing a particular task. We illustrate these ideas in the following example. Example 4.9 : We examine the problem of maze searching. This domain, which has received considerable attention in the past (e.g. [Bud75, BK78]) allows us to illustrate the use of our formal language in a nontrivial application. More importantly, we shall show that existing work in this area, due to Blum and Kozen [BK78] can be best understood as performing a knowledge complexity analysis of this domain that naturally fits within the ....

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M. Blum and D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In Proc 19th Symp. Found. Comp. Sci., pages 132--142, 1978.

....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 D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs. In Proceedings of the Nineteenth Annual Symposium on Foundations of Computer Science, pages 132--142, October 1978.

....else learn one of Kff 1 ; Kff n ; Figure 2: A maze This a special class of SKBPs in which not all conditions are positive. In general, we believe that negative conditions play precisely this role, acting as learning subroutines. Example 4. 9 : The work of Blum and Kozen on maze searching [BK78] provides an excellent example for illustrating the use of our formal language. Donald [Don94] was the first to point out the relevance of their work to this line of research, and this example was an important motivating force behind some of our definitions. We now show that Blum and Kozen s ....

M. Blum and D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In Proc 19th Symp. Found. Comp. Sci., pages 132--142, 1978.

....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 D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs. In Proceedings of the Nineteenth Annual Symposium on Foundations of Computer Science, pages 132-- 142, October 1978.

....is the number of times st crosses the ith obstacle, and L i is the perimeter of the ith obstacle. For convex obstacles, BUG2 is essentially optimal in their model. See also [129] for some further work on an extension of the Lumelsky Stepanov model. Other papers on maze traversal strategies include [74, 329], as well as the surveys of Lumelsky [265, 266, 267] While the Lumelsky Stepanov result gives a worst case additive error bound on the robot s path length, it does not give a bound on the ratio between the robot s path length and the (true) shortest path length, d(s; t; P ) in P . In order to ....

M. Blum and D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In Proc. 19th Annu. IEEE Sympos. Found. Comput. Sci., pages 132--142, 1978.

.... by Erdmann s monograph on sensor design [Erd3] and the information invariants that Erdmann introduced to the robotics community in 1989 [Erd2] We also observe that rigorous examples of information invariants can be found in the theoretical literature from as far back as 1978 (see, for example, BK, Koz] We note that many interesting lower bounds (in the complexity theoretic sense) have been obtained for motion planning questions (see, eg, Reif, HSS, Nat, CR] see, eg, Erd1, Don2, Can, Bri] for upper bounds) Rosenschein has developed a theory of synthetic automata which explore the ....

Blum, M. and Kozen, D. On the power of the compass (or, why mazes are easier to search than graphs), Proc. 19 th Symp. Found. Computer Science, Ann Arbor, MI, pp. 132-42 (1978).

.... Erdmann s monograph on sensor design [Erd3] and the information invariants that Erdmann introduced to the robotics community in 1989 [Erd2] We also observe that rigorous examples of information invariants can be found in the theoretical literature from as far back as 1978 (see, for example, BK, Koz] We note that many interesting lower bounds (in the complexitytheoretic sense) have been obtained for motion planning questions (see, eg, Reif, HSS, Nat, CR] see, eg, Erd1, Don2, Can, Bri] for upper bounds) Rosenschein has developed a theory of synthetic automata which explore the ....

Blum, M. and Kozen, D. On the power of the compass (or, why mazes are easier to search than graphs), Proc. 19 th Symp. Found. Computer Science, Ann Arbor, MI, pp. 132-42 (1978).

....the type matrix. A m Theta n type matrix is constructed for a block of text of m lines and n columns. If t ij = t i 0 j 0 the data in row i and column j and the data in row i 0 and column j 0 are ffl similar. The type matrix for the table in Figure 4 is given in Figure 7. A GCD algorithm [BK] can be used to determine the type, if any, of the overall matrix and thus to decide whether the matrix represents a table. We provide for error tolerance in the typing of each column by supplying an error parameter ffl r . This parameter specifies the amount of noise in the pattern that ....

....at different levels of detail. To do this, we need a formal framework for analyzing what information is necessary for performing a task. Such a framework, based on the notion of information invariants, has been discussed in the robotics context by [Don, DJR] and in the theoretical literature by [BK]. Our long term goal is to computationally characterize methods such as statistics over character sequences [SM, PN] statistics over word occurrence, layout and geometry, and other notions of structure with respect to information content. There are many important questions that arise in the ....

M. Blum and D. Kozen, On the power of the compass (or, why mazes are easier to search than graphs), in Proceedings of the Symposium on Foundations of Computer Science, pp 132-142, 1978.

....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 model, but not a weak ....

....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 values of P . In ....

M. Blum and D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In 19th Annual Symposium on Foundations of Computer Science, pages 132--142, Ann Arbor, MI, Oct. 1978. IEEE.

No context found.

M. Blum and D. Kozen. On the power of a compass (or, why mazes are easier to search than graphs). In Proc. 19th Symp. Foundations of Comp. Sc. (FOCS), 1978.

No context found.

M. Blum and D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In Proceedings of the Nineteenth Annual Symposium on Foundations of Computer Science, pages 132--142, October 1978. 26

....mapping 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 nding ecient solutions to variants of the problem, with restrictive assumptions on the form of the environment (cf. [16, 15, 22, 31, 17, 35, 10, 6, 2]) In this paper, we consider a model that makes very limited assumptions about the environment, and give ecient 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 ....

....two robot algorithm. 4 Most early work on graph exploration assumed that the robot is a nite automaton. Rabin [32] rst 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 [38, 16, 15, 5, 33]. For a survey on automata exploring labyrinths, see [29] Deng and Papadimitriou [22] propose and study the problem of exploring an unknown directed graph having labeled vertices. Albers and Henzinger [2] give improved algorithms for this problem. These works study exploration from the ....

M. Blum and D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In Proceedings of the Nineteenth Annual Symposium on Foundations of Computer Science, pages 132-142, October 1978. 26

....or the convexity of the obstacles. Their algorithm does not minimize the ratio #. Several papers (see [25, 28, 29] and references therein) give algorithms for building up a map of a scene by exploring it entirely. Maze traversal has received considerable attention in the past in various papers [5, 19, 27], none of which considers the ratio metric. The reader is referred to [20] for a comprehensive survey of the results in these papers. The ratio measure #(R, n) has close connections to the competitiveness measure used in the study of on line algorithms [6, 23, 31] indeed, our problem resembles ....

....hence or otherwise obtain an algorithm for point to point navigation with such obstacles. Extend all of the above to three dimensions. Give an algorithm that achieves a provably good ratio for three dimensional scenes with nonconvex obstacles (three dimensional mazes) Blum and Kozen [5] show that a planar maze can be traversed in a number of steps polynomial in the number of vertices in the maze by a deterministic automaton using two pebbles. We have seen that the deterministic algorithm of Rao et al. achieves an optimal ratio but is memory intensive, whereas the random walk ....

M. Blum and D. Kozen, On the power of the compass (or, why mazes are easier to search than graphs), in Proc. 19th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1978, pp. 132--142.

.... 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 other and ....

Manuel Blum and Dexter Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In 19th Annual Symposium on Foundations of Computer Science, pages 132--142. IEEE, 1978.

....or the convexity of the obstacles. Their algorithm does not minimize the ratio ae. Several papers (see [25, 28, 29] and references therein) give algorithms for building up a map of a scene by exploring it entirely. Maze traversal has received considerable attention in the past in various papers [5, 19, 27], none of which considers the ratio metric. The reader is referred to [20] for a comprehensive survey of the results in these papers. The ratio measure ae(R; n) has close connections to the competitiveness measure used in the study of on line algorithms [6, 23, 31] indeed, our problem resembles ....

....or otherwise obtain an algorithm for point to point navigation with such obstacles. ffl Extend all of the above to three dimensions. ffl Give an algorithm that achieves a provably good ratio for three dimensional scenes with non convex obstacles (three dimensional mazes) ffl Blum and Kozen [5] show that a planar maze can be traversed in a number of steps polynomial in the number of vertices in the maze, by a deterministic automaton using two pebbles. We have seen that the deterministic algorithm of Rao et al. achieves an optimal ratio but is memory intensive, whereas the random walk ....

M. Blum and D. Kozen, On the power of the compass (or, why mazes are easier to search than graphs), Proceedings of the 19th Annual Symposium on Foundations of Computer Science, pp. 132--142, October 1978.

....with just four pebbles can completely search any 2 dimensional finite maze, and that a single automaton with seven pebbles can completely search any 2 dimensional infinite maze. They also prove, however, that no collection of finite automata can search every 3 dimensional maze. Blum and Kozen [BK78] improve this result to show that a single automaton with 2 pebbles can search a finite, 2 dimensional maze. Their results imply that mazes are strictly easier to search than planar graphs, since they also show that no single automaton with pebbles can search all planar graphs. Savitch [Sav73] ....

Manuel Blum and Dexter Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In 19th Annual Symposium on Foundations of Computer Science, pages 132--142. IEEE, 1978.

No context found.

M. Blum and D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In Proc. Annual Symposium on Foundations of Computer Science, pages 132--142, 1978.

No context found.

M. Blum and D. Kozen. On the power of the compass (or, why mazes are easier to search than graphs). In Proc. Annual Symposium on Foundations of Computer Science, pages 132--142, 1978.

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