Optimal Deterministic Protocols for Mobile Robots on a Grid

This paper studies a system of m robots operating in a set of n work locations connected by aisles in a × grid, where m≤n. From time to time the robots need to move along the aisles, in order to visit disjoint sets of locations. The movement of the robots must comply with the following constraints: (1) no two robots can collide at a grid node or traverse a grid edge at the same time; (2) a robot's sensory capability is limited to detecting the presence of another robot at a neighboring node. We present a deterministic protocol that, for any small constant ε>0, allows m≤(1-ε)n robots to visit their target locations in O( ) time, where each robot visits no more than d≤n targets and no target is visited by more than one robot. We also prove a lower bound showing that our protocol is optimal. Prior to this paper, no optimal protocols were known for d>1. For d=1, optimal protocols were known only for m≤ , while for general m≤n only a suboptimal randomized protocol was known.     

Formative Processes with Applications to the Decision Problem in Set Theory: I. Powerset and Singleton Operators

This paper introduces formative processes, composed by transitive partitions. Given a family of sets, a formative process ending in the Venn partition Σ of is shown to exist. Sufficient criteria are also singled out for a transitive partition to model (via a function from set variables to unions of sets in the partition) all set-literals modeled by Σ. On the basis of such criteria a procedure is designed that mimics a given formative process by another where sets have finite rank bounded by C(|Σ|), with C a specific computable function. As a by-product, one of the core results on decidability in computable set theory is rediscovered, namely the one that regards the satisfiability of unquantified set-theoretic formulae involving Boolean operators, the singleton-former, and the powerset operator. The method described (which is able to exhibit a set-solution when the answer is affirmative) can be extended to solve the satisfiability problem for broader fragments of set theory.     

Reduction of Permutation-Invariant Polynomials: A Noncommutative Case Study

Let R be a commutative ring with 1, let RX₁,…,X_n/I be the polynomial algebra in the n≥4 noncommuting variables X₁,…,X_n over R modulo the set of commutator relations I={(X₁+···+X_n)*X_i=X_i*(X₁+···+X_n)|1≤i≤n}. Furthermore, let G be an arbitrary group of permutations operating on the indeterminates X₁,…,X_n, and let RX₁,…,X_n/I^G be the R-algebra of G-invariant polynomials in RX₁,…,X_n/I. The first part of this paper is about an algorithm, which computes a representation for any fRX₁,…,X_n/I^G as a polynomial in multilinear G-invariant polynomials, i.e., the maximal variable degree of the generators of RX₁,…,X_n/I^G is at most 1. The algorithm works for any ring R and for any permutation group G. In addition, we present a bound for the number of necessary generators for the representation of all G-invariant polynomials in RX₁,…,X_n/I^G with a total degree of at most d. The second part contains a first but promising analysis of G-invariant polynomials of solvable polynomial rings.     

The Solution Operator of the Korteweg-de Vries Equation is Computable

The initial value problem of the Korteweg-de Vries (KdV) equation posted on the real line R: defines a nonlinear map K from the space H^s( ) to the space C( , H^s( )) for any given real numbers s ≥ = 0. In this paper we prove that the map K_R is computable for any integer s ≥ = 3.     

Simulation Preorder over Simple Process Algebras

We consider the problem of simulation preorder/equivalence between infinite-state processes and finite-state ones. First, we describe a general method how to utilize the decidability of bisimulation problems to solve (certain instances of) the corresponding simulation problems. For certain process classes, the method allows us to design effective reductions of simulation problems to their bisimulation counterparts and some new decidability results for simulation have already been obtained in this way. Then we establish the decidability border for the problem of simulation preorder/equivalence between infinite-state processes and finite-state ones w.r.t. the hierarchy of process rewrite systems. In particular, we show that simulation preorder (in both directions) and simulation equivalence are decidable in EXPTIME between pushdown processes and finite-state ones. On the other hand, simulation preorder is undecidable between PA and finite-state processes in both directions. These results also hold for those PA and finite-state processes which are deterministic and normed, and thus immediately extend to trace preorder. Regularity (finiteness) w.r.t. simulation and trace equivalence is also shown to be undecidable for PA. Finally, we prove that simulation preorder (in both directions) and simulation equivalence are intractable between all classes of infinite-state systems (in the hierarchy of process rewrite systems) and finite-state ones. This result is obtained by showing that the problem whether a BPA (or BPP) process simulates a finite-state one is PSPACE-hard and the other direction is co -hard; consequently, simulation equivalence between BPA (or BPP) and finite-state processes is also co -hard.     

Lower Bounds for the Weak Pigeonhole Principle and Random Formulas beyond Resolution

We work with an extension of Resolution, called Res(2), that allows clauses with conjunctions of two literals. In this system there are rules to introduce and eliminate such conjunctions. We prove that the weak pigeonhole principle PHP^cn_n and random unsatisfiable CNF formulas require exponential-size proofs in this system. This is the strongest system beyond Resolution for which such lower bounds are known. As a consequence to the result about the weak pigeonhole principle, Res(log) is exponentially more powerful than Res(2). Also we prove that Resolution cannot polynomially simulate Res(2) and that Res(2) does not have feasible monotone interpolation solving an open problem posed by Krají ek.     

Lower Bounds for Merging Networks

A lower bound theorem is established for the number of comparators in a merging network. Let M(m, n) be the least number of comparators required in the (m, n)-merging networks, and let C(m, n) be the number of comparators in Batcher's (m, n)-merging network, respectively. We prove for n≥1 that M(4, n)=C(4, n) for n≡0, 1, 3 mod 4, M(4, n)≥C(4, n)−1 for n≡2 mod 4, and M(5, n)=C(5, n) for n≡0, 1, 5 mod 8. Furthermore Batcher's (6, 8k+6)-, (7, 8k+7)-, and (8, 8k+8)-merging networks are optimal for k≥0. Our lower bound for (m, n)-merging networks, m≤n, has the same terms as C(m, n) has as far as n is concerned. Thus Batcher's (m, n)-merging network is optimal up to a constant number of comparators, where the constant depends only on m. An open problem posed by Yao and Yao (Lower bounds on merging networks, J. Assoc. Comput. Mach.23, 566–571) is solved: lim_n→∞M(m, n)/n=log m/2+m/2^{log m}.     

Recognizing Hand-Printed Digits with a Distance Quasi-Metric

A distance quasi-metric for pattern recognition is presented. The “quasi” modifier distinguishes the metric from “true” distance metrics which obey a set of standard constraints. By relaxing one of the constraints and coupling it with a fast multidimensional search technique, the metric demonstrates improved accuracy and efficiency compared to other metrics in recognizing hand-written digit samples. A high-level design of a fast optical comparator for computing the distance in O( ) is also presented.     

The Undecidability of the First-Order Theories of One Step Rewriting in Linear Canonical Systems

By reduction from the halting problem for Minsky's two-register machines we prove that there is no algorithm capable of deciding the -theory of one step rewriting of an arbitrary finite linear confluent finitely terminating term rewriting system (weak undecidability). We also present a fixed such system with undecidable *-theory of one step rewriting (strong undecidability). This improves over all previously known results of the same kind.     

On identification secret sharing schemes

Let be a set of participants sharing a secret from a set of secrets. A secret sharing scheme is a protocol such that any qualified subset of can determine the secret by pooling their shares, the messages which they receive, without error, whereas non-qualified subsets of cannot obtain any knowledge about the secret when they pool what they receive. In (optimal) schemes, the sizes of shared secrets depend on the sizes of shares given to the participants. Namely the former grow up exponentially as the latter increase exponentially. In this paper, instead of determining the secret, we require the qualified subsets of participants to identify the secret. This change would certainly make no difference from determining secret if no error for identification were allowed. So here we relax the requirement to identification such that an error may occur with a vanishing probability as the sizes of the secrets grow up. Under relaxed condition this changing allows us to share a set of secrets with double exponential size as the sizes of shares received by the participants exponentially grow. Thus much longer secret can be shared. On the other hand, by the continuity of Shannon entropy we have that the relaxation makes no difference for (ordinary) secret sharing schemes. We obtain the characterizations of relations of sizes of secrets and sizes of the shares for identification secret sharing schemes without and with public message. Our idea originates from Ahlswede–Dueck’s awarded work in 1989, where the identification codes via channels were introduced.     

An Algebra of Discrete Channels That Involve Combinations of Three Basic Error Types

Recently, the author introduced a nonprobabilistic mathematical model of discrete channels, the BEE channels, that involve the error-types substitution, insertion, and deletion. This paper defines an important class of BEE channels, the SID channels, which include channels that permit a bounded number of scattered errors and, possibly at the same time, a bounded burst of errors in any segment of predefined length of a message. A formal syntax is defined for generating channel expressions, and appropriate semantics is provided for interpreting a given channel expression as a communication channel (SID channel) that permits combinations of substitutions, insertions, and deletions of symbols. Our framework permits one to generalize notions such as error correction and unique decodability, and express statements of the form “The code K can correct all errors of type ξ” and “it is decidable whether the code K is uniquely decodable for the channel described by ξ”, where ξ is any SID channel expression.     

A Probabilistic Model for Recovering Camera Translation

This paper describes the mathematical basis and application of a probabilistic model for recovering the direction of camera translation (heading) from optical flow. According to the theorem that heading cannot lie between two converging points in a stationary environment, one can compute the posterior probability distribution of heading across the image and choose the heading with maximum a posteriori (MAP). The model requires very simple computation, provides confidence level of the judgments, applies to both linear and curved trajectories, functions in the presence of camera rotations, and exhibited high accuracy up to 0.1°–0.2° in random dot simulations.     

Attentional Scene Segmentation: Integrating Depth and Motion

We present an approach to attention in active computer vision. The notion of attention plays an important role in biological vision. In recent years, and especially with the emerging interest in active vision, computer vision researchers have been increasingly concerned with attentional mechanisms as well. The basic principles behind these efforts are greatly influenced by psychophysical research. That is the case also in the work presented here, which adapts to the model of Treisman (1985, Comput. Vision Graphics Image Process. Image Understanding31, 156–177), with an early parallel stage with preattentive cues followed by a later serial stage where the cues are integrated. The contributions in our approach are (i) the incorporation of depth information from stereopsis, (ii) the simple implementation of low level modules such as disparity and flow by local phase, and (iii) the cue integration along pursuit and saccade mode that allows us a proper target selection based on nearness and motion. We demonstrate the technique by experiments in which a moving observer selectively masks out different moving objects in real scenes.     

The most nonelementary theory

We give a direct proof by generic reduction that testing validity of formulas in a decidable rudimentary theory Ω of finite typed sets (Henkin, Fundamenta Mathematicæ 52 (1963) 323–344) requires space and time exceeding infinitely often(1)where n denotes the length of input. This gives the highest currently known lower bound for a decidable logical theory and affirmatively settles Problem 10.13 from (Compton and Henson, Ann. Pure Applied Logic 48 (1990) 1–79):

Is there a “natural” decidable theory with a lower bound of the form exp_∞(f(n)), where f is not linearly bounded?

The highest previously known lower (and upper) bounds for “natural” decidable theories, like WS1S, S2S, are of the form exp_∞(dn), with just linearly growing stacks of twos.Originally, the lower bound (1) for Ω was settled in (12th Annual IEEE Symposium on Logic in Computer Science (LICS’97), 1997, 294–305) using the powerful uniform lower bounds method due to Compton and Henson, and probably would never be discovered otherwise. Although very concise, the original proof has certain gaps, because the method was pushed out of the limits it was originally designed and intended for, and some hidden assumptions were violated. This results in slightly weaker bounds—the stack of twos in (1) grows subexponentially, but superpolynomially, namely, as for formulas with fixed quantifier prefix, or as 2^cn/log(n) for formulas with varying prefix. The independent direct proof presented in this paper closes the gaps and settles the originally claimed lower bound (1) for the minimally typed, succinct version of Ω.     

Compact Recognizers of Episode Sequences

Given two strings X=a₁…a_n and P=b₁…b_m over an alphabet Σ, the problem of testing whether P occurs as a subsequence of X is trivially solved in linear time. It is also known that a simple O(n log |Σ|) time preprocessing of X makes it easy to decide subsequently, for any P and in at most |P| log |Σ| character comparisons, whether P is a subsequence of X. These problems become more complicated if one asks instead whether P occurs as a subsequence of some substring Y of X of bounded length. This paper presents an automaton built on the textstring X and capable of identifying all distinct minimal substrings Y of X having P as a subsequence. By a substring Y being minimal with respect to P, it is meant that P is not a subsequence of any proper substring of Y. For every minimal substring Y, the automaton recognizes the occurrence of P having the lexicographically smallest sequence of symbol positions in Y. It is not difficult to realize such an automaton in time and space O(n²) for a text of n characters. One result of this paper consists of bringing those bounds down to linear or O(n log n), respectively, depending on whether the alphabet is bounded or of arbitrary size, thereby matching the corresponding complexities of automata constructions for offline exact string searching. Having built the automaton, the search for all lexicographically earliest occurrences of P in X is carried out in time O(∑_i=1^mrocc_i·i) or O(n+∑_i=1^mrocc_i·i· log n), depending on whether the alphabet is fixed or arbitrary, where rocc_i is the number of distinct minimal substrings of X having b₁…b_i as a subsequence (note that each such substring may occur many times in X but is counted only once in the bound). All log factors appearing in the above bounds can be further reduced to log log by resorting to known integer-handling data structures.     

The ** Part of the Theory of Ground Term Algebra Modulo an AC Symbol is Undecidable

We show that the ** part of the equational theory modulo an AC symbol is undecidable. This solves the open problem 25 from the RTA list. We show that this result holds also for the equational theory modulo an ACI symbol.     

Optimal Computing the Chessboard Distance Transform on Parallel Processing Systems

Thedistance transform(DT) is an image computation tool which can be used to extract the information about the shape and the position of the foreground pixels relative to each other. It converts a binary image into a grey-level image, where each pixel has a value corresponding to the distance to the nearest foreground pixel. The time complexity for computing the distance transform is fully dependent on the different distance metrics. Especially, the more exact the distance transform is, the worse execution time reached will be. Nowadays, quite often thousands of images are processed in a limited time. It seems quite impossible for a sequential computer to do such a computation for the distance transform in real time. In order to provide efficient distance transform computation, it is considerably desirable to develop a parallel algorithm for this operation. In this paper, based on the diagonal propagation approach, we first provide anO(N²) time sequential algorithm to compute thechessboard distance transform(CDT) of anN×Nimage, which is a DT using the chessboard distance metrics. Based on the proposed sequential algorithm, the CDT of a 2D binary image array of sizeN×Ncan be computed inO(logN) time on the EREW PRAM model usingO(N²/logN) processors,O(log logN) time on the CRCW PRAM model usingO(N²/log logN) processors, andO(logN) time on the hypercube computer usingO(N²/logN) processors. Following the mapping as proposed by Lee and Horng, the algorithm for the medial axis transform is also efficiently derived. The medial axis transform of a 2D binary image array of sizeN×Ncan be computed inO(logN) time on the EREW PRAM model usingO(N²/logN) processors,O(log logN) time on the CRCW PRAM model usingO(N²/log logN) processors, andO(logN) time on the hypercube computer usingO(N²/logN) processors. The proposed parallel algorithms are composed of a set of prefix operations. In each prefix operation phase, only increase (add-one) operation and minimum operation are employed. So, the algorithms are especially efficient in practical applications.     

Optimal Robot Localization in Trees

The problem of localization, that is, of a robot finding its position on a map, is an important task for autonomous mobile robots. It has applications in numerous areas of robotics ranging from aerial photography to autonomous vehicle exploration. In this paper we present a new strategy LPS (Localize-by-Placement-Separation) for a robot to find its position on a map, where the map is represented as a geometric tree of bounded degree. Our strategy exploits to a high degree the self-similarities that may occur in the environment. We use the framework of competitive analysis to analyze the performance of our strategy. In particular, we show that the distance traveled by the robot is at most O( ) times longer than the shortest possible route to localize the robot, where n is the number of vertices of the tree. This is a significant improvement over the best known previous bound of O(n^2/3). Moreover, since there is a lower bound of Ω( ), our strategy is optimal up to a constant factor. Using the same approach we can also show that the problem of searching for a target in a geometric tree, where the robot is given a map of the tree and the location of the target but does not know its own position, can be solved by a strategy with a competitive ratio of O( ), which is again optimal up to a constant factor.     

2DRMP: A suite of two-dimensional R-matrix propagation codes

The R-matrix method has proved to be a remarkably stable, robust and efficient technique for solving the close-coupling equations that arise in electron and photon collisions with atoms, ions and molecules. During the last thirty-four years a series of related R-matrix program packages have been published periodically in CPC. These packages are primarily concerned with low-energy scattering where the incident energy is insufficient to ionise the target. In this paper we describe 2DRMP, a suite of two-dimensional R-matrix propagation programs aimed at creating virtual experiments on high performance and grid architectures to enable the study of electron scattering from H-like atoms and ions at intermediate energies.

Program summary

Program title: 2DRMPCatalogue identifier: AEEA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEA_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 196 717No. of bytes in distributed program, including test data, etc.: 3 819 727Distribution format: tar.gzProgramming language: Fortran 95, MPIComputer: Tested on CRAY XT4 [1]; IBM eServer 575 [2]; Itanium II cluster [3]Operating system: Tested on UNICOS/lc [1]; IBM AIX [2]; Red Hat Linux Enterprise AS [3]Has the code been vectorised or parallelised?: Yes. 16 cores were used for small test runClassification: 2.4External routines: BLAS, LAPACK, PBLAS, ScaLAPACKSubprograms used: ADAZ_v1_1Nature of problem: 2DRMP is a suite of programs aimed at creating virtual experiments on high performance architectures to enable the study of electron scattering from H-like atoms and ions at intermediate energies.Solution method: Two-dimensional R-matrix propagation theory. The (r₁,r₂) space of the internal region is subdivided into a number of subregions. Local R-matrices are constructed within each subregion and used to propagate a global R-matrix, ℜ, across the internal region. On the boundary of the internal region ℜ is transformed onto the IERM target state basis. Thus, the two-dimensional R-matrix propagation technique transforms an intractable problem into a series of tractable problems enabling the internal region to be extended far beyond that which is possible with the standard one-sector codes. A distinctive feature of the method is that both electrons are treated identically and the R-matrix basis states are constructed to allow for both electrons to be in the continuum. The subregion size is flexible and can be adjusted to accommodate the number of cores available.Restrictions: The implementation is currently restricted to electron scattering from H-like atoms and ions.Additional comments: The programs have been designed to operate on serial computers and to exploit the distributed memory parallelism found on tightly coupled high performance clusters and supercomputers. 2DRMP has been systematically and comprehensively documented using ROBODoc [4] which is an API documentation tool that works by extracting specially formatted headers from the program source code and writing them to documentation files.Running time: The wall clock running time for the small test run using 16 cores and performed on [3] is as follows: bp (7 s); rint2 (34 s); newrd (32 s); diag (21 s); amps (11 s); prop (24 s).References:

• [1] 

  HECToR, CRAY XT4 running UNICOS/lc, http://www.hector.ac.uk/, accessed 22 July, 2009.

• [2] 

  HPCx, IBM eServer 575 running IBM AIX, http://www.hpcx.ac.uk/, accessed 22 July, 2009.

• [3] 

  HP Cluster, Itanium II cluster running Red Hat Linux Enterprise AS, Queen s University Belfast, http://www.qub.ac.uk/directorates/InformationServices/Research/HighPerformanceComputing/Services/Hardware/HPResearch/, accessed 22 July, 2009.

• [4] 

  Automating Software Documentation with ROBODoc, http://www.xs4all.nl/~rfsber/Robo/, accessed 22 July, 2009.