decentralised consensus based formation tracking of multiple differential drive robots abstractthis article investigates the control problem for formation tracking of multiple nonholonomic robots under distributed manner which means each robot only needs local information exchange a class of general state and input transform is introduced to convert the formation tracking issue of multi robot systems into the consensus like problem with time varying reference the distributed observer based protocol with nonlinear dynamics is developed for each robot to achieve the consensus tracking of the new system which namely means a group of nonholonomic mobile robots can form the desired formation configuration with its centroid moving along the predefined reference trajectory the finite time stability of observer and control law is analysed rigorously by using the lyapunov direct method algebraic graph theory and matrix analysis numerical examples are finally provided to illustrate the effectiveness of the theory results proposed in this paper

two time space tradeoffs for element distinctness two time space tradeoffs for element distinctness are given the first one shows t2s n3 on a branching program using minimum operations by a result of yao 1982 this implies the same bounds for linear queries the second result extends one by duris and galil 1984 who constructed a boolean function that requires t2s n3 on a k headed turing machine here it is shown that their proof also holds for element distinctness a more natural problem

expanders and time restricted branching programs the replication number of a branching program is the minimum number r such that along every accepting computation at most r variables are tested more than once the sets of variables re tested along different computations may be different for every branching program this number lies between 0 read once programs and the total number n of variables general branching programs the best results so far were exponential lower bounds on the size of branching programs with r o n logn we improve this to email protected email protected for a constant e 0 this also gives an alternative and simpler proof of an exponential lower bound for email protected n time branching programs for a constant e 0 we prove these lower bounds for quadratic functions of ramanujan graphs

reciprocal transitive matrices over abelian linearly ordered groups we consider reciprocal matrices over an abelian linearly ordered group in this way we provide a general framework including multiplicative additive and fuzzy matrices in a multi criteria decision making context a pairwise comparison matrix a a i j is a reciprocal matrix that represents a useful tool for determining a weighting vector w for a set x of decision elements but when a is inconsistent the weighting vector usually proposed in literature may provide a ranking on x that does not agree with the expressed preference intensities a i j thus it is unreliable we analyze a condition of transitivity for a reciprocal matrix a a i j over an abelian linearly ordered group that whenever a is a pairwise comparison matrix allows us to state a qualitative dominance ranking on x and obtain ordinal evaluation vectors in this way we get a first tool for checking the reliability of a weighting vector we also provide tools to check the transitivity

a space lower bound for st connectivity on node named jags abstract given a directed graph g and two of its nodes s and t the directed st connectivity problem stcon is to decide whether there is a directed path from s to t in g establishing a super logarithmic space lower bound for stcon on a general computation model such as turing machine or branching program has been a big challenge in complexity theory as an intermediate step cook and rackoff siam j comput 9 3 1980 636 652 introduce a structured model called jag jumping automaton for graphs and prove a space lower bound of log 2 n log log n on this model berman and simon proc 24th annual symp on foundations of computer science ieee press new york tucson az november 1983 show a similar lower bound for a probabilistic jag we take a step further by introducing a stronger model called nn jag node named jag and obtain the same space lower bound of log 2 n log log n on a probabilistic nnjag we show that s log 2 n log log n log log t where s is the space and t is the expected time used respectively this gives the best expected time lower bound on this model when the space s is o log 2 n log log n

a general sequential time space tradeoff for finding unique elements an optimal omega n 2 lower bound is shown for the time space product of any r branching program that determines those values which occur exactly once in a list of n integers in the range 1 r where r geqq n this omega n 2 tradeoff also applies to the sorting problem and thus improves the previous time space tradeoffs for sorting because the r way branching program is such a powerful model these time space product tradeoffs also apply to all models of sequential computation that have a fair measure of space such as off line multitape turing machines and off line log cost random access machines rams

distributed formation control of multi agent systems using complex laplacian the paper concentrates on the fundamental coordination problem that requires a network of agents to achieve a specific but arbitrary formation shape a new technique based on complex laplacian is introduced to address the problems of which formation shapes specified by inter agent relative positions can be formed and how they can be achieved with distributed control ensuring global stability concerning the first question we show that all similar formations subject to only shape constraints are those that lie in the null space of a complex laplacian satisfying certain rank condition and that a formation shape can be realized almost surely if and only if the graph modeling the inter agent specification of the formation shape is 2 rooted concerning the second question a distributed and linear control law is developed based on the complex laplacian specifying the target formation shape and provable existence conditions of stabilizing gains to assign the eigenvalues of the closed loop system at desired locations are given moreover we show how the formation shape control law is extended to achieve a rigid formation if a subset of knowledgable agents knowing the desired formation size scales the formation while the rest agents do not need to re design and change their control laws

time space lower bounds for satisfiability we establish the first polynomial time space lower bounds for satisfiability on general models of computation we show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic random access turing machine can solve satisfiability in time nc and space nd where d approaches 1 when c does on conondeterministic instead of deterministic machines we prove the same for any constant c less than 2radic our lower bounds apply to nondeterministic linear time and almost all natural np complete problems known in fact they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space n1 c our proofs follow the paradigm of indirect diagonalization we also use that paradigm to prove time space lower bounds for languages higher up in the polynomial time hierarchy

communication complexity of simultaneous messages in the multiparty communication game cfl game of chandra furst and lipton proceedings of the 15th annual acm symposium on theory of computing boston ma 1983 pp 94 99 k players collaboratively evaluate a function f x0 xk 1 in which player i knows all inputs except xi the players have unlimited computational power the objective is to minimize communication in this paper we study the simultaneous messages sm model of multiparty communication complexity the sm model is a restricted version of the cfl game in which the players are not allowed to communicate with each other instead each of the k players simultaneously sends a message to a referee who sees none of the inputs the referee then announces the function value we prove lower and upper bounds on the sm complexity of several classes of explicit functions our lower bounds extend to randomized sm complexity via an entropy argument a lemma establishing a tradeoff between average hamming distance and range size for transformations of the boolean cube might be of independent interest our lower bounds on sm complexity imply an exponential gap between the sm model and the cfl model for up to log n 1 epsilon players for any epsilon 0 this separation is obtained by comparing the respective complexities of the generalized addressing function gafg k where g is a group of order n we also combine our lower bounds on sm complexity with the ideas of hastad and goldmann comput complexity 1 1991 pp 113 129 to derive superpolynomial lower bounds for certain depth 2 circuits computing a function related to the gaf function we prove some counterintuitive upper bounds on sm complexity we show that sf gaf mathbb z 2 t 3 has sm complexity o n 0 92 when the number of players is at least c log n for some constant c 0 our sm protocol for sf gaf mathbb z 2 t k has polylog n complexity we also examine a class of functions defined by certain depth 2 circuits this class includes the generalized inner product function and majority of majorities when the number of players is at least 2 log n we obtain polylog n upper bounds for this class of functions

tight lower bounds for st connectivity on the nnjag model directed st connectivity is the problem of deciding whether or not there exists a path from a distinguished node s to a distinguished node t in a directed graph we prove a time space lower bound on the probabilistic nnjag model of poon proc 34th annual symposium on foundations of computer science palo alto ca 1993 pp 218 227 let n be the number of nodes in the input graph and s and t be the space and time used by the nnjag respectively we show that for any delta 0 if an nnjag uses space s in o n 1 delta then t in 2 omega log 2 n s otherwise t in 2 omega log 2 n log n over s log log n times ns log n 1 2 in a preliminary version of this paper by edmonds and poon proc 27th annual acm symposium on theory of computing las vegas nv 1995 pp 147 156 a lower bound of t in 2 omega log 2 n log n over s log log n times ns log n 1 2 was proved our result greatly improves the previous lower bound of st in omega n 2 log n on the jag model by barnes and edmonds proc 34th annual symposium on foundations of computer science palo alto ca 1993 pp 228 237 and that of s 1 3 t in omega n 4 3 on the nnjag model by edmonds time space lower bounds for undirected and directed st connectivity on jag models ph d thesis university of toronto toronto on canada 1993 our lower bound is tight for s in o n 1 delta for any delta 0 matching the upper bound of barnes etal proc 7th annual ieee conference on structure in complexity theory boston ma 1992 pp 27 33 as a corollary of this improved lower bound we obtain the first tight space lower bound of omega log 2 n on the nnjag model no tight space lower bound was previously known even for the more restricted jag model

hierarchy theorems for k obdds and k ibdds almost the same types of restricted branching programs or binary decision diagrams bdds are considered in complexity theory and in applications like hardware verification these models are read once branching programs free bdds and certain types of oblivious branching programs ordered and indexed bdds with k layers the complexity of the satisfiability problem for these restricted branching programs is investigated and tight hierarchy results are proved for the classes of functions representable by k layers of ordered or indexed bdds of polynomial size

tight bounds on computing error correcting codes by bounded depth circuits with arbitrary gates we bound the minimum number w of wires needed to compute any asymptotically good error correcting code c 0 1 n 0 1 n with minimum distance n using unbounded fan in circuits of depth d with arbitrary gates our main results are 1 if d 2 then w n lgn lglgn 2 2 if d 3 then w nlglgn 3 if d 2k or d 2k 1 for some integer k 2 then w n k n where 1 n lgn i 1 n i n and the operation gives how many times one has to iterate the function i to reach a value at most 1 from the argument n and 4 if d lg n then w o n for depth d 2 our n lgn lglgn 2 lower bound gives the largest known lower bound for computing any linear map the upper bounds imply that a necessarily dense generator matrix for our code can be written as the product of two sparse matrices using known techniques we also obtain similar but not tight bounds for computing pairwise independent hash functions our lower bounds are based on a superconcentrator like condition that the graphs of circuits computing good codes must satisfy this condition is provably intermediate between superconcentrators and their weakenings considered before

quantum branching programs and space bounded nonuniform quantum complexity in this paper the space complexity of non uniform quantum algorithms is investigated using the model of quantum branching programs qbps in order to clarify the relationship between qbps and non uniform quantum turing machines simulations between these two models are presented which allow to transfer upper and lower bound results exploiting additional insights about the connection between the running time and the precision of amplitudes it is shown that non uniform quantum turing machines with algebraic amplitudes and qbps with a suitable analogous set of amplitudes are equivalent in computational power if both models work with bounded or unbounded error furthermore quantum ordered binary decision diagrams qobdds are considered which are restricted qbps that can be regarded as a non uniform analog of one way quantum finite automata upper and lower bounds are proved that allow a classification of the computational power of qobdds in comparison to usual deterministic and randomized variants of the model finally an extension of qbps is proposed where the performed unitary operation may depend on the result of a previous measurement a simulation of randomized bps by this generalized qbp model as well as exponential lower bounds for its ordered variant are presented

nondeterministic space is closed under complementation in this paper we show that nondeterministic space s n is closed under complementation for s n greater than or equal to log n it immediately follows that the context sensitive languages are closed under complementation thus settling a question raised by kuroda inform and control 7 1964 pp 207 233

a time space tradeoff for element distinctness in a time space tradeoff for sorting on non oblivious machines borodin et al b 81 proved that to sort n elements requires ts n2 where t time and s space on a comparison based branching problem although element distinctness and sorting are equivalent problems on a computation tree the stated tradeoff result does not immediately follow for element distinctness or indeed for any decision problem in this paper we are able to show that ts n3 2 for deciding element distinctness or the sign of a permutation

the geometric consistency index approximated thresholds abstract crawford and williams journal of mathematical psychology 29 1985 387 suggested for the row geometric mean method rgmm one of the most extended ahp s priorization procedure a measure of the inconsistency based on stochastic properties of a subjacent model in this paper we formalize this inconsistency measure hereafter called the geometric consistency index gci and provide the thresholds associated with it these thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by saaty multicriteria decision making the analytic hierarchy process new york 1980 for the consistency ratio cr used with the right eigenvector method in conventional ahp

formation stability analysis of unmanned multi vehicles under interconnection topologies in this paper the overall formation stability of an unmanned multi vehicle is mathematically presented under interconnection topologies a novel definition of formation error is first given and followed by the proposed formation stability hypothesis based on this hypothesis a unique extension decomposition aggregation scheme is then employed to support the stability analysis for the overall multi vehicle formation under a mesh topology it is proved that the overall formation control system consisting of n number of nonlinear vehicles is not only asymptotically stable but also exponentially stable in the sense of lyapunov within a neighbourhood of the desired formation this technique is shown to be applicable for a mesh topology but is equally applicable for other topologies a simulation study of the formation manoeuvre of multiple aerosonde uavs unmanned aerial vehicles in 3 d space is finally carried out verifying the achieved formation stability result

rough set approach to multi attribute decision analysis abstract we review the methodology of the rough set analysis of multi attribute decision problems rough set theory proved to be a useful tool for analysis of a vague description of decision situations it answers two basic questions related to multi attribute decision problems one about explanation of a decision situation and another about prescription of some decisions basing on analysis of a decision situation we define four classes of multi attribute decision problems depending on the structure of their representation its interpretation and the kind of questions related then we characterize the rough set methodology for each particular class of decision problems we use simple practical examples to illustrate this presentation a review of related literature is made throughout the paper

linear hash functions consider the set h of all linear or affine transformations between two vector spaces over a finite field f we study how good h is as a class of hash functions namely we consider hashing a set s of size n into a range having the same cardinality n by a randomly chosen function from h and look at the expected size of the largest hash bucket h is a universal class of hash functions for any finite field but with respect to our measure different fields behave differently if the finite field f has n elements then there is a bad set s f 2 of size n with expected maximal bucket size h n 1 3 if n is a perfect square then there is even a bad set with largest bucket size always at least n this is worst possible since with respect to a universal class of hash functions every set of size n has expected largest bucket size below n 1 2 if however we consider the field of two elements then we get much better bounds the best previously known upper bound on the expected size of the largest bucket for this class was o 2 log n we reduce this upper bound to o log n log log n note that this is not far from the guarantee for a random function there the average largest bucket would be t log n log log n in the course of our proof we develop a tool which may be of independent interest suppose we have a subset s of a vector space d over z 2 and consider a random linear mapping of d to a smaller vector space r if the cardinality of s is larger than c e r log r then with probability 1 e the image of s will cover all elements in the range

on the descriptive and algorithmic power of parity ordered binary decision diagrams the main result of this paper is a polynomial time algorithm that minimizes the number of nodes in a parity obdd ordered binary decision diagram moreover we prove that the synthesis and the equivalence test for obdds which are the fundamental operations for circuit verification have polynomial time deterministic solutions we conclude that it takes deterministic polynomial time to decide whether the parity of clauses implicants is satisfiable several functions typically used as examples in theory e g the indirect storage access function have exponential obdd size but are of polynomial size if obdds are used

on the transitivity and consistency approximated thresholds of some consistency indices for pairwise comparison matrices abstract several measures have been proposed in the literature to verify the rationality and consistency of judgements expressed by means of pairwise comparisons however some of these indices lack a meaningful interpretation because of the absence of thresholds associated with them in this paper we first introduce the consistency approximated thresholds for the salo hamalainen index by deriving an inconsistency tolerance level which is analogous to that proposed for the consistency ratio by saaty for the geometric consistency index by aguaron moreno jimenez and for the consistency measure by koczkodaj then via a simulation we propose a transitivity threshold for n n matrices which may give meaningful information about the degree of misclassification and the reliability of preferences while avoiding the need to revise the judgements on varying n we derive the transitivity thresholds for salo hamalainen and for the other consistency indices proposed in literature

computing random consistency indices and assessing priority vectors reliability abstract the paper deals with two crucial steps in multi criteria decision analysis that are consistency of the judgments and priority vectors for alternatives criteria from a side several consistency indices are proposed for measuring the consistency of a pairwise comparison matrix from another one conditions weaker than consistency such as transitivity and weak consistency are proposed for representing further levels of coherence of a decision maker when he she expresses his her preferences by means of reciprocal pairwise comparison matrix firstly in this paper a simulation is performed in order to establish a relation between random consistency index and coherence level then since weak consistency ensures reliability to priority vectors proposed in literature a second simulation is performed in order to measure in case of no weak consistency the reliability of these priority vectors

a lower bound for integer multiplication with read once branching programs we prove that read once branching programs computing integer multiplication require size 2 omega sqrt n this is the first nontrivial lower bound for multiplication on branching programs that are not oblivious by the appropriate problem reductions we obtain the same lower bound for other arithmetic functions

fusion trees can be implemented with ac 0 instructions only addressing a problem of fredman and willard we implement fusion trees in deterministic linear space using ac o instructions only more precisely we show that a subset of 0 2 w 1 of size n

ensuring reliability of the weighting vector in the context of pairwise comparison matrices pcms defined over abelian linearly ordered group consistency and transitivity represent a full coherence of the decision maker dm and the minimal logical requirement that dm s preferences should satisfy respectively moreover the mean vector wm is proposed as weighting vector for the decision elements related to the pcm in this paper we investigate the effects of inconsistency of a transitive pcm on wm and in order to ensure its reliability as weighting vector we provide the notion of weak consistency it is weaker than consistency and stronger than transitivity and ensures that vectors associated with a pcm by means of a strictly increasing synthesis functional are reliable for assigning a preference order on the set of related decision elements the mean vector wm is associated with a pcm by means of one of these functionals finally we introduce an order relation on the rows of the pcm that is a simple order if and only if the condition of weak consistency is satisfied the simple order allows us to easily determine the actual ranking on the set of related decision elements

capacity achieving codes with bounded graphical complexity and maximum likelihood decoding in this paper the existence of capacity achieving codes for memoryless binary input output symmetric mbios channels under maximum likelihood ml decoding with bounded graphical complexity is investigated graphical complexity of a code is defined as the number of edges in the graphical representation of the code per information bit and is proportional to the decoding complexity per information bit per iteration under iterative decoding irregular repeat accumulate ira codes are studied first utilizing the asymptotic average weight distribution aawd of these codes and invoking divsalar s bound on the binary input additive white gaussian noise biawgn channel it is shown that simple nonsystematic ira ensembles outperform systematic ira and regular low density parity check ldpc ensembles with the same graphical complexity and are at most 0 124 db away from the shannon limit however a conclusive result as to whether these nonsystematic ira codes can really achieve capacity cannot be reached motivated by this inconclusive result a new family of codes is proposed called low density parity check and generator matrix ldpc gm codes which are serially concatenated codes with an outer ldpc code and an inner low density generator matrix ldgm code it is shown that these codes can achieve capacity on any mbios channel using ml decoding and also achieve capacity on any bec using belief propagation bp decoding both with bounded graphical complexity moreover it is shown that under certain conditions these capacity achieving codes have linearly increasing minimum distances and achieve the asymptotic gilbert varshamov bound for all rates

an ensemble method using credal decision trees supervised classification learning can be considered as an important tool for decision support in this paper we present a method for supervised classification learning which ensembles decision trees obtained via convex sets of probability distributions also called credal sets and uncertainty measures our method forces the use of different decision trees and it has mainly the following characteristics it obtains a good percentage of correct classifications and an improvement in time of processing compared with known classification methods it not needs to fix the number of decision trees to be used and it can be parallelized to apply it on very large data sets

bounds on monotone switching networks for directed connectivity we separate monotone analogues of l and nl by proving that any monotone switching network solving directed connectivity on n vertices must have size at least n lg n

coding theorems for turbo code ensembles this paper is devoted to a shannon theoretic study of turbo codes we prove that ensembles of parallel and serial turbo codes are good in the following sense for a turbo code ensemble defined by a fixed set of component codes subject only to mild necessary restrictions there exists a positive number spl gamma sub 0 such that for any binary input memoryless channel whose bhattacharyya noise parameter is less than spl gamma sub 0 the average maximum likelihood ml decoder block error probability approaches zero at least as fast as n sup spl beta where spl beta is the interleaver gain exponent defined by benedetto et al in 1996

a time space tradeoff for undirected graph traversal by walking automata we prove a time space tradeoff for traversing undirected graphs using a structured model that is a nonjumping variant of cook and rackoff s jumping automata for graphs

the computational complexity of universal hashing any implementation of carter wegman universal hashing from n bit strings to m bit strings requires a time space tradeoff of ts nm the bound holds in the general boolean branching program model and thus in essentially any model of computation as a corollary computing a b c in any field f requires a quadratic time space tradeoff and the bound holds for any representation of the elements of the field other lower bounds on the complexity of any implementation of universal hashing are given as well quadratic at2 bound for vlsi implementation logn parallel time bound on a crew pram and exponential size for constant depth circuits

and like uninorm based consistency analysis and optimized fuzzy weight closed form solution of triangular fuzzy additive preference relations abstract this paper focuses on multiplicative consistency of triangular fuzzy additive preference relations tfaprs and deriving a closed form solution of optimized triangular fuzzy weights tfws from tfaprs and like uninorm based indices are introduced to measure increasing part vagueness decreasing part vagueness and overall vagueness for 0 1 valued triangular fuzzy preferences and tfaprs an index is then defined to measure row vagueness proportionality of tfaprs an and like uninorm based method is further proposed to generate multiplicatively consistent tfaprs from 0 1 valued tfws by discussing equivalency of 0 1 valued tfw vectors the paper presents two frameworks of normalized tfws called multiplicatively modal value normalized tfws and multiplicatively support interval normalized tfws based on crucial properties of consistent tfaprs a logarithmic least square lls model is established to seek multiplicatively support interval normalized tfws from tfaprs by decomposing its goals and constraints the lls model is transformed into two least square models whose closed form solutions are found by the lagrange multiplier method on basis of the obtained closed form solution of tfws an algorithm including acceptable consistency checking is developed for decision making with tfaprs the rationality and advantages of the presented models are illustrated by a numerical example with five tfaprs and a comparative analysis

on the complexity of branching programs and decision trees for clique functions exponential lower bounds on the complexity of computing the clique functions in the boolean decision tree model are proved for one time only branching programs large polynomial lower bounds are proved for k clique functions if k is fixed and exponential lower bounds if k increases with n finally the hierarchy of the classes bp d p of all sequences of boolean functions that may be computed by d times only branching programs of polynomial size is introduced it is shown constructively that bp 1 p is a proper subset of bp 2 p

two familiar transitive closure algorithms which admit no polynomial time sublinear space implementations any boolean straight line program which computes the transitive closure of an n times n boolean matrix by successive squaring requires time exceeding any polynomial in n if the space used is o n this is the first demonstration of a natural algorithm which 1 has a polynomial time implementation and 2 has a small e g o log 2 n space implementation but 3 has no implementation running in polynomial time and small space simultaneously it is also shown that any implementation of warshall s transitive closure algorithm requires omega n space and that many familiar sorting algorithms exhibit similar behavior

a further discussion of a semiring based study of judgment matrices properties and models information sciences 181 2011 2166 2176 abstract in the paper a semiring based study of judgment matrices properties and models information sciences 181 2011 2166 2176 the semiring is applied to the discussion of judgment matrix properties this paper shows that the suggested semirings could be replaced by a more convenient algebraic structure

quantum walk algorithm for element distinctness we use quantum walks to construct a new quantum algorithm for element distinctness and its generalization for element distinctness the problem of finding two equal items among n given items we get an o n 2 3 query quantum algorithm this improves the previous o n 3 4 quantum algorithm of buhrman et al siam j comput 34 2005 pp 1324 1330 and matches the lower bound of aaronson and shi j acm 51 2004 pp 595 605 we also give an o n k k 1 query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among n items

time space trade off lower bounds for randomized computation of decision problems we prove the first time space lower bound trade offs for randomized computation of decision problems the bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs our techniques are extension of those used by ajtai and by beame jayram and saks that applied to deterministic branching programs our results also give a quantitative improvement over the previous results previous time space trade off results for decision problems can be divided naturally into results for functions with boolean domain that is each input variable is 0 1 valued and the case of large domain where each input variable takes on values from a set whose size grows with the number of variables in the case of boolean domain ajtai exhibited an explicit class of functions and proved that any deterministic boolean branching program or ram using space s o n requires superlinear time t to compute them the functional form of the superlinear bound is not given in his paper but optimizing the parameters in his arguments gives t n log log n log log log n for s o n1 e for the same functions considered by ajtai we prove a time space trade off for randomized branching programs with error of the form t n s log n s log log n s in particular for space o n1 e this improves the lower bound on time to ns log n log log n in the large domain case we prove lower bounds of the form t ns log n s log log n s for randomized computation of the element distinctness function and lower bounds of the form t n log n s for randomized computation of ajtai s hamming closeness problem and of certain functions associated with quadratic forms over large fields

a general decomposition approach for multi criteria decision trees in this paper we address the dynamic and multi criteria decision making problems under uncertainty generally represented by multi criteria decision trees decision making consists of choosing at each period a decision that maximizes the decision maker outcomes these outcomes should often be measured against a set of heterogeneous and conflicting criteria generating the set of non dominated solutions is a common approach considered in the literature to solve the multi criteria decision trees but it becomes very challenging for large problems we propose a new approach to solve multi criteria decision trees without generating the set of all non dominated solutions which should reduce the computation time and the cardinality of the solution set in particular the proposed approach combines the advantages of decomposition with the application of multi criteria decision aid mcda methods at each decision node a generalization of the bellman s principle of decomposition to the multi criteria context is put forward a decomposition theorem is therefore proposed under the sufficient conditions stated by the theorem the principle of decomposition will generate the set of best compromise strategies seven mcda methods are then characterized lexicographic weighted sum multi attribute value theory topsis electre iii and promethee ii against the conditions of the theorem of decomposition and against other properties neutrality anonymous fidelity dominance independency in order to confirm or infirm their applicability with the proposed decomposition principle moreover the relationship between independency and temporal consistence is discussed as well as the effects of incomparableness rank reversals and use of thresholds two conjectures resulted from this characterization

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

geometric analysis of the formation problem for autonomous robots in the formation control problem for autonomous robots a distributed control law steers the robots to the desired target formation a local stability result of the target formation can be derived by methods of linearization and center manifold theory or via a lyapunov based approach besides the target formation the closed loop dynamics of the robots feature various other undesired invariant sets such as nonrigid formations this note addresses a global stability analysis of the closed loop formation control dynamics we pursue a differential geometric approach and derive purely algebraic conditions for local stability of invariant embedded submanifolds these theoretical results are then applied to the well known example of a cyclic triangular formation and result in instability of all invariant sets other than the target formation

oracle branching programs and logspace versus p abstract we define the notion of an oracle branching program in order to investigate space bounded computation within this new framework we examine the p complete problem gen which consists of determining membership in a subalgebra of a general not necessarily associative binary algebra input as a multiplication table our work begins with the statement of a conceptually simple conjecture highlighting the combinatorics which underlie the relationship between logspace and p we show that natural subclasses of p can be expressed as natural subproblems for gen finally we prove optimal lower bounds on the size of branching programs for gen with certain natural oracles

lower bounds for depth restricted branching programs abstract we present a new method for proving lower bounds on the complexity of branching programs and consider k times only branching programs while exponential and nearly exponential lower bounds on the complexity of one time only branching programs were proved for many problems there are still missing methods of proving lower bounds for k times only programs k 1 we prove exponential lower bounds for k times only branching programs which have the additional restriction that the input bits are read k times yet blockwise and in each block in the same order this is done both for the algebraic decision problem polyn d n n prime d n whether a given mapping g f n f n is a polynomial over f n of degree at most d and for the corresponding monotone problem over quadratic boolean matrices as a consequence we obtain a sharp bound of order n log n on the communication complexity of polyn n 0 1 2

a time space tradeoff for sorting on a general sequential model of computation in a general sequential model of computation no restrictions are placed on the way in which the computation may proceed except that parallel operations are not allowed we show that in such an unrestricted environment text time cdot text space omega n 2 log n in order to sort n integers each in the range 1 n 2

constant depth circuits fourier transform and learnability in this paper boolean functions in 4c0 are studied using harmonic analysis on the cube the main result is that an aco boolean function has almost all of its power spectrum on the low order coefficients an important ingredient of the proof is hastad s switching lemma 8 this result implies several new properties of functions in 4c functions in ac have low average sensitivity they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function generators perhaps the most interesting application is an o n poiyiog n time algorithm for learning func tions in aco the algorithm observes the behavior of an ac function on o npo y og n randomly chosen inputs and derives a good approximation for the fourier transform of the function this approximation allows the algorithm to predict with high probability the value of the function on other randomly chosen inputs

time space tradeoffs for set operations abstract this paper considers time space tradeoffs for various set operations denoting the time requirement of an algorithm by t and its space requirement by s it is shown that ts n 2 for set complementation and ts n 3 2 for set intersection in the r way branching program model in the more restricted model of comparison branching programs the paper provides two additional types of results a tradeoff of ts n 2 e n derived from yao s lower bound for element distinctness is shown for set disjointness set union and set intersection where e n o log n 1 2 a bound of ts n 3 2 is shown for deciding set equality and set inclusion finally a classification of set operations is presented and it is shown that all problems of a large naturally arising class are as hard as the problems bounded in this paper

finding median in read only memory on integer input starting with munro and paterson 1980 13 the selection or median finding problem has been extensively studied in the read only memory model and in streaming models munro and paterson s deterministic algorithm and its subsequent refinements require at least polylogarithmic or logarithmic space whereas the algorithms by munro and raman 1996 14 and raman and ramnath 1999 15 can be made to use just o 1 storage cells but take o n 1 e time for an arbitrarily small constant e 0 in this paper we initiate the first study on this problem when the input is a sequence of integers we show that faster selection algorithms in read only memory are possible if the input is a sequence of integers for example one of our algorithms uses o 1 storage cells and takes o n lg u 1 time where u is the universe size another algorithm uses o 1 storage cells and takes o n lg n lg lg u time a combination of the two yields an algorithm that uses o 1 words of space and takes o n lg 1 n time a bound independent of u we also describe an o n time algorithm for finding an approximate median using o lg e u storage cells all our algorithms are simple and deterministic interestingly one of our algorithms is inspired by centroids of binary trees and finds an approximate median by repeatedly invoking a textbook algorithm for the majority problem this technique could be of independent interest

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

axiomatic property based consistency analysis and decision making with interval multiplicative reciprocal preference relations abstract this paper focuses on obtaining an interval extension of saaty s consistency and eliciting normalized interval weights in analytic form from interval multiplicative reciprocal preference relations imrprs as well as checking acceptability of imrprs four axiomatic properties are presented to characterize multiplicative consistency of imrprs the paper shows that six existing consistency definitions fail to satisfy all the four axiomatic properties a constrained interval multiplication based transitivity equation is devised and an interval matrix is constructed to capture original preferences and row uncertainty proportionalities of an imrpr based on the constructed interval matrix an ordinary interval multiplication based transitivity equation is developed to define consistency of imrprs a logarithmic least square model is established to find normalized interval weights from imrprs and its analytic solution is obtained by the lagrangian multiplier method the paper introduces an index to measure uniformity of row uncertainties in an imrpr and proposes a geometric consistency index to measure inconsistency of imrprs a novel method is put forward to determine acceptability of imrprs by checking acceptable consistency and acceptable uncertainty the proposed models are illustrated by eight numerical examples and a hierarchical multi criteria decision making problem dealing with recommending undergraduate students to graduate admission

symmetric space bounded computation abstract a symmetric turing machine is one such that the yields relation between configurations is symmetric the space complexity classes for such machines are found to be intermediate between the corresponding deterministic and nondeterministic space complexity classes certain natural problems are shown to be complete for symmetric space complexity classes and the relationship of symmetry to determinism and nondeterminism is investigated

performance of ldpc codes under faulty iterative decoding departing from traditional communication theory where decoding algorithms are assumed to perform without error a system where noise perturbs both computational devices and communication channels is considered here this paper studies limits in processing noisy signals with noisy circuits by investigating the effect of noise on standard iterative decoders for low density parity check ldpc codes concentration of decoding performance around its average is shown to hold when noise is introduced into message passing and local computation density evolution equations for simple faulty iterative decoders are derived in one model computing nonlinear estimation thresholds shows that performance degrades smoothly as decoder noise increases but arbitrarily small probability of error is not achievable probability of error may be driven to zero in another system model the decoding threshold again decreases smoothly with decoder noise as an application of the methods developed an achievability result for reliable memory systems constructed from unreliable components is provided

a critical analysis of the eigenvalue method used to derive priorities in ahp abstract a lot of research has been devoted to the critical analysis of the analytic hierarchy process ahp from various perspectives however as far as we know no one has addressed a fundamental problem discussed in this paper concerning the meaning of the priority vector derived from the principal eigenvalue method used in ahp the role of ahp s consistency ratio is also analysed

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

bi criteria optimization of decision trees with applications to data analysis abstract this paper is devoted to the study of bi criteria optimization problems for decision trees we consider different cost functions such as depth average depth and number of nodes we design algorithms that allow us to construct the set of pareto optimal points pops for a given decision table and the corresponding bi criteria optimization problem these algorithms are suitable for investigation of medium sized decision tables we discuss three examples of applications of the created tools the study of relationships among depth average depth and number of nodes for decision trees for corner point detection such trees are used in computer vision for object tracking study of systems of decision rules derived from decision trees and comparison of different greedy algorithms for decision tree construction as single and bi criteria optimization algorithms

endcoding complexity versus minimum distance a bound on the minimum distance of a binary error correcting code is established given constraints on the computational time space complexity of its encoder where the encoder is modeled as a branching program the bound obtained asserts that if the encoder uses linear time and sublinear memory in the most general sense then the minimum distance of the code cannot grow linearly with the block length when the rate is nonvanishing that is the minimum relative distance of the code tends to zero in such a setting the setting is general enough to include nonserially concatenated turbo like codes and various generalizations our argument is based on branching program techniques introduced by ajtai the case of constant depth and or circuit encoders with unbounded fanins are also considered

fast learning requires good memory a time space lower bound for parity learning we prove that any algorithm for learning parities requires either a memory of quadratic size or an exponential number of samples this proves a recent conjecture of steinhardt et al 2016 and shows that for some learning problems a large storage space is crucial more formally in the problem of parity learning an unknown string x 0 1 n was chosen uniformly at random a learner tries to learn x from a stream of samples a1 b1 a2 b2 where each at is uniformly distributed over 0 1 n and bt is the inner product of at and x modulo 2 we show that any algorithm for parity learning that uses less than n2 25 bits of memory requires an exponential number of samples previously there was no non trivial lower bound on the number of samples needed for any learning problem even if the allowed memory size is o n where n is the space needed to store one sample we also give an application of our result in the field of bounded storage cryptography we show an encryption scheme that requires a private key of length n as well as time complexity of n per encryption decryption of each bit and is provably and unconditionally secure as long as the attacker uses less than n2 25 memory bits and the scheme is used at most an exponential number of times previous works on bounded storage cryptography assumed that the memory size used by the attacker is at most linear in the time needed for encryption decryption

stabilisation of infinitesimally rigid formations of multi robot networks this article considers the design of a formation control for multivehicle systems that uses only local information the control is derived from a potential function based on an undirected infinitesimally rigid graph that specifies the target formation a potential function is obtained from the graph from which a gradient control is derived under this controller the target formation becomes a manifold of equilibria for the multivehicle system it is shown that infinitesimal rigidity is a sufficient condition for local asymptotical stability of the equilibrium manifold a complete study of the stability of the regular polygon formation is presented and results for directed graphs are presented as well finally the controller is validated experimentally

bounded controllers for formation stabilization of mobile agents with limited sensing ranges a constructive method is presented to design bounded cooperative controllers that force a group of n mobile agents with limited sensing ranges to stabilize at a desired location and guarantee no collisions between the agents the control development is based on new general potential functions which attain the minimum value when the desired formation is achieved and are equal to infinity when a collision occurs a p times differential bump function is introduced and embedded into the potential functions to deal with the agent limited sensing ranges an alternative to barbalat s lemma is developed to analyze stability of the closed loop system extension to formation tracking is also addressed

lyapunov based approach to multiagent systems with switching jointly connected interconnection this note addresses a coordination problem of a multiagent system with jointly connected interconnection topologies neighbor based rules are adopted to realize local control strategies for these continuous time autonomous agents described by double integrators although the interagent connection structures vary over time and related graphs may not be connected a sufficient condition to make all the agents converge to a common value is given for the problem by a proposed lyapunov based approach and related space decomposition technique

a technical note on two inconsistency indices for preference relations a case of functional relation abstract different representations of valued preferences have been studied and used over the years however one common factor has been the prominence of the concept of consistency inconsistency indices have been introduced to estimate the deviation of preferences from a fully consistent form in this note we shall recall two types of preference relations reciprocal relations and multiplicative preference relations and show that two inconsistency indices introduced in these two different frameworks are functionally related besides this main result some reflections on the consequences of being functionally related are presented

on oblivious branching programs of linear length abstract input oblivious decision graphs of linear length are considered among other concerns the computational complexity of three graph accessibility problems and the word problem of the free group are investigated several exponential lower bounds are proved

leader following formation control of multi agent systems under fixed and switching topologies in this article two kinds of leader following formation control problems for second order nonlinear multi agent systems are investigated that is the cases with fixed topology and with switching topology for the former by constructing an appropriate lyapunov functional and utilising linear matrix inequality lmi method we propose a formation control algorithm which makes the nonlinear multi agent systems converge to a desired formation in addition a formation control algorithm is also developed for coupled double integrators with a constant reference velocity then we extend these results to the case when the interaction topology is switching numerical simulations are presented finally to demonstrate the effectiveness of the proposed results

boundary properties of the inconsistency of pairwise comparisons in group decisions abstract this paper proposes an analysis of the effects of consensus and preference aggregation on the consistency of pairwise comparisons we define some boundary properties for the inconsistency of group preferences and investigate their relation with different inconsistency indices some results are presented on more general dependencies between properties of inconsistency indices and the satisfaction of boundary properties in the end given three boundary properties and nine indices among the most relevant ones we will be able to present a complete analysis of what indices satisfy what properties and offer a reflection on the interpretation of the inconsistency of group preferences

amplifying lower bounds by means of self reducibility we observe that many important computational problems in nc1 share a simple self reducibility property we then show that for any problem a having this self reducibility property a has polynomial size tc0 circuits if and only if it has tc0 circuits of size n1 for every 0 counting the number of wires in a circuit as the size of the circuit as an example of what this observation yields consider the boolean formula evaluation problem bfe which is complete for nc1 and has the self reducibility property it follows from a lower bound of impagliazzo paturi and saks that bfe requires depth d tc0 circuits of size n1 d if one were able to improve this lower bound to show that there is some constant 0 independent of the depth d such that every tc0 circuit family recognizing bfe has size at least n1 then it would follow that tc0 nc1 we show that proving lower bounds of the form n1 is not ruled out by the natural proof framework of razborov and rudich and hence there is currently no known barrier for separating classes such as acc0 tc0 and nc1 via existing natural approaches to proving circuit lower bounds we also show that problems with small uniform constant depth circuits have algorithms that simultaneously have small space and time bounds we then make use of known time space tradeoff lower bounds to show that sat requires uniform depth d tc0 and ac0 6 circuits of size n1 c for some constant c depending on d

a construction method for optimally universal hash families and its consequences for the existence of rbibds we introduce a method for constructing optimally universal hash families and equivalently rbibds as a consequence of our construction we obtain minimal optimally universal hash families if the cardinalities of the universe and the range are powers of the same prime a corollary of this result is that the necessary conditions for the existence of an rbibd with parameters v k namely v 0 mod k and v 1 0 mod k 1 are sufficient if v and k are powers of the same prime as an application of our construction we show that the k maxcut algorithm of hofmeister and lefmann a combinatorial design approach to maxcut random struct algorithms 9 1996 163 173 can be implemented such that it has a polynomial running time in the case that the number of vertices and k are powers of the same prime