The Complexity of Learning According to Two Models of a Drifting Environment

We show that a bound on the rate of drift of the distribution generating the examples is sufficient for agnostic learning to relative accuracy , where c > 0 is a constant; this matches a known necessary condition to within a constant factor. We establish a sufficient condition for the realizable case, also matching a known necessary condition to within a constant factor. We provide a relatively simple proof of a bound of + on the sample complexity of agnostic learning in a fixed environment.     

Finding tree patterns consistent with positive and negative examples using queries

This paper is concerned with the problem of finding a hypothesis in consistent with given positive and negative examples. The hypothesis class consists of all sets of at most two tree patterns and represents the class of unions of at most two tree pattern languages. Especially, we consider the problem from the point of view of the consistency problem for . The consistency problem is a problem for deciding whether there exists a consistent hypothesis with given positive and negative examples within some fixed hypothesis space. Efficient solvability of that problem is closely related to the possibility of efficient machine learning or machine discovery. Unfortunately, however, the consistency problem is known to be NP-complete for many hypothesis spaces. In this paper, the problem for the class is also shown to be NP-complete. In order to overcome this computational hardness, we try to use additional information obtained by making queries. First, we give an algorithm that, using restricted subset queries, solves the consistency problem for in time polynomial in the total size of given positive and negative examples. Next, we show that each subset query made by the algorithm can be replaced by several membership queries under some condition on a set of function symbols. As a result, we have that the consistency problem for is solved in polynomial time using membership queries. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stability of Linear Difference and Differential Inclusions

Any given n×n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N×N matrix B of spectral radius (B) arbitrarily close to (A). A difference inclusion , where is a compact set of matrices, is asymptotically stable if and only if can be extended to a set of nonnegative matrices B with or . Similar results are derived for differential inclusions.     

Tracking the Best Disjunction

Littlestone developed a simple deterministic on-line learning algorithm for learning k-literal disjunctions. This algorithm (called ) keeps one weight for each of then variables and does multiplicative updates to its weights. We develop a randomized version of and prove bounds for an adaptation of the algorithm for the case when the disjunction may change over time. In this case a possible target disjunction schedule is a sequence of disjunctions (one per trial) and the shift size is the total number of literals that are added/removed from the disjunctions as one progresses through the sequence.We develop an algorithm that predicts nearly as well as the best disjunction schedule for an arbitrary sequence of examples. This algorithm that allows us to track the predictions of the best disjunction is hardly more complex than the original version. However, the amortized analysis needed for obtaining worst-case mistake bounds requires new techniques. In some cases our lower bounds show that the upper bounds of our algorithm have the right constant in front of the leading term in the mistake bound and almost the right constant in front of the second leading term. Computer experiments support our theoretical findings.     

Lange and Wiehagen's pattern language learning algorithm: An average-case analysis with respect to its total learning time

The present paper deals with the best-case, worst-case and average-case behavior of Lange and Wiehagen's (1991) pattern language learning algorithm with respect to its total learning time. Pattern languages have been introduced by Angluin (1980) and are defined as follows: Let be any non-empty finite alphabet containing at least two elements. Furthermore, let be an infinite set of variables such that . Patterns are non-empty strings over . L(π), the language generated by pattern π, is the set of strings which can be obtained by substituting non-null strings from for the variables of the pattern π. Lange and Wiehagen's (1991) algorithm learns the class of all pattern languages in the limit from text. We analyze this algorithm with respect to its total learning time behavior, i.e., the overall time taken by the algorithm until convergence. For every pattern π containing k different variables it is shown that the total learning time is in the best-case and unbounded in the worst-case. Furthermore, we estimate the expectation of the total learning time. In particular, it is shown that Lange and Wiehagen's algorithm possesses an expected total learning time of with respect to the uniform distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Finiteness in Infinite-Valued Łukasiewicz Logic

In this paper we deepen Mundici's analysis on reducibility of the decision problem from infinite-valued ukasiewicz logic to a suitable m-valued ukasiewicz logic _m , where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in if and only if it is also valid in _m. We also reduce the notion of logical consequence in to the same notion in a suitable finite set of finite-valued ukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued ukasiewicz logic.     

A semantically complete extension sequence of the system $$\mathcal{L}^* $$

In this paper, the method of well-combined semantics and syntax proposed by Pavelka is applied to the research of the propositional calculus formal system . The partial constant values are taken as formulas, formulas are fuzzified in two manners of semantics and syntax, and inferring processes are fuzzified. A sequence of new extensions { } of the system is proposed, and the completeness of is proved.     

An Algebraic Approach to Propositional Fuzzy Logic

We investigate the variety corresponding to a logic (introduced in Esteva and Godo, 1998, and called there), which is the combination of ukasiewicz Logic and Product Logic, and in which Gödel Logic is interpretable. We present an alternative (and slightly simpler) axiomatization of such variety. We also investigate the variety, called the variety of algebras, corresponding to the logic obtained from by the adding of a constant and of a defining axiom for one half. We also connect algebras with structures, called f-semifields, arising from the theory of lattice-ordered rings, and prove that every algebra can be regarded as a structure whose domain is the interval [0, 1] of an f-semifield , and whose operations are the truncations of the operations of to [0, 1]. We prove that such a structure is uniquely determined by up to isomorphism, and we establish an equivalence between the category of algebras and that of f-semifields.     

Staircase visibility and computation of kernels

Let be some set of orientations, that is, . We consider the consequences of defining visibility based on curves that are monotone with respect to the orientations in . We call such curves -staircases. Two points p andq in a polygonP are said to -see each other if an -staircase fromp toq exists that is completely contained inP. The -kernel of a polygonP is then the set of all points which -see all other points. The -kernel of a simple polygon can be obtained as the intersection of all {}-kernels, with . With the help of this observation we are able to develop an algorithm to compute the -kernel of a simple polygon, for finite . We also show how to compute theexternal -kernel of a polygon in optimal time . The two algorithms are combined to compute the ( -kernel of a polygon with holes in time .This work was supported by the Deutsche Forschungsgemeinschaft under Grant No. Ot 64/5-4 and the Natural Sciences and Engineering Research Council of Canada and Information Technology Research Centre of Ontario.     

Universal asynchronous iterative arrays of Mealy automata

Summary Asynchronous two-dimensional iterative arrays of automata will be introduced where the underlying automata are not of Moore-type but of Mealy-type. We will prove that there exists a Mealy automaton, , with only two states and one input and output for each of its four distinguished directions, such that any given Mealy-automaton can be realized by an iterative array with only for its component-machines. It is known that loop-free nets cannot be as powerful as Mealy automata; however, we will show that any Mealy automaton can be realized by a network, N, with very restrictive component machines, where no signal may pass a loop in N. Using this fact asynchronous iterative arrays can be built up with one component machine, such that any given Mealy automaton can be realized under the restriction that no signal passes a loop more than once. contains only four states and one input and output for each direction.     

Biologically Inspired Autonomous Rover Control

Robotic missions beyond 2013 will likely be precursors to a manned habitat deployment on Mars. Such missions require robust control systems for long duration activities. Current single rover missions will evolve into deployment of multiple, heterogeneous cooperating robotic colonies. This paper describes the map-making memory and action selection mechanism of BISMARC ( iologically nspired ystem for ap-based utonomous over ontrol) that is currently under development at the Jet Propulsion Laboratory in Pasadena, CA (Huntsberger and Rose, Neutral Networks, 11(7/8):1497–1510). BISMARC is an integrated control system for long duration missions involving robots performing cooperative tasks.     

Computing the Kantorovich Distance for Images

Computing the Kantorovich distance for images is equivalent to solving a very large transportation problem. The cost-function of this transportation problem depends on which distance-function one uses to measure distances between pixels.In this paper we present an algorithm, with a computational complexity of roughly order (N2), where N is equal to the number of pixels in the two images, in case the underlying distance-function is the L1-metric, an approximation of the L2-metric or the square of the L2-metric; a standard algorithm would have a computational complexity of order (N3). The algorithm is based on the classical primal-dual algorithm.The algorithm also gives rise to a transportation plan from one image to the other and we also show how this transportation plan can be used for interpolation and possibly also for compression and discrimination.     

Computational Finite-Element Schemes for Optimal Control of an Elliptic System with Conjugation Conditions

New optimal control problems are considered for distributed systems described by elliptic equations with conjugate conditions and a quadratic minimized function. Highly accurate computational discretization schemes are constructed for the case where a feasible control set coincides with the full Hilbert space of controls.     

A Parallel Logarithmic Order Algorithm for General Multibody System Dynamics

A novel optimal order optimal resource parallel multibody algorithm with general system applicability is derived directly from the sequential recursive methods and the most recent developments in recursive constraint treatments. This new Recursive Coordinate Reduction Parallelism (RCRP) is the first optimal order parallel direct method with a sequential implementation that is exactly the efficient algorithm. Consequently, the RCRP sets new benchmarks for performance over a wide range of problem size and parallel resources. Comparisons to existing methods also demonstrate that the RCRP is presently the best general parallel method.     

Modelling Asynchrony with a Synchronous Model

The I/O automaton paradigm of Lynch and Tuttle models asynchrony through an interleaving parallel composition. The recognition that such interleaving models in fact can be viewed as special cases of synchronous parallel composition has been very limited. Let be any set of finite-state I/O automata drawing actions from a fixed finite set containing a subset . In this article we establish a translation T : to a class of -automata closed under a synchronous parallel composition, for which T is monotonic with respect to implementation relative to , and linear with respect to composition. Thus, for A¹, ..., A, B¹, ..., B and A = A¹ ... A, B = B¹ ... B, if is the set of actions common to both A and B, then A implements B (in the sense of I/O automata) if and only if the -automaton language containment (T(A¹) ... T(A)) (T(B¹) ... T(B)) obtains, where denotes the interleaving parallel composition on and denotes the synchronous parallel composition on . For the class , we use the L-process model of -automata. This result enables one to verify systems specified by I/O automata through model-checkers such as COSPAN or SMV, that operate on models with synchronous parallel composition. The translation technique generalizes to other interleaving models, although in each case, the translation map must match the specific model.     

Fast Range Searching with Delaunay Triangulations

This paper studies the idea of answering range searching queries using simple data structures. The only data structure we need is the Delaunay Triangulation of the input points. The idea is to first locate a vertex of the (arbitrary) query polygon and walk along the boundary of the polygon in the Delaunay Triangulation and report all the points enclosed by the query polygon. For a set of uniformly distributed random points in 2-D and a query polygon the expected query time of this algorithm is O(n ^1/3 + Q + E K + L _r n ^1/2), where Q is the size of the query polygon , {\bf E}K = O(n\bcdot area is the expected number of output points, L _r is a parameter related to the shape of the query polygon and n, and L _r is always bounded by the sum of the edge lengths of . Theoretically, when L _r = O(1/n^1/6) the expected query time is O(n^1/3 + Q + E K), which improves the best known average query time for general range searching. Besides the theoretical meaning, the good property of this algorithm is that once the Delaunay Triangulation is given, no additional preprocessing is needed. In order to obtain empirical results, we design a new algorithm for generating random simple polygons within a given domain. Our empirical results show that the constant coefficient of the algorithm is small, at least for the special (practical) cases when the query polygon is either a triangle (simplex range searching) or an axis-parallel box (orthogonal range searching) and for the general case when the query polygons are generated by our new polygon-generating algorithms and their sizes are relatively small.     

Combining the Perceptron Algorithm with Logarithmic Simulated Annealing

We present results of computational experiments with an extension of the Perceptron algorithm by a special type of simulated annealing. The simulated annealing procedure employs a logarithmic cooling schedule , where is a parameter that depends on the underlying configuration space. For sample sets S of n-dimensional vectors generated by randomly chosen polynomials , we try to approximate the positive and negative examples by linear threshold functions. The approximations are computed by both the classical Perceptron algorithm and our extension with logarithmic cooling schedules. For and , the extension outperforms the classical Perceptron algorithm by about 15% when the sample size is sufficiently large. The parameter was chosen according to estimations of the maximum escape depth from local minima of the associated energy landscape.