Polynomial permutations on bounded commutative directoids with an antitone involution

Main results of Dorninger and L?nger (J Pure Appl Math 40:441–449, 2007) concerning polynomial permutations on bounded lattices with an antitone involution are generalized to the case of bounded commutative directoids.     

Exponential stability of discrete-time filters for bounded observation noise

This paper proves exponential asymptotic stability of discrete-time filters for the estimation of solutions to stochastic difference equations, when the observation noise is bounded. No assumption is made on the ergodicity of the signal. The proof uses the Hilbert projective metric, introduced into filter stability analysis by Atar and Zeitouni [1,2]. It is shown that when the signal noise is sufficiently regular, boundedness of the observation noise implies that the filter update operation is, on average, a strict contraction with respect to the Hilbert metric. Asymptotic stability then follows.     

Extended adaptive lattice filters for disturbance input processes

The adaptive lattice filter is characterized as an input-dependent filter lying in an impulse disturbance. A modified recursive least-squares algorithm for bad data input is obtained by ‘dead zone’ control of the so-called reflection coefficients. The constraint-input constraint-output for stability analysis of the adaptive filter is specified as a sufficient condition.     

Unification in partially commutative semigroups

Unification algorithms have been constructed for semigroups and commutative semigroups. This paper considers the intermediate case of partially commutative semigroups. We introduce classesN and of such semigroups and justify their use. We present an equation-solving algorithm for any member of the classN. This algorithm is relative to having an algorithm to determine all non-negative solutions of a certain class of diophantine equations of degree 2 which we call -equations. The difficulties arising when attempting to solve equations in members of the class are discussed, and we present arguments that strongly suggest that unification in these semigroups is undecidable.     

Representations of commutative asynchronous automata

Ito (1976, 1978) [14], [17] provided representations of strongly connected automata by group-matrix type automata. This shows the close connection between strongly connected automata with their automorphism groups. In this paper we deal with commutative asynchronous automata. In particular, we introduce and study normal commutative asynchronous automata and cyclic commutative asynchronous automata. Some properties on endomorphism monoids of these automata are given. Also, the representations of normal commutative asynchronous automata and cyclic commutative asynchronous automata are provided by S-automata and regular S-automata, respectively. The cartesian composition $\mathbf{A}°\mathbf{B}$ of a strongly connected automaton A and a cyclic commutative asynchronous automaton B is studied. It is shown that the endomorphism monoid $E(\mathbf{A}°\mathbf{B})$ of automaton $\mathbf{A}°\mathbf{B}$ is a Clifford monoid. Finally, a representation of $\mathbf{A}°\mathbf{B}$ is provided by regular Clifford monoid matrix-type automaton. This generalizes and extends the representations of strongly connected automata given by Ito (1976) [14].     

On commutative DT0L systems

Restricted versions of DT0L systems, so-called commutative DT0L systems, are considered. In these systems the length of a word derived is independent of the order of tables used. It turns out that many interesting length sets or languages, such as the set of composite numbers, are generated by these systems. Moreover, this approach makes it possible to give new (and slightly generalized) proofs for some undecidability results concerning DT0L functions.     

Design of closed-loop systems with bounded transmission or bounded sensitivity

This paper considers the design, in the frequency domain, of controllers for SISO plants, so that the closed-loop system is asymptotically stable, has a sensitivity function with zeros at given points in Re (s) ?0, and has a transmission or a sensitivity function bounded by a given function. The design requirements are reduced to an interpolation in Re (s) ?0, and a necessary and sufficient condition for its solution is presented. An algorithm is given for the construction of the required controllers.     

Systems of linear difference equations with bounded and robustly bounded solutions

In this paper, we investigate higher-order systems of linear difference equations where the associated characteristic matrix polynomial is self-inversive. We consider classes of equations with bounded solutions. It is known that stability properties of higher-order systems of linear difference equations are determined by the characteristic values of the corresponding matrix polynomials. All solutions are bounded (in both time directions) if the spectrum of the corresponding matrix polynomial lies on the unit circle, and moreover if the characteristic values of modulus 1 are semisimple. If the corresponding matrix polynomial is self-inversive, then one can use the inner radius of the numerical range to obtain a criterion for boundedness of solutions. We show that all solutions are bounded if the inner radius is greater than 1. In the case of matrix polynomials with positive definite coefficient matrices, we derive a computable lower bound for the inner radius and we obtain a criterion for robust boundedness.     

Symbolic bounded synthesis

Synthesizing finite-state systems from full linear-time temporal logic (LTL) is an ambitious way to tackle the challenge of constructing correct-by-construction systems. One particularly promising approach in this context is bounded synthesis, originally proposed by Schewe and Finkbeiner, which in turn builds upon Safraless synthesis, as described by Kupferman and Vardi. Previous implementations of these approaches performed the computation either in an explicit way or used symbolic data structures other than binary decision diagrams (BDDs). In this paper, we reconsider BDDs as state space representation and use it as data structure for bounded synthesis. The key to this construction is the application of two novel optimisation techniques that decrease the number of state bits in such a representation significantly. The first technique uses signalling bits to connect sub-games representing the safety- and non-safety parts of the specification. The second technique is based on a closer analysis of the step of building a safety game from a universal automaton and uses a sufficient condition to remove some so-called counters from the state space of the game.     

Optimal bounded disorder

In this paper we present for the Bounded Disorder file organization a model that relates the secondary overflow bucket size to the primary bucket size of the data nodes (supposedly different from that of overflow) and the whole size of a data node, by considering the tree index. This result is obtained by minimizing the cost of inserting data into the structure and enables to compute the optimal overflow bucket size. Both the normal and partial expansion cases are considered and a model for the optimal relation of the parameters involved is developed.     

Learning commutative deterministic finite state automata in polynomial time

We consider the problem of learning the commutative subclass of regular languages in the on-line model of predicting {0,1∼-valued functions from examples and reinforcements due to Littlestone [7,4]. We show that the entire class of commutative deterministic finite state automata (CDFAs) of an arbitrary alphabet sizek is predictable inO(s ^k) time with the worst case number of mistakes bounded above byO(s ^kk logs), wheres is the number of states in the target DFA. As a corollary, this result implies that the class of CDFAs is also PAC-learnable from random labeled examples in timeO(s ^k) with sample complexity, using a different class of representations. The mistake bound of our algorithm is within a polynomial, for a fixed alphabet size, of the lower boundO(s+k) we obtain by calculating the VC-dimension of the class. Our result also implies the predictability of the class of finite sets of commutative DFAs representing the finite unions of the languages accepted by the respective DFAs. Part of this work was supported by the Office of Naval Research under contract number N00014-87-K-0401 while the author was at the Department of Computer and Information Science, University of Pennsylvania, and N0014-86-K-0454 while at the Department of Computer and Information Sciences, U.C. Santa Cruz. The author’s email address is abe@IBL.CL.nec.co.jp     

Properties of commutative transformations of data models

Cybernetics and Systems Analysis -     

A note on commutative multivariate rational series

Direct arguments are presented showing that for rational series in several commuting variables, the rational series problem is undecidable, and closure under Hadamard product fails.     

Remarks on commutative n-rational series

Commutative N-rational formal power series are considered. Series giving polynomial functions as their coefficients are characterized, and a monotonicity result for commutative series is proved. As an application the division problem for N-rational formal power series is solved.     

Asynchronous deterministic rendezvous in bounded terrains

Two mobile agents (robots) have to meet in an a priori unknown bounded terrain modeled as a polygon, possibly with polygonal obstacles. Robots are modeled as points, and each of them is equipped with a compass. Compasses of robots may be incoherent. Robots construct their routes, but the actual walk of each robot is decided by the adversary that may, e.g., speed up or slow down the robot. We consider several scenarios, depending on three factors: (1) obstacles in the terrain are present, or not, (2) compasses of both robots agree, or not, (3) robots have or do not have a map of the terrain with their positions marked. The cost of a rendezvous algorithm is the worst-case sum of lengths of the robots’ trajectories until they meet. For each scenario, we design a deterministic rendezvous algorithm and analyze its cost. We also prove lower bounds on the cost of any deterministic rendezvous algorithm in each case. For all scenarios these bounds are tight.     

Efficient spiral search in bounded spaces

This correspondence defines approaches for the efficient generation of a spiral-like search pattern within bounded rectangularly tessellated regions. The defined spiral-like search pattern grows outward from a given source in a two-dimensional space, thus tending to minimize search time in many sequential tracking tasks. Efficient spiral generation is achieved by minimizing the number of operations required for interaction with boundaries. Algorithms are developed for both rectangular search regions and for arbitrary convex search regions.