Infinite Alphabet Research Articles

On Regular Paths with Counting and Data Tests

Regular path expressions represent the navigation core of the XPath query language for semi-structured data (XML), and it has been characterized as the First Order Logic with Two Variables (FO2). Data tests refers to (dis)equality comparisons on data tree models, which are unranked trees with two kinds of labels, propositions from a finite alphabet, and data values from a possibly infinite alphabet. Node occurrences on tree models can be constrained by counting/arithmetic constructors. In this paper, we identify an EXPTIME extension of regular paths with data tests and counting operators. This extension is characterized in terms of a closed under negation Presburger tree logic. As a consequence, the EXPTIME bound also applies for standard query reasoning (emptiness, containment and equivalence).

On the pertinence to Physics of random walks induced by random dynamical systems: a survey

Let be an abstract space and a denumerable (finite or infinite) alphabet. Suppose that is a family of functions such that for all we have and a family of transformations . The pair ((Sa)a, (pa)a) is termed an iterated function system with place dependent probabilities. Such systems can be thought as generalisations of random dynamical systems. As a matter of fact, suppose we start from a given ; we pick then randomly, with probability pa (x), the transformation Sa and evolve to Sa (x). We are interested in the behaviour of the system when the iteration continues indefinitely.Random walks of the above type are omnipresent in both classical and quantum Physics. To give a small sample of occurrences we mention: random walks on the affine group, random walks on Penrose lattices, random walks on partially directed lattices, evolution of density matrices induced by repeated quantum measurements, quantum channels, quantum random walks, etc.In this article, we review some basic properties of such systems and provide with a pathfinder in the extensive bibliography (both on mathematical and physical sides) where the main results have been originally published.

Multiuser Random Coding Techniques for Mismatched Decoding

This paper studies multiuser random coding techniques for channel coding with a given (possibly suboptimal) decoding rule. For the mismatched discrete memoryless multiple-access channel, an error exponent is obtained that is tight with respect to the ensemble average, and positive within the interior of Lapidoth's achievable rate region. This exponent proves the ensemble tightness of the exponent of Liu and Hughes in the case of maximum-likelihood decoding. An equivalent dual form of Lapidoth's achievable rate region is given, and the latter is shown to extend immediately to channels with infinite and continuous alphabets. In the setting of single-user mismatched decoding, similar analysis techniques are applied to a refined version of superposition coding, which is shown to achieve rates at least as high as standard superposition coding for any set of random-coding parameters.

Monadic Datalog and Regular Tree Pattern Queries

Containment of monadic datalog programs over trees is decidable. The situation is more complex when tree nodes carry labels from an infinite alphabet that can be tested for equality. It then matters whether the descendant relation is allowed or not: the descendant relation can be eliminated easily from monadic programs only when label equalities are not used. With descendant, even containment of linear monadic programs in unions of conjunctive queries is undecidable, and positive results are known only for bounded-depth trees. We show that without descendant, containment of connected monadic programs is decidable over ranked trees, but over unranked trees it is so only for linear programs. With descendant, it becomes decidable over unranked trees under restriction to downward programs: each rule only moves down from the node in the head. This restriction is motivated by regular tree pattern queries, a recent formalism in the area of ActiveXML, which we show to be equivalent to linear downward programs.

Sliding block codes between shift spaces over infinite alphabets

Recently Ott, Tomforde and Willis introduced a notion of one‐sided shifts over infinite alphabets and proposed a definition for sliding block codes between such shift spaces. In this work we propose a more general definition for sliding block codes between Ott–Tomforde–Willis shift spaces and then we prove Curtis–Hedlund–Lyndon type theorems for them, finding sufficient and necessary conditions under which the class of the sliding block codes coincides with the class of continuous shift‐commuting maps.

Continuity properties of a factor of Markov chains

Abstract Starting from a Markov chain with a finite or a countable infinite alphabet, we consider the chain obtained when all but one symbol are indistinguishable for the practitioner. We study conditions on the transition matrix of the Markov chain ensuring that the image chain has continuous or discontinuous transition probabilities with respect to the past.

Reordered Elias Gamma Error Correction Codes for the Near-Capacity Transmission of Multimedia Information

A novel joint source and channel coding (JSCC) scheme is proposed, which we refer to as the reordered Elias gamma error correction (REGEC) code. Like the recently proposed unary error correction (UEC) code and EGEC code, the proposed code facilitates the practical near-capacity transmission of source symbol values that are randomly selected from a large or infinite alphabet. However, in contrast to the UEC code, both the EGEC and our proposed REGEC codes are universal codes, facilitating the transmission of source symbol values that are randomly selected using any monotonic probability distribution. However, the EGEC code has a complicated structure comprising two parts, where unequal error protection is required to balance the two parts with the aid of a specific parameterization that must be tailored to the source distribution, preventing its employment for unknown or non-stationary sources. By contrast, the proposed REGEC code does not need unequal error protection, and hence its parameterization does not have to be tailored to the particular source distribution, and thus the REGEC code is a more attractive scheme. More explicitly, our REGEC code has a simple structure comprising only a single part, which does not suffer from the delay and loss of synchronization that are associated with the two parts of the EGEC code. In a particular practical scenario, where the source symbols obey a specific Zeta probability distribution, our REGEC scheme is shown to offer gains of up to 0.9 dB over the best of JSCC and separate source and channel coding (SSCC) benchmarkers, when QPSK modulation is employed for transmission over an uncorrelated narrowband Rayleigh fading channel. In the scenario where the source symbols obey the distribution produced by the H.265 video codec, our REGEC scheme is shown to offer a gain of 0.7 dB over the SSCC benchmarker. These gains are achieved for free, without increasing the required transmit-duration, transmit-bandwidth, transmit-energy, or decoding complexity.

Fast

Tree automata and transducers are used in a wide range of applications in software engineering. While these formalisms are of immense practical use, they can only model finite alphabets. To overcome this problem we augment tree automata and transducers with symbolic alphabets represented as parametric theories. Admitting infinite alphabets makes these models more general and succinct than their classic counterparts. Despite this, we show how the main operations, such as composition and language equivalence, remain computable given a decision procedure for the alphabet theory. We introduce a high-level language called F ast that acts as a front-end for the preceding formalisms.

Distributed Kalman Filtering With Quantized Sensing State

This paper studies a Quantized Gossip-based Interactive Kalman Filtering (QGIKF) algorithm implemented in a wireless sensor network, where the sensors exchange their quantized states with neighbors via inter-sensor communications. We show that, in the countable infinite quantization alphabet case, the network can still achieve weak consensus with the information loss due to quantization, i.e., the estimation error variance sequence at a randomly selected sensor can converge weakly (in distribution) to a unique invariant measure. To prove the weak convergence, we first interpret error variance sequences as interacting particles, then model each sequence evolution as a Random Dynamical System (RDS), and further prove its stochastically bounded nature. Moreover, based on the analysis for the countable infinite quantization alphabet case, we also prove that under certain conditions the network can also achieve weak consensus, when the quantization alphabet is finite, which is more restricted and practical.

Extended symbolic finite automata and transducers

Symbolic finite automata and transducers augment classic automata and transducers with symbolic alphabets represented as parametric theories. This extension enables to succinctly represent large and potentially infinite alphabets while preserving closure and decidability properties. Extended symbolic finite automata and transducers further extend these objects by allowing transitions to read consecutive input elements in a single step. In this paper we study the properties of these models. In contrast to the case of finite alphabets, we show how reading multiple symbols increases the expressiveness of the models, which causes some closure properties to stop holding and most decision problems to become undecidable. In particular we show how extended symbolic finite transducers are not closed under composition, and the equivalence problem is undecidable for both extended symbolic finite automata and transducers. We then introduce the subclass of Cartesian extended symbolic finite transducers in which guards are limited to conjunctions of unary predicates and we propose an equivalence algorithm for this subclass in the single-valued case. We also present a heuristic algorithm for composing extended symbolic finite transducers that works for many practical cases. Finally, we model real world programs with Cartesian extended symbolic finite transducers and use the proposed algorithms to prove their correctness.

Non-autonomous conformal iterated function systems and Moran-set constructions

We study non-autonomous conformal iterated function systems with finite or countable infinite alphabet alike. These differ from the usual (autonomous) iterated function systems in that the contractions applied at each step in time are allowed to vary. (In the case where all maps are affine similarities, the resulting system is also called a “Moran set construction”.) We shall show that, given a suitable restriction on the growth of the number of contractions used at each step, the Hausdorff dimension of the limit set of such a system is determined by an equation known as Bowen’s formula. We also give examples that show the optimality of our results. In addition, we prove Bowen’s formula for a class of infinite alphabet-systems and deal with Hausdorff measures for finite systems, as well as continuity of topological pressure and Hausdorff dimension for both finite and infinite systems. In particular we strengthen the existing continuity results for infinite autonomous systems. As a simple application of our results, we show that, for a transcendental meromorphic function f f , the Hausdorff dimension of the set of transitive points (i.e., those points whose orbits are dense in the Julia set) is bounded from below by the hyperbolic dimension of f f .

On the source-channel separation theorem for infinite source alphabets

The single-user source-channel separation theorem has been proved for many classes of sources and channels, including sources with finite or countably infinite alphabets. Typically, the source-channel separation theorem is first proved for sources with a finite alphabet, and then the results are extended to sources with a countably infinite alphabet. This paper considers the direct extension of the source-channel separation theorem for some classes of sources with a finite alphabet to a countably infinite alphabet. Specifically, we provide a solution for memoryless sources and arbitrary channels. It is then discussed how this approach may be extended to the case of general sources and arbitrary channels.

Approximate periodicity

Finding an approximate period in a given string S of length n is defined as follows. Let S′ be a periodic string closest to S under some distance metric, find the smallest period of S′. This period is called an approximate period of S under the given metric. Let the distance between the input string S and a closest periodic string under the Hamming distance S′ be k. We develop algorithms that construct an approximate period of S under the Hamming distance in time O(nklog⁡log⁡n) and under the swap distance in time O(n2). Finally, we show an O(nlog⁡n) algorithm for finite alphabets, and an O(nlog3⁡n) algorithm for infinite alphabets, that approximate the minimum number of mismatches between the input string and a closest periodic string under the Hamming distance.

Logical Theory of the Monoid of Languages over a Non Tally Alphabet

We consider the first-order theory of the monoid PA* of languages over a finite or infinite alphabet A with at least two letters endowed solely with concatenation lifted to sets: no set theoretical predicate or function, no constant. Coding a word u by the submonoid u* it generates, we prove that the operation u*, v* → uv* and the predicate {u*,X | e ∈ X, u ∈ X} are definable in . This allows to interpret the second-order theory of in the first-order theory of and prove the undecidability of the Π8 fragment of this last theory. These results involve technical difficulties witnessed by the logical complexity of the obtained definitions: the above mentioned predicates are respectively Δ5 and Δ7.

A Robust Class of Data Languages and an Application to Learning

We introduce session automata, an automata model to process data words, i.e., words over an infinite alphabet. Session automata support the notion of fresh data values, which are well suited for modeling protocols in which sessions using fresh values are of major interest, like in security protocols or ad-hoc networks. Session automata have an expressiveness partly extending, partly reducing that of classical register automata. We show that, unlike register automata and their various extensions, session automata are robust: They (i) are closed under intersection, union, and (resource-sensitive) complementation, (ii) admit a symbolic regular representation, (iii) have a decidable inclusion problem (unlike register automata), and (iv) enjoy logical characterizations. Using these results, we establish a learning algorithm to infer session automata through membership and equivalence queries.

Unshared Secret Key Cryptography

Current security techniques can be implemented with either secret key exchange or physical-layer wiretap codes. In this paper, we investigate an alternative solution for MIMO wiretap channels. Inspired by the artificial noise (AN) technique, we propose the unshared secret key (USK) cryptosystem, where the AN is redesigned as a one-time pad secret key aligned within the null space between a transmitter and a legitimate receiver. The proposed USK cryptosystem is a new physical-layer cryptographic scheme, which was obtained by combining traditional network-layer cryptography and physical-layer security. Unlike previously studied AN techniques, rather than ensuring nonzero secrecy capacity, the USK is valid for an infinite lattice input alphabet and guarantees Shannon's ideal secrecy and perfect secrecy without the need for secret key exchange. We then show how ideal secrecy can be obtained for finite lattice constellations with an arbitrarily small outage.

Partitions of groups into large subsets

Abstract Let G be a group and let κ be a cardinal. A subset A of G is called left (right) κ-large if there exists a subset F of G such that |F| < κ and G = F A (G = A F). We say that A is κ-large if A is left and right κ-large. It is known that every infinite group G can be partitioned into countably many ℵ0-large subsets. On the other hand, every amenable (in particular, Abelian) group G cannot be partitioned into more than ℵ0 ℵ0-large subsets. We prove that every group G of cardinality κ can be partitioned into κ ℵ1-large subsets and every free group F κ in the infinite alphabet κ can be partitioned into κ 4-large subsets but cannot be partitioned into three 3-large subsets.

Generating models of infinite-state communication protocols using regular inference with abstraction

In order to facilitate model-based verification and validation, effort is underway to develop techniques for generating models of communication system components from observations of their external behavior. Most previous such work has employed regular inference techniques which generate modest-size finite-state models. They typically suppress parameters of messages, although these have a significant impact on control flow in many communication protocols. We present a framework, which adapts regular inference to include data parameters in messages and states for generating components with large or infinite message alphabets. A main idea is to adapt the framework of predicate abstraction, successfully used in formal verification. Since we are in a black-box setting, the abstraction must be supplied externally, using information about how the component manages data parameters. We have implemented our techniques by connecting the LearnLib tool for regular inference with an implementation of session initiation protocol (SIP) in ns-2 and an implementation of transmission control protocol (TCP) in Windows 8, and generated models of SIP and TCP components.

A PTAS for the Square Tiling Problem

The Square Tiling Problem was recently introduced as equivalent to the problem of reconstructing an image from patches and a possible general-purpose indexing tool. Unfortunately, the Square Tiling Problem was shown to be NP-hard. A 1/2-approximation is known.We show that if the tile alphabet is fixed and finite, there is a Polynomial Time Approximation Scheme (PTAS) for the Square Tiling Problem with approximation ratio of (1−ϵ2log⁡n) for any given ϵ≤1.Another topic handled in this paper is the NP-hardness of the Tiling problem with an infinite alphabet. We show that when the alphabet is not bounded, even the decision version for rectangles of size 3n is NP-Complete.

Universal coding for memoryless sources with countably infinite alphabets

We present an asymptotically efficient coding strategy for a stationary countably infinite source determined over a set of nonnegative integers. If the kth moment µ k of the source data is finite, then asymptotic average coding redundancy for length-n blocks, n ? ?, is upper bounded by C (log n/n) k/(k+1), where C is a nonnegative constant. The coding efficiency is demonstrated via an example of scalar quantization of random variables with generalized Gaussian distribution.