Recursive List Research Articles

Recursive Decoding of Reed-Muller Codes Starting With the Higher-Rate Constituent Code

Recursive list decoding of Reed-Muller (RM) codes, with moderate list size, is known to approach maximum-likelihood (ML) performance of short length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\leq 256)$ </tex-math></inline-formula> RM codes. Recursive decoding employs the Plotkin construction to split the original code into two shorter RM codes with different rates. In contrast to the standard approach which decodes the lower-rate code first, the method in this paper decodes the higher-rate code first. This modification enables an efficient permutation-based decoding technique, with permutations being selected on the fly from the automorphism group of the code using soft information from a channel. Simulation results show that the error-rate performance of the proposed algorithms, enhanced by a permutation selection technique, is close to that of the automorphism-based recursive decoding algorithm with similar complexity for short RM codes, while our decoders perform better for longer RM codes. In particular, it is demonstrated that the proposed algorithms achieve near-ML performance for short RM codes and for RM codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m}$ </tex-math></inline-formula> and order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m-3$ </tex-math></inline-formula> with reasonable complexity.

Decoding Reed-Muller Codes With Successive Codeword Permutations

A novel recursive list decoding (RLD) algorithm for Reed-Muller (RM) codes based on successive permutations (SP) of the codeword is presented. A low-complexity SP scheme applied to a subset of the symmetry group of RM codes is first proposed to carefully select a good codeword permutation on the fly. Then, the proposed SP technique is integrated into an improved RLD algorithm that initializes different decoding paths with random codeword permutations, which are sampled from the full symmetry group of RM codes. Finally, efficient latency and complexity reduction schemes are introduced that virtually preserve the error-correction performance of the proposed decoder. Simulation results demonstrate that at the target frame error rate of 10−3 for the RM code of length 256 with 163 information bits, the proposed decoder reduces 6% of the computational complexity and 22% of the decoding latency of the state-of-the-art semi-parallel simplified successive-cancellation decoder with fast Hadamard transform (SSC-FHT) that uses 96 permutations from the full symmetry group of RM codes, while relatively maintaining the error-correction performance and memory consumption of the semi-parallel permuted SSC-FHT decoder.

AN APPROACH FOR AUTOMATING DESIGN OF WEB, ITS METRICS &amp; KNOWLEDGE-BASE

In view of the fact that web applications are significantly large, complex, highly dynamic & critical software systems, it is desirable that processes of web design and development be systematic & be automated as far as possible. The task of automating an engineering activity is quite complex because it involves two types of contrasting intelligences: (i) Human intelligence which is rooted in informal expression, judgmental evaluation, inductive reasoning and commonsense, and (ii) The machine intelligence which is essentially based on formal expression, formal rule-based evaluation & deductive reasoning etc. Human intelligence is required for understanding the problem domains & their environments, and then for using the understanding for designing & developing solutions in such a form that it is understood and executed by the contrasting intelligence, viz. the machine intelligence. The proposed approach takes care of the facts of human capabilities being rooted in informality & of machine capabilities being purely formal. For this purpose, solutions conceived by human experts—which are generally expressed in some natural language—are, first of all, translated to semi-formal mathematical entities: recursive lists. These recursive lists then are easily translated to fully formal entities in some functional programming language like LISP or Haskell. In this communication, the approach is applied to structural aspects of web and is explained through sufficient exemplars. The dynamics of web will be discussed subsequently.

Termination of canonical context-sensitive rewriting and productivity of rewrite systems

Termination of programs, i.e., the absence of infinite computations, ensures the existence of normal forms for all initial expressions, thus providing an essential ingredient for the definition of a normalization semantics for functional programs. In lazy functional languages, though, infinite data structures are often delivered as the outcome of computations. For instance, the list of all prime numbers can be returned as a neverending stream of numerical expressions or data structures. If such streams are allowed, requiring termination is hopeless. In this setting, the notion of productivity can be used to provide an account of computations with infinite data structures, as it "captures the idea of computability, of progress of infinite-list programs" (B.A. Sijtsma, On the Productivity of Recursive List Definitions, ACM Transactions on Programming Languages and Systems 11(4):633-649, 1989). However, in the realm of Term Rewriting Systems, which can be seen as (first-order, untyped, unconditional) functional programs, termination of Context-Sensitive Rewriting (CSR) has been showed equivalent to productivity of rewrite systems through appropriate transformations. In this way, tools for proving termination of CSR can be used to prove productivity. In term rewriting, CSR is the restriction of rewriting that arises when reductions are allowed on selected arguments of function symbols only. In this paper we show that well-known results about the computational power of CSR are useful to better understand the existing connections between productivity of rewrite systems and termination of CSR, and also to obtain more powerful techniques to prove productivity of rewrite systems.

THE NON-AXIOMATIZABILITY OF O-MINIMALITY

Abstract Fix a language extending the language of ordered fields by at least one new predicate or function symbol. Call an L-structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures. We show that for any recursive list of L-sentences , there is a real closed field satisfying which is not pseudo-o-minimal. This shows that the theory To−min consisting of those -sentences true in all o-minimal -structures, also called the theory of o-minimality (for L), is not recursively axiomatizable. And, in particular, there are locally o-minimal, definably complete expansions of real closed fields which are not pseudo-o-minimal.

Soft-decision decoding of Reed-Muller codes: recursive lists

Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes of length 512, these algorithms approach maximum-likelihood (ML) performance within a margin of 0.1 dB. As a result, we present tight experimental bounds on ML performance for these codes

A recursive list paradigm with Java and C++ implementations

This paper presents a LISP-like recursive list paradigm with implementations in the form of Java and C++ classes. The efficacy of this approach is demonstrated in both languages with the presentati...

Learning Prolog in a schema-based environment

A set of basic construct Prolog schemata are presented as the basic constructs of a structured Prolog for recursive list processing. Prolog schema hierarchies are given that group classes of Prolog programs within each of the basic construct Prolog schemata via most-specific schemata which high-light their similarities and differences. A schema-based instructional approach based upon this classification of Prolog programs is provided as a method of introducing recursion to novice Prolog programmers. The incorporation of this approach to teaching recursive Prolog programming in an intelligent tutoring system is presented. Finally, a schema-based Prolog programming environment embedding this tutoring system in addition to a standard Prolog development system is proposed which promotes the acquisition and use of structured programming constructs in Prolog.

On the productivity of recursive list definitions

Several related notions of the productivity are presented for functional languages with lazy evaluation. The notion of productivity captures the idea of computability, of progress of infinite-list programs. If an infinite-list program is productive , then every element of the list can be computed in finite “time.” These notions are used to study recursive list definitions, that is, lists defined by l where l = fl . Sufficient conditions are given in terms of the function f that either guarantee the productivity of the list or its unproductivity. Furthermore, a calculus is developed that can be used in verifying that lists defined by l where l = f I are productive. The power and the usefulness of our theory are demonstrated by several nontrivial examples. Several observations are given in conclusion.