Links between complexity theory and constrained block coding

The theory of computational complexity is based on the tradeoff between two computational resources, memory space and computer time. This paper investigates the physical counterparts of these resources. Memory space is the number of bits of information required by a computer over the course of a computation. Physically, information is intimately connected with entropy: entropy can be thought of as random bits of information. The physical analogue of the amount of memory space required for a computation, Ф- SPACE, is the difference between the maximum entropy Smax of the computing system and its actual entropy, S.Ф-SPACE = S - Smax is the thermodynamic depth of the computation. Ф-SPACE is the maximum number of bits of memory space available for a physical system to perform computation. In conventional computational complexity theory, computer time is identified as the number of elementary logical operations (AND, OR, NOT, COPY) required to perform the computation. The physical analogue of time computational complexity is simply the time required for the physical computing system to perform the computation. In order to obtain a number that measures the number of physical operations that can be performed by the system over time t, we define Ф-TIME to be proportional to the computation time multiplied by the energy E of the computing system over the course of the computation: Ф-TIME = 2Et/πħ. Here, the proportionality factor 2/πħ is chosen to match the time it takes a computing system with energy E to flip a bit in time t. Ф-TIME is thus the number of bit flips that can take place in an information processing system. In addition, we propose a novel measure of physical complexity that combines these measures. The free energy F = E - TS of a system is the amount of energy that is available to do useful work - as opposed to the energy bound up in the thermal fluctuations of heat. We define the free complexity to be the free energy × time product, 2Ft/πħ required by a physical system to perform a computation. Physical spatial and temporal complexity provide upper bounds on the amount of computation that can be performed by any physical system. We discuss the implications of physical complexity for the computational complexity hierarchy, in particular, we show that in the presence of errors in the computation, Ф-SPACE is proportional Ф-TIME. We apply the resulting theory of physical computational complexity to analyze the computational power of black holes: exactly half of the energy of a black hole is available for physical computation.

Kauffman nets, also known as N-K models, have been studied extensively because their dynamics can be used to model a variety of interesting dynamical processes. This paper investigates the properties of the problem of determining whether or not a given configuration of a Kauffman net has a predecessor. Here it is shown that when the parameter K that governs the number of connections grows as ln(N) , where N is the number of elements, the problem of finding a solution is extremely sensitive to small changes in the problem statement. This result has implications for studies of the physics of random systems and also may have applications for questions in computational complexity theory.

THE INTERFACE BETWEEN LANGUAGE THEORY AND COMPLEXITY THEORY

The notion of NP-completeness has helped researchers in several fields to argue that some problems are intrinsically difficult to solve computationally, even though proof of that is beyond the current state of our knowledge. The study of properties of complete sets and reducibilities has led to the development of a branch of complexity theory called structural complexity theory. This paper is a brief survey of recent results in structural complexity theory and is aimed at researchers who know something about computational complexity theory but do not work in structural complexity theory. To describe the theory in a way that gives the reader some concrete applications, the results are described in terms of their relationship to P =?NP and other problems.

This paper is an introduction to the computational complexity theory. I believe that the standard courses in complexity theory make some things much more important than they really are, while things which I find extremely interesting are marginalized. Thus, it can be seen as my subjective view on what is important and interesting in computational complexity, and what is not. We state the basic facts from computational complexity theory using the blackboard computations (one-dimensional cellular automata) as the standard computation model; we argue the benefits of such approach. This article is somewhat based on my series of talks from Warsztaty Logiczne 2014 in Białka Tatrzańska, although these talks covered more topics, such as NL=coNL, and links between BPP and the polynomial hierarchy.

Herbert A. Simon and Alan Newell won the Turing Award jointly in Computer Science for foundational work on Artificial Intelligence. Simon also won the Nobel Prize in Economics for the concept of “bounded rationality.” In both cases, the same heuristic was deemed fundamental: “Search till a satisfactory solution is found.” We argue that behavioral strategy can learn a great deal from the Theory of Computational Complexity and Artificial Intelligence. These fields can provide a sounder theoretical grounding for bounded rationality and for the necessity and usefulness of heuristics. Finally, a concept of “organizational intractability” based roughly on the metaphor provided by the Theory of Computational Complexity may be useful in determining what analytical decision technologies are actually intractable in real organizations with constraints on time and managerial attention.

The decoding of random linear codes is one of the most fundamental problems in both computational complexity theory and algorithmic cryptanalysis. Specifically, the best attacks known against existing code-based cryptosystems, such as McEliece, are unstructural, i.e., these attacks directly use generic decoding algorithms that treat the hidden binary codes as random linear codes. This topic is also attracting increasing interest in a post-quantum context as this area becomes increasingly active. In an attempt to solve this problem, several algorithms and their variants have recently been proposed, with increasingly lower time complexities. However, their memory complexities, which are even more important in practice for real attacks, are neglected.

This paper propose a new flexible low-density parity-check (LDPC) decoding architecture that can be dynamically reconfigured according to a parity-check matrix which changes as code rate or block length varies. Since the LDPC codes are adopted in standards such as DVB-S2, IEEE 802.1 In and IEEE 802.16e and each standard allows various code rates and block lengths, an LDPC decoder architecture that could support multiple code rates and code block lengths is needed. The proposed architecture employs Benes network to implement configurable interconnection network. In the proposed architecture the number of levels and the delay of the interconnection network is reduced by using broadcasting technique which transmits reduced messages to save the complexity of the interconnection network. The number of processing units in the proposed partially parallel architecture is determined by investigating the relation between hardware complexity and throughput. By applying pipelining to the processing units, decoding throughput is increased. To verify the proposed architecture, a flexible LDPC decoder is implemented using a 0.18 mum CMOS process. The decoder occupies an area of 16.261 mm and runs correctly at the frequency of 212 MHz resulting in 1 Gbps decoding throughput.

Approximate counting with uniform constant-depth circuits by M. Ajtai On strong separations from $AC^0$ by E. Allender and V. Gore Parallel matching complexity of Ramsey's theorem by J. Beck On algorithms for simple stochastic games by A. Condon Locally random reductions in interactive complexity theory by J. Feigenbaum An application of game-theoretic techniques to cryptography by M. J. Fischer and R. N. Wright Composition of the universal relation by J. Ha stad and A. Wigderson Practical perfect cryptographic security by U. M. Maurer Fair games against an all-powerful adversary by R. Ostrovsky, R. Venkatesan, and M. Yung Factoring integers and computing discrete logarithms via diophantine approximation by C. P. Schnorr A new lower bound theorem for read-only-once branching programs and its applications by J. Simon and M. Szegedy On the E -isomorphism problem by J. Wang.

Current methods for optimization of low-density parity-check (LDPC) codes analyze the degree distribution pair asymptotically as block length approaches infinity. This effectively ignores the discrete nature of the space of valid degree distribution pairs for LDPC codes of finite block length. While large codes are likely to conform reasonably well to the infinite block length analysis, shorter codes have no such guarantee. We present and analyze an algorithm for completely enumerating the space of all valid degree distribution pairs for a given block length, code rate, maximum variable node degree, and maximum check node degree. We then demonstrate this algorithm on an example LDPC code of finite block length. Finally, we discuss how the result of this algorithm can be utilized by discrete optimization routines to form novel methods for the optimization of small block length LDPC codes.

Wolfram's Cellular Automata and Complexity, an anthology of some of his best work in the area, was published in 1994, and it seems fitting to attempt to explain how his seminal work can be put to good use in laboratory‐based computer science education. This article documents the advantages of using cellular automata (CA) to model complexity theory, chaos, and some aspects of the theory of computation. It explains how CA are constructed, it details the fourfold classification scheme advocated by Wolfram, and it explores in some depth issues associated with induction and Class 2 CA. Several CA are examined with an eye to determining what kinds of empirical evidence count for and against such classifications. The argument is that such lab explorations are pedagogically effective and engaging as an introduction to computer modeling, complexity theory, and related issues in the theory of computation. Software to run the CA simulations has been written by the author and can be downloaded from America Online and...

We consider transmissions of secure messages over a burst erasure wiretap channel under decoding delay constraint. For block codes we introduce and study delay optimal secure burst erasure correcting (DO-SBE) codes that provide perfect security and recover a burst of erasures of a limited length with minimum possible delay. Our explicit constructions of DO-SBE block codes achieve maximum secrecy rate. We also consider a model of a burst erasure wiretap channel for the streaming setup, where in any sliding window of a given size, in a stream of encoded source packets, the eavesdropper is able to observe packets in an interval of a given size. For that model we obtain an information theoretic upper bound on the secrecy rate for delay optimal streaming codes. We show that our block codes can be used for construction of delay optimal burst erasure correcting streaming codes which provide perfect security and meet the upper bound for a certain class of code parameters.

본 논문에서는 개선된 정규화 최소합(improved normalized min-sum) 복호 알고리듬을 적용한 LDPC 복호기를 설계하였다. 설계된 LDPC 복호기는 IEEE 802.16e 모바일 WiMAX 표준의 19가지 블록길이(576~2304)에 따른 6가지 부호율(1/2, 2/3A, 2/3B, 3/4A, 3/4B, 5/6)과 IEEE 802.11n 무선 랜 표준의 3가지 블록길이(648, 1296, 1944)에 따른 4가지 부호율(1/2, 2/3, 3/4, 5/6)을 지원한다. INMS 복호 알고리듬과 SM(sign-magnitude) 수체계 연산을 기반으로 하는 DFU(decoding function unit)을 구현하여 하드웨어 복잡도와 복호 성능을 최적화시켰다. 설계된 LDPC 복호기는 0.18-<TEX>${\mu}m$</TEX> CMOS 셀 라이브러리를 이용하여 100 MHz 동작 주파수로 합성한 결과, 284,409 게이트와 62,976 비트의 메모리로 구현되었으며, FPGA 구현을 통해 하드웨어 동작을 검증하였다. 1.8V 전원전압에서 100 MHz로 동작 가능할 것으로 평가되며, 부호율과 블록길이에 따라 약 82~218 Mbps의 성능을 가질 것으로 예상된다. A hardware design of LDPC decoder which is based on the improved normalized min-sum(INMS) decoding algorithm is described in this paper. The designed LDPC decoder supports 19 block lengths(576~2304) and 6 code rates(1/2, 2/3A, 2/3B, 3/4A, 3/4B, 5/6) of IEEE 802.16e mobile WiMAX standard and 3 block lengths(648, 1296, 1944) and 4 code rates(1/2, 2/3, 3/4, 5/6) of IEEE 802.11n WLAN standard. The decoding function unit(DFU) which is a main arithmetic block is implemented using sign-magnitude(SM) arithmetic and INMS decoding algorithm to optimize hardware complexity and decoding performance. The LDPC decoder synthesized using a 0.18-<TEX>${\mu}m$</TEX> CMOS cell library with 100 MHz clock has 284,409 gates and RAM of 62,976 bits, and it is verified by FPGA implementation. The estimated performance depending on code rate and block length is about 82~218 Mbps at 100 MHz@1.8V.

This ethno-biographical account centres around a designer’s curiosity about complexity theories. It explores the common ground between 30 years of practice in planning and landscape architecture, and system theory in all its expressions in this period. The paper sketches out the use and misuse of systems theory, dissipative structures and the notion of uncertainty, complexity theories and finally of the new thermodynamics. The author’s changing view on nature, nature conservation and nature development as a planner and a landscape architect is a subtext throughout. He makes observations on the recurring problems faced by planners and designers that use analogies from other disciplines and sciences to conceptualize design strategies. Ross Ashby’s law of the requisite variety, for instance, was adopted by Dutch national planning in the 1970s and brought the whole planning machine to a grinding halt by the sheer complexity of the org-ware trying to mimic the complexity of society. The argument leads to the conclusion, supported by examples, that very complex problems can be tackled by applying simple rules. Examples range from the occupation of the Dutch countryside by newcomers and alternative ways for the effective taxing of CO2 to developmental simulation in silico for delta cities. Complexity theories have a lot to offer to the field. Alongside the ‘Santa-Fe’ school of natural science oriented complexity theory, the European school of Social Complexity theory as epitomized by Edgar Morin seems promising for solving planning and design problems. ‘Predict and Control’ can and must be exchanged for a far more flexible way of planning. We can put our trust in, and work with, the self-organizational energy in society and nature without becoming noncommittal. Simple rules can indeed give rise to rich results and an emerging order. Moreover, systems thinking could very well prove to be the strong countervailing force against the prevailing reductionism; we need to re-establish the role of design as the synthesizing activity in these complex processes.

Orthogonal space-time block codes (OSTBCs) have several desirable properties for delay limited communication over a coherent narrowband (/sub at/) fading multiple antenna channel. In particular, they provide full diversity, and maximum likelihood detection can be achieved via linear processing and symbol-by-symbol detection. As a result, the maximal symbol rate and minimal latencies of OSTBCs have been well studied. In this paper, we construct orthogonal block codes for frequency-selective multiple antenna channels, and study their symbol rate and latency properties. Our code construction is based on direct application of the orthogonality constraints. This enables us to show that the maximum symbol rate of an orthogonal block code for a channel with memory length L is 1/L times the maximum rate of an OSTBC for the same antenna configuration. Orthogonal codes that achieve this maximal rate can be simply constructed via the Kronecker product of a rate maximal OSTBC with an identity matrix of size L. This Kronecker design scheme can also be applied to construct orthogonal block codes for frequency-selective channels with other desirable properties, such as minimal latency.