Morphological Computation and Control Complexity

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.

This special issue of Theory of Computing Systems contains articles which relate to talks given at the 8th International Conference on Computability, Complexity and Randomness (CCR) in Moscow from the 23rd to the 27th of September 2015. The CCR conference series is devoted to algorithmic randomness, Kolmogorov complexity, and their relationship with computability theory, complexity theory, logic and reverse mathematics. The contributions to this special issue reflect both the interests of reserachers in Russia, in particular Kolmogorov and computational complexity, and new directions in the field of algorithmic randomness such as the focus on normal numbers and connections to ergodic theory. No conference proceedings have been published; this special issue is the only publication relating to the confernece. All contributions are a significant extension of the conference presentations, and underwent the usual rigorous journal reviewing process. The article Polynomial-time algorithms for checking some properties of Boolean functions given by polynomials by Svetlana Selezneva and Anton Bukhman presents a polynomial time algorithm, which tests whether a Boolean function represented by a multivariate polynomial over GF(2) is even, that is, has zero derivative in the direction of (1, . . . , 1). The approach is based on the notion of the characteristic of monomials in the polynomial representation of Boolean functions. In Normality and finite-state dimension of Liouville numbers Satyadev Nandakumar gives a new construction of a Liouville number which is normal for a given base. Vladimir V’yugin in On Stability of Probability Laws with Respect to Small Violations of Algorithmic Randomness examines the robustness of ergodic theorems when the assumption of randomness is relaxed only a little. For example he shows that

Computability and computational complexity were first considered over the fields of real and complex numbers and generalized to arbitrary algebraic systems. This article approaches the theory of computational complexity over an arbitrary algebraic system by taking computability over the list extension for a computational model of it. We study the resultant polynomial complexity classes and mention some NP-complete 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.

THE INTERFACE BETWEEN LANGUAGE THEORY AND COMPLEXITY THEORY

Lyapunov design has never been systematic. In the adaptive control of complex multi-input non-linear systems, physical considerations, such as conservation of energy or entropy increase, represent one of the major tools in building Lyapunov-like functions and providing stability and performance guarantees. In this paper we show that a physically motivated Lyapunov-like function based on the concept of total information can be derived for large classes of non-linear physical systems. We study how this function may be used for designing estimation, adaptation and learning mechanisms for such systems. In the process we revisit familiar notions such as controllability and observability from an information perspective, which in turns allows us to define 'natural' space-time scales at which to observe and control a given complex system. By formulating control problems in algorithmic form, we emphasize the importance of computability and computational complexity for issues of control. Generic control problems are shown to be NP-hard: each additional complication, such as the presence of noise or the absence of complete system identification, moves the control problem further up the polynomial hierarchy of computational complexity. In some cases, requirements of 'optimality' may be unrealistic or irrelevant, since the solution to the problem of finding the optimal algorithm for control is uncomputable.

This issue of the Journal of Functional Programming is dedicated to work presented at the Workshop on Implicit Computational Complexity in Programming Languages, affiliated with the 1998 meeting of the International Conference on Functional Programming in Baltimore. Several machine-independent approaches to computational complexity have been developed in recent years; they establish a correspondence linking computational complexity to conceptual and structural measures of complexity of declarative programs and of formulas, proofs and models of formal theories. Examples include descriptive complexity of finite models, restrictions on induction in arithmetic and related first order theories, complexity of set-existence principles in higher order logic, and specifications in linear logic. We refer to these approaches collectively as Implicit Computational Complexity. This line of research provides a framework for a streamlined incorporation of computational complexity into areas such as formal methods in software development, programming language theory, and database theory. A fruitful thread in implicit computational complexity is based on exploring the computational complexity consequences of introducing various syntactic control mechanisms in functional programming, including restrictions (akin to static typing) on scoping, data re-use (via linear modalities), and iteration (via ramification of data). These forms of control, separately and in combination, can certify bounds on the time and space resources used by programs. In fact, all results in this area establish that each restriction considered yields precisely a major computational complexity class. The complexity classes thus obtained range from very restricted ones, such as NC and Alternating logarithmic time, through the central classes Poly-Time and Poly-Space, to broad classes such as the Elementary and the Primitive Recursive functions. Considerable effort has been invested in recent years to relax as much as possible the structural restrictions considered, allowing for more exible programming and proof styles, while still guaranteeing the same resource bounds. Notably, more exible control forms have been developed for certifying that functional programs execute in Poly-Time. The 1998 workshop covered both the theoretical foundations of the field and steps toward using its results in various implemented systems, for example in controlling the computational complexity of programs extracted from constructive proofs. The five papers included in this issue nicely represent this dual concern of theory and practice. As they are going to print, we should note that the field of Implicit Computational Complexity continues to thrive: successful workshops dedicated to it were affiliated with both the LICS'99 and LICS'00 conferences. Special issues, of Information and Computation dedicated to the former, and of Theoretical Computer Science to the latter, are in preparation.

We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multi-quantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multi-quantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic distinctions between quantified reciprocal sentences. We show a computational dichotomy between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility of revising the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multi-quantifier sentences. The paper not only contributes to the field of formal semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science.

This paper examines the pedagogical benefits of introducing the history of algebra at the outset of a Theory of Computation course. By tracing the evolution from the abstraction of numbers and the invention of computational tools to the development of abstract concepts such as imaginary numbers, students are guided to recognize that the essence of computation resides in the abstraction and symbolic representation of human thought. This historical approach provides an intuitive and intellectually rich foundation for understanding core topics, including formal languages, computability, and computational complexity, while also encouraging students to reflect on the origins, development, and future trajectory of computational science.

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...

Predictive fast Motion Estimation (ME) algorithms have been widely used in video CODECs due to their performance efficiency and low computational complexity. In this thesis, a new block-based fast motion estimation technique named Dynamic Predictive Search Algorithm (DPSA) is developed, which can be considered in predictive zonal search category. The proposed approach is based on the observation that temporally and spatially adjacent macro-blocks are not just statically correlated, but also dynamic alterations in their motion content are highly coherent. DPSA introduces a new set of six candidate predicted motion vectors. For early termination criteria, DPSA modifies termination procedure of already existing EPZS algorithm. Performance of this newly proposed algorithm has been compared to four other state-of-the-art algorithms implemented on JVT, H.264 standard software platform. Experimental results have proven that DPSA accomplishes up to 38% compression ratio enhancement achieved by a process with more 14.75% less computational complexity and up to0.47 dB higher PSNR values over the EPZS. It also manages to have up to 13% speed up over EPZS algorithm. Because of its simplicity and low computational complexity DPSA is energy efficient for portable video processing in computation- or power-constrained applications and easy to be implemented on both FPGA- and Microcontroller-based embedded systems. Also, higher compression ratio makes DPSA more compatible with limited capacity storage media, and limited band-width transmission networks.

An artificial neural network method used for analog IC parameter fault diagnosis is presented in this paper. It is fast and accurate. Therefore it has boundless prospects in the field of analog IC parameter fault diagnosis. With the rapid development in IC technology, the fault diagnosis problem of analog IC has become more acute. The traditional methods' computation complexity and inaccuracy of results make most of them still unacceptable. We therefore research and develop an artificial neural network system to resolving the low velocity and low measurability problem of the traditional methods.

The central goal of this manuscript is to survey the relationships between fundamental physics and computer science. We begin by providing a short historical review of how different concepts of computer science have entered the field of fundamental physics, highlighting the claim that the universe is a computer. Following the review, we explain why computational concepts have been embraced to interpret and describe physical phenomena. We then discuss seven arguments against the claim that the universe is a computational system and show that those arguments are wrong because of a misunderstanding of the extension of the concept of computation. Afterwards, we address a proposal to solve Hempel’s dilemma using the computability theory but conclude that it is incorrect. After that, we discuss the relationship between the proposals that the universe is a computational system and that our minds are a simulation. Analysing these issues leads us to proposing a new physical principle, called the principle of computability, which claims that the universe is a computational system (not restricted to digital computers) and that computational power and the computational complexity hierarchy are two fundamental physical constants. On the basis of this new principle, a scientific paradigm emerges to develop fundamental theories of physics: the computer-theoretic framework (CTF). The CTF brings to light different ideas already implicit in the work of several researchers and provides a new view on the universe based on computer theoretic concepts that expands the current view. We address different issues regarding the development of fundamental theories of physics in the new paradigm. Additionally, we discuss how the CTF brings new perspectives to different issues, such as the unreasonable effectiveness of mathematics and the foundations of cognitive science.

We present a deterministic $O(\log n \log \log n)$ space algorithm for undirected st-connectivity. It is based on a space-efficient simulation of the deterministic EREW algorithm of Chong and Lam [J. Algorithms, 18 (1995), pp. 378-402], an approach suggested by Prof. Vijaya Ramachandran, and uses the universal exploration sequences for trees constructed by Koucký in [Proceedings of the 16th Annual IEEE Conference on Computational Complexity, 2001, pp. 21-27]. Our result improves the $O(\log^{4/3} n)$ bound of Armoni et al. in [Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 1997, pp. 230-239] and is a big step towards the optimal $O(\log n)$. Independently of our result and using a different set of techniques, the optimal bound was achieved by Reingold in [Proceedings of the 37th Annual ACM Symposium on Theory of Computing, 2005, pp. 376-385].

Hybrid deep learning architecture for skin disease classification