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# Topic: Computational complexity theory

###### In the News (Mon 18 Feb 19)

 Wiley::Theory of Computational Complexity Theory of Computational Complexity offers a thorough presentation of the fundamentals of complexity theory, including NP-completeness theory, the polynomial-time hierarchy, relativization, and the application to cryptography. It also examines the theory of nonuniform computational complexity, including the computational models of decision trees and Boolean circuits, and the notion of polynomial-time isomorphism. The theory of probabilistic complexity, which studies complexity issues related to randomized computation as well as interactive proof systems and probabilistically checkable proofs, is also covered. www.wiley.com /WileyCDA/WileyTitle/productCd-0471345067.html   (285 words)

 Computational complexity theory - Wikipedia, the free encyclopedia In computer science, computational complexity theory is the branch of the theory of computation that studies the resources, or cost, of the computation required to solve a given problem. Complexity theory differs from computability theory, which deals with whether a problem can be solved at all, regardless of the resources required. The time complexity of a problem is the number of steps that it takes to solve an instance of the problem as a function of the size of the input (usually measured in bits), using the most efficient algorithm. en.wikipedia.org /wiki/Computational_complexity_theory   (1290 words)

 Complexity theory - Wikipedia, the free encyclopedia Computational complexity theory: a field in theoretical computer science and mathematics dealing with the resources required during computation to solve a given problem. The theoretical treatment of Kolmogorov complexity of a string is studied in algorithmic information theory by identifying the length of the shortest binary program which can output that string. Complexity Theory is sometimes used as a broad term addressing the study of complex systems, including subjects such as chaos theory, artificial life, and genetic algorithms. en.wikipedia.org /wiki/Complexity_theory   (209 words)

 Computational complexity theory   (Site not responding. Last check: 2007-10-26) Complexity theory is part of the theoryof computation dealing with the resources required during computation to solve a given problem. The time complexity of a problem is the number of steps that it takes to solve an instance of the problem, asa function of the size of the input, (usually measured in bits) using themost efficient algorithm. The complexity class NP is the set of decision problems thatcan be solved by a non-deterministic machine in polynomial time. www.therfcc.org /computational-complexity-theory-23762.html   (884 words)

 Dieter van Melkebeek - Research on Computational Complexity Theory Complexity theory aims to answer this question by describing how many resources we need to compute the solution as a function of the problem size. Torenvliet established that large complexity classes like doubly exponential space have complete languages that are not autoreducible, whereas the complete languages of smaller classes like exponential time all share the property of autoreducibility. The Kolmogorov complexity of a string is the length of its shortest description; various complexity restrictions on the descriptions lead to various notions of Kolmogorov complexity. www.cs.wisc.edu /~dieter/Research/complexity.html   (2933 words)

 Complexity Theory and Finite Model Theory Computational complexity theory is concerned with the investigation of the resources (usually time and space) required for the computational solution of problems. Finite model theory is the model theory of finite structures, where model theory (of finite and infinite structures) is the branch of mathematical logic dealing with the relationship between a formal language and its interpretation in mathematical structures: it has a long and varied history. The model theory of finite structures is very underdeveloped although interest in the subject has now exploded, mainly due to the intimate relationship between finite model theory and complexity theory. www.swan.ac.uk /compsci/ResearchGroups/TheoryGroups/CTFMT.html   (178 words)

 Computational Complexity: Defining Theory Computational complexity and other fun stuff in math and computer science as viewed by Lance Fortnow. The field of theoretical computer science is interpreted broadly so as to include algorithms, data structures, complexity theory, distributed computation, parallel computation, VLSI, machine learning, computational biology, computational geometry, information theory, cryptography, quantum computation, computational number theory and algebra, program semantics and verification, automata theory, and the study of randomness. This would take into account that theory covers not only 'internal' topics (as suggested by the "analysis of computation/information processing" definition), but is also sometimes applied to understand and develop tools for 'external' areas that have a computational character (such as e-commerce, comp. weblog.fortnow.com /2005/06/defining-theory.html   (1852 words)

 CSCE 424/896 - Introduction to Computational Complexity Theory   (Site not responding. Last check: 2007-10-26) Complexity theory investigates the reasons behind this important phenomenon. Computational problems are at the heart of computer science; and since complexity theory studies the nature of computational problems, it is a foundational area of computer science. In particular, complexity theory has many applications to various subareas of computer science including machine learning, computer security, distributed computing and bio-informatics. csce.unl.edu /~vinod/CSCE424   (290 words)

 Introduction to Complexity Theory The Space Complexity of a decision problem, f, is the amount of memory used by the `best' algorithm A for f. A major open problem in Computational Complexity Theory is to develop arguments by which important computational problems can have their time complexity described exactly. Computational Complexity Theory involves a large number of subfields each of which is ultimately concerned with problems such as those above, e.g. www.csc.liv.ac.uk /~ped/teachadmin/algor/complex.html   (816 words)

 Research Areas | CCS-5 Basic and Applied Science | Decision Applications Division The goal of this project is to study the intrinsic computational complexity of problems. We are investigating the complexity of decision, counting and approximate optimization of combinatorial problems. - to ellucidate the connection between computational complexity and the existence of a phase transition. www.ccs.lanl.gov /ccs5/research/compcomplex.shtml   (487 words)

 Newsroom   (Site not responding. Last check: 2007-10-26) They will concentrate on the power of randomness in computation, the complexity of proofs and search for proofs, and the power and limitations of various computational models. “Computational complexity research here at the Institute for Advanced Study is guided by a few clear questions, deeply motivated by scientific, practical, and philosophical concerns,” notes Wigderson. Wigderson’s research interests are in complexity theory, algorithms, randomness, and cryptography. www.ias.edu /Newsroom/announcements/Uploads/view.php?cmd=view&id=243   (335 words)

 COLT: Computational Learning Theory Computational Learning Theory (COLT) is a research field devoted to studying the design and analysis of algorithms for making predictions about the future based on past experiences. As a field with roots in theoretical computer science, COLT is largely concerned with computational and data efficiency. COLT has strongly encouraged interaction with other fields that work on problems of prediction such as applied machine learning, statistics, information theory, pattern recognition and statistical physics, as well as other areas of computer science such as artificial intelligence, complexity theory and cryptography. www.learningtheory.org   (163 words)

 [No title]   (Site not responding. Last check: 2007-10-26) Computational complexity theory has been one of the most exciting fields of scientific research over the last few decades. This research studies the power of feasible computation, and is guided by a few clear and focused questions, deeply motivated on scientific, practical and philosophical grounds, like the P vs NP problem, and the questions on the power of randomized and quantum computation. While these problems are far from resolved, Complexity Theory was able to offer fresh rigorous definitions to some central notions which naturally (or less so) arise from these questions, and unveil many rich and beautiful connections between them. www.cs.buffalo.edu /pub/WWW/Events/Colloquia_Info/Avi.htm   (164 words)

 Computational Complexity Theory (ResearchIndex)   (Site not responding. Last check: 2007-10-26) Abstract: INTRODUCTION Computational complexity is the study of the resources, such as time and space (memory), required to solve computational problems. Computability theory establishes the existence of undecidable problems that cannot be solved in principle, regardless of the amount of time invested. In contrast, complexity theory establishes the existence of decidable problems that, although... citeseer.ist.psu.edu /500303.html   (155 words)

 Average-case Computational Complexity Theory - Wang (ResearchIndex) The theory of average-case computational complexity, initiated by Levin about ten years ago, is devoted to studying this problem. 254 Elements of the Theory of Computation (context) - Lewis, Papadimitriou - 1981 ACM 57 the computational complexity of algorithms (context) - Hartmanis, Stearns - 1965 citeseer.ist.psu.edu /252343.html   (946 words)

 Computational Complexity at WIS   (Site not responding. Last check: 2007-10-26) Computational Complexity Theory is a central field of Theoretical Computer Science, with a remarkable list of celebrated achievements as well as a very vibrant present research activity. The field is concerned with the study of the intrinsic complexity of computational tasks, and this study tend to aim at generality: It focuses on natural computational resources, and the effect of limiting those on the class of problems that can be solved. The Department of Computer Science at the Weizmann Institute of Science maintains a wide range of activities in Computational Complexity. www.wisdom.weizmann.ac.il /~oded/wis-cc.html   (181 words)

 Advances in Complexity Theory   (Site not responding. Last check: 2007-10-26) Computational Complexity Theory is the field that studies the efficiency of computation. The famed "P versus NP" problem (one of the seven open problems of the Clay Institute) is the central problem of this field and, despite three decades of active research, this problem has eluded resolution. The goal of this workshop is to examine these and other recent achievements in complexity theory, exchange ideas, formulate open problems, and identify new directions of research. www.pims.math.ca /birs/workshops/2004/04w5100   (280 words)

 AT02.20 Theory of Computation To provide an exposure to the theory of formal languages, automata and complexity theory. Theory of Computer Science: Automata, Language and Computation, Prentice Hall, 1995. A Recursive Introduction to the Theory of Computation, Springer Verlag, 1994. www.cs.ait.ac.th /~on/AT02.20.shtml   (108 words)

 Advanced Computational Complexity Theory The course covers fundamental concepts of Computational Complexity Theory, in particular complexity-classes and their relationships. First, figure out which complexity class is Universality complete for, then prove Universality in that class and hard for it. As for the hardness result, given a TM M and input x, consider the language of all malformed or non-legal or non-accepting computations of M on x (represented as a sequence of configurations), and construct an NFA that accepts that language. www.math.tau.ac.il /~safra/ACT   (1760 words)

 The complexity theory companion   (Site not responding. Last check: 2007-10-26) The book's theme is that simple algorithms are at the heart of complexity theory. In fact, to most clearly highlight the role of algorithmic techniques in complexity theory, the book is organized by technique rather than by topic. In particular, and in contrast to the organization of previous textbooks on complexity, each chapter of this book focuses on one technique - what it is, and what results and applications it has yielded in complexity theory. www.cs.ioc.ee /~bibi/kyber/Contents/hemaspaandra.html   (139 words)

 Computational Complexity Theory   (Site not responding. Last check: 2007-10-26) A fundamental area of theoretical computer science is complexity theory, the analysis of the resources needed to solve computational problems. Researchers in this area define computational models, such as Turing machines, Boolean Circuits, Parallel Random Access Machines, etc., and resource measures such as space, parallel time, amount of hardware, etc. A complexity class is then the set of problems solvable in a particular model under particular resource constraints. Computational complexity was originally defined in terms of the natural entities of time and space, and the term complexity was used to denote the time or space used in the computation. www.cs.umass.edu /~immerman/complexity_theory.html   (295 words)

 OUP: Computational Complexity Theory: Rudich   (Site not responding. Last check: 2007-10-26) Computational complexity theory is a major research area in mathematics and computer science, the goal of which is to set the formal mathematical foundations for efficient computation. It has evolved to encompass a broad variety of computational tasks by a diverse set of computational models, such as randomized, interactive, distributed, and parallel computations. The volume is recommended for independent study and is suitable for graduate students and researchers interested in computational complexity. www.oup.co.uk /isbn/0-8218-2872-X   (431 words)

 About CSAIL The MIT Computer Science and Artificial Intelligence Laboratory, or CSAIL, was formed on July 1st, 2003 by the merger of the Artificial Intelligence Lab and the Laboratory for computer Science, each with four decades of rich history. The primary mission of CSAIL is research in both computer science and artificial intelligence, broadly construed. Theory: This area of research studies the mathematics of computation and its consequences. www.csail.mit.edu /about/about.html   (300 words)

 Computational Complexity   (Site not responding. Last check: 2007-10-26) The initial purpose of Computational Complexity is to make precise the intuitive idea of a computationally feasible problem. The birth of the theory of Computational Complexity can be set in mid 60s when the first users of electronic computers started to pay increasing attention to the performances of their programs. The main goals of Computational Complexity theory are to introduce classes of problems that have similar complexity with respect to a specific computation model and to study the intrinsic properties of such classes. alpha.fdu.edu /~sharma/images/Computational_Complexity.html   (117 words)

 Computability and Complexity - Introduction The answer is that such results are proved with respect to abstract models of computation that capture all the essential elements of a typical computer. So a number is computable if there is such a TM that will stop after printing n number of 1’s on an input tape of zeroes. But Turing says that these computable numbers constitute a small part of the set of all numbers and for the most part the great percentage of numbers are not computable. hypatia.math.uri.edu /~kulenm/mth381pr/comput/computab.html   (1914 words)

 Complexity class - Psychology Central   (Site not responding. Last check: 2007-10-26) In computational complexity theory, a complexity class is a set of problems of related complexity. For example, the class NP is the set of decision problems that can be solved by a non-deterministic Turing machine in polynomial time, while the class PSPACE is the set of decision problems that can be solved by a deterministic Turing machine in polynomial space. The Blum axioms can be used to define complexity classes without referring to a concrete computational model. psychcentral.com /psypsych/Complexity_class   (371 words)

 An Introduction to Computational Complexity - Part A   (Site not responding. Last check: 2007-10-26) The goal of this theory is therefore to assist algorithm designers in directing their efforts towards promising areas and avoid impossible tasks. Example #1 illustrated a very important point: Algorithms which have a polynomial or sub-polynomial time complexity (that is, they take time O(g(n)) where g(n) is either a polynomial or a function bounded by a polynomial), are practical. A second argument is that, polynomial-time algorithms, once they are discovered, undergo a series of improvements, that is, reductions in the constants in their complexity functions and the degree itself. users.forthnet.gr /ath/kimon/CC/CCC1b.htm   (1501 words)

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