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NPcomplete problems and physical reality
 ACM SIGACT News Complexity Theory Column, March. ECCC
, 2005
"... Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Mal ..."
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Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, MalamentHogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing. ” The section on soap bubbles even includes some “experimental ” results. While I do not believe that any of the proposals will let us solve NPcomplete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1
The tractable cognition thesis
 Cognitive Science: A Multidisciplinary Journal
, 2008
"... The recognition that human minds/brains are finite systems with limited resources for computation has led some researchers to advance the Tractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. This thesis, if true, serves cognitive psychology by constra ..."
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The recognition that human minds/brains are finite systems with limited resources for computation has led some researchers to advance the Tractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. This thesis, if true, serves cognitive psychology by constraining the space of computationallevel theories of cognition. To utilize this constraint, a precise and workable definition of “computational tractability ” is needed. Following computer science tradition, many cognitive scientists and psychologists define computational tractability as polynomialtime computability, leading to the PCognition thesis. This article explains how and why the PCognition thesis may be overly restrictive, risking the exclusion of veridical computationallevel theories from scientific investigation. An argument is made to replace the PCognition thesis by the FPTCognition thesis as an alternative formalization of the Tractable Cognition thesis (here, FPT stands for fixedparameter tractable). Possible objections to the Tractable Cognition thesis, and its proposed formalization, are discussed, and existing misconceptions are clarified.
Computational universes
 Chaos, Solitons & Fractals
, 2006
"... Suspicions that the world might be some sort of a machine or algorithm existing “in the mind ” of some symbolic number cruncher have lingered from antiquity. Although popular at times, the most radical forms of this idea never reached mainstream. Modern developments in physics and computer science h ..."
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Cited by 9 (5 self)
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Suspicions that the world might be some sort of a machine or algorithm existing “in the mind ” of some symbolic number cruncher have lingered from antiquity. Although popular at times, the most radical forms of this idea never reached mainstream. Modern developments in physics and computer science have lent support to the thesis, but empirical evidence is needed before it can begin to replace our contemporary world view.
Quantum computation and quantum information
 International Journal of Parallel, Emergent and Distributed Systems
, 2006
"... The paper is intended to be a survey of all the important aspects and results that have shaped the eld of quantum computation and quantum information. The reader is rst familiarized with those features and principles of quantum mechanics providing a more e cient and secure information processing. Th ..."
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Cited by 3 (3 self)
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The paper is intended to be a survey of all the important aspects and results that have shaped the eld of quantum computation and quantum information. The reader is rst familiarized with those features and principles of quantum mechanics providing a more e cient and secure information processing. Their applications to the general theory of information, cryptography, algorithms, computational complexity and errorcorrection are then discussed. Prospects for building a practical quantum computer are also analyzed. 1 Introduction and
Why philosophers should care about computational complexity
 In Computability: Gödel, Turing, Church, and beyond (eds
, 2012
"... One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that ..."
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One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that studies the resources (such as time, space, and randomness) needed to solve computational problems—leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume’s problem of induction, Goodman’s grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing
Quantum computability and complexity and the limits of quantum computation. http://www.cs.berkeley.edu/ kamil/quantum/qc4.pdf
, 2003
"... This paper will present an overview of the current beliefs of quantum complexity theorists and discuss in detail the impacts these beliefs may have on the future of the field. In section one we give a brief fundamental overview of classical complexity theory, defining the time and space hierarchies ..."
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This paper will present an overview of the current beliefs of quantum complexity theorists and discuss in detail the impacts these beliefs may have on the future of the field. In section one we give a brief fundamental overview of classical complexity theory, defining the time and space hierarchies and providing examples of problems that fit into several important categories thereon. In section two we introduce quantum complexity and discuss its relationship to classical complexity. Section three presents an overview of the successful existing quantum algorithms, and section four presents a discussion on the actual practical gains that might be realized by quantum computation. Finally in section five we speculate briefly on the direction we believe the field to be headed, and what might reasonably be expected in the future. 1 Introduction to Computability and Complexity In computability and complexity theory, all problems are reduced to language recognition, where a language is defined as a set of strings over some alphabet, typically {0, 1}. For example, L1 = {0, 00, 000, · · · } is the language that contains all strings that contain only the letter 0. Once we have a language L, there are some interesting questions we can ask about it. Given an input x, can a machine decide if x ∈ L? If so, how
The Complexity of Tensor Circuit Evaluation (Extended Abstract)
"... Martin Beaudry 2 Markus Holzer 3 Abstract The study of tensor calculus over semirings in terms of complexity theory was initiated by Damm et al. in [9]. Here we first look at tensor circuits, a natural generalization of tensor formulas; we show that the problem of asking whether the output of su ..."
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Martin Beaudry 2 Markus Holzer 3 Abstract The study of tensor calculus over semirings in terms of complexity theory was initiated by Damm et al. in [9]. Here we first look at tensor circuits, a natural generalization of tensor formulas; we show that the problem of asking whether the output of such circuits is nonzero is complete for the class NE = NTIME(2 O(n) ) for circuits over the boolean semiring, E for the field F2 , and analogous results for other semirings. Commonsense restrictions such as imposing a logarithmic upper bound on circuit depth are also discussed. Second, we analyze other natural problems concerning tensor formulas and circuits over various semirings, such as asking whether the output matrix is diagonal, or a null matrix. 1
COCOON'01 Submission
"... Grover's quantum search algorithm finds one of t solutions in N candidates by using N/t basic steps. It is, however, necessary to know the number t of solutions in advance for using the Grover's algorithm directly. On the other hand, Boyer etal proposed a randomized application of Grover's algorit ..."
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Grover's quantum search algorithm finds one of t solutions in N candidates by using N/t basic steps. It is, however, necessary to know the number t of solutions in advance for using the Grover's algorithm directly. On the other hand, Boyer etal proposed a randomized application of Grover's algorithm, which runs, on average, in O( N/t) basic steps (more precisely, (9/4) N/t steps) without knowing t in advance. Here we show a simple (almost trivial) deterministic application of Grover's algorithm also works and finds a solution in O( N/t) basic steps (more precisely, (8#/3) N/t steps) on average. 1.