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Hierarchies Of Generalized Kolmogorov Complexities And Nonenumerable Universal Measures Computable In The Limit
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 2000
"... The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable m ..."
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Cited by 44 (21 self)
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The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly in nite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briey mention consequences for universes sampled from such priors.
Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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Cited by 44 (1 self)
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
SelfEvolution in a Constructive Binary String System
 Artificial Life
, 1998
"... This paper focuses on the phenomena of evolution whose appearance is notable because no explicit mutation, recombination or artificial selection operators are introduced. We call the system selfevolving because every variation is performed by the objects themselves in their machine form. Keywords: ..."
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Cited by 39 (16 self)
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This paper focuses on the phenomena of evolution whose appearance is notable because no explicit mutation, recombination or artificial selection operators are introduced. We call the system selfevolving because every variation is performed by the objects themselves in their machine form. Keywords: artificial chemistry, autocatalytic reaction system, molecular computing, prebiotic evolution, selforganization, selfprogramming 1
Computational Logic and Human Thinking: How to be Artificially Intelligent
, 2011
"... The mere possibility of Artificial Intelligence (AI) – of machines that can think and act as intelligently as humans – can generate strong emotions. While some enthusiasts are excited by the thought that one day machines may become more intelligent than people, many of its critics view such a prosp ..."
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Cited by 37 (10 self)
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The mere possibility of Artificial Intelligence (AI) – of machines that can think and act as intelligently as humans – can generate strong emotions. While some enthusiasts are excited by the thought that one day machines may become more intelligent than people, many of its critics view such a prospect with horror. Partly because these controversies attract so much attention, one of the most important accomplishments of AI has gone largely unnoticed: the fact that many of its advances can also be used directly by people, to improve their own human intelligence. Chief among these advances is Computational Logic. Computational Logic builds upon traditional logic, which was originally developed to help people think more effectively. It employs the techniques of symbolic logic, which has been used to build the foundations of mathematics and computing. However, compared with traditional logic, Computational Logic is much more powerful; and compared with symbolic logic, it is much simpler and more practical. Although the applications of Computational Logic in AI require the use of mathematical notation, its human applications do not. As a consequence, I have written the main part of this book informally, to reach as wide an audience as possible. Because human thinking is also the subject of study in many other fields, I have drawn upon related studies in Cognitive Psychology, Linguistics, Philosophy, Law, Management Science and English
Algorithmic Theories Of Everything
, 2000
"... The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lac ..."
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Cited by 36 (15 self)
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The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lacking a short description, and study the spectrum of TOEs spanned by two Ps, one reflecting the most compact constructive descriptions, the other the fastest way of computing everything. The former derives from generalizations of traditional computability, Solomonoff’s algorithmic probability, Kolmogorov complexity, and objects more random than Chaitin’s Omega, the latter from Levin’s universal search and a natural resourceoriented postulate: the cumulative prior probability of all x incomputable within time t by this optimal algorithm should be 1/t. Between both Ps we find a universal cumulatively enumerable measure that dominates traditional enumerable measures; any such CEM must assign low probability to any universe lacking a short enumerating program. We derive Pspecific consequences for evolving observers, inductive reasoning, quantum physics, philosophy, and the expected duration of our universe.
A Theory for Mentally Developing Robots
"... This paper introduces a theory about mentally developing robots. The limitation of the traditional agent model is raised and a new SASE agent is proposed, based on our SAIL developmental robot. We formulate the manual development paradigm and autonomous development paradigm. The performance of a dev ..."
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Cited by 31 (5 self)
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This paper introduces a theory about mentally developing robots. The limitation of the traditional agent model is raised and a new SASE agent is proposed, based on our SAIL developmental robot. We formulate the manual development paradigm and autonomous development paradigm. The performance of a developmental robot is then formulated as reaching the norm of a human age group. The framework of autonomously generating brain 1 representation is investigated in mathematical terms. Some techniques of such a representation are provided based on our SAIL2 developmental algorithm. We establish the conceptual limitation of symbolic representation and from the limitation we propose that no developmental robot can use a symbolic representation. Finally, the completeness of developmental robot is investigated conditioned on five factors.
Driven by Compression Progress: A Simple Principle Explains Essential Aspects of Subjective Beauty, Novelty, Surprise, Interestingness, Attention, Curiosity, Creativity, Art, Science, Music, Jokes
, 2009
"... I argue that data becomes temporarily interesting by itself to some selfimproving, but computationally limited, subjective observer once he learns to predict or compress the data in a better way, thus making it subjectively simpler and more beautiful. Curiosity is the desire to create or discover m ..."
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Cited by 30 (7 self)
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I argue that data becomes temporarily interesting by itself to some selfimproving, but computationally limited, subjective observer once he learns to predict or compress the data in a better way, thus making it subjectively simpler and more beautiful. Curiosity is the desire to create or discover more nonrandom, nonarbitrary, regular data that is novel and surprising not in the traditional sense of Boltzmann and Shannon but in the sense that it allows for compression progress because its regularity was not yet known. This drive maximizes interestingness, the first derivative of subjective beauty or compressibility, that is, the steepness of the learning curve. It motivates exploring infants, pure mathematicians, composers,
A New Method for Undecidability Proofs of First Order Theories
 Journal of Symbolic Computation
, 1992
"... this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction ..."
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Cited by 30 (6 self)
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this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction
Ultimate Cognition à la Gödel
 COGN COMPUT
, 2009
"... "All life is problem solving," said Popper. To deal with arbitrary problems in arbitrary environments, an ultimate cognitive agent should use its limited hardware in the "best" and "most efficient" possible way. Can we formally nail down this informal statement, and der ..."
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Cited by 30 (12 self)
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"All life is problem solving," said Popper. To deal with arbitrary problems in arbitrary environments, an ultimate cognitive agent should use its limited hardware in the "best" and "most efficient" possible way. Can we formally nail down this informal statement, and derive a mathematically rigorous blueprint of ultimate cognition? Yes, we can, using Kurt Gödel’s celebrated selfreference trick of 1931 in a new way. Gödel exhibited the limits of mathematics and computation by creating a formula that speaks about itself, claiming to be unprovable by an algorithmic theorem prover: either the formula is true but unprovable, or math itself is flawed in an algorithmic sense. Here we describe an agentcontrolling program that speaks about itself, ready to rewrite itself in arbitrary fashion once it has found a proof that the rewrite is useful according to a userdefined utility function. Any such a rewrite is necessarily globally optimal—no local maxima!—since this proof necessarily must have demonstrated the uselessness of continuing the proof search for even better rewrites. Our selfreferential program will optimally speed up its proof searcher and other program parts, but only if the speed up’s utility is indeed provable—even ultimate cognition has limits of the Gödelian kind.