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41
Optimal Ordered Problem Solver
, 2002
"... We present a novel, general, optimally fast, incremental way of searching for a universal algorithm that solves each task in a sequence of tasks. The Optimal Ordered Problem Solver (OOPS) continually organizes and exploits previously found solutions to earlier tasks, eciently searching not only the ..."
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Cited by 61 (20 self)
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We present a novel, general, optimally fast, incremental way of searching for a universal algorithm that solves each task in a sequence of tasks. The Optimal Ordered Problem Solver (OOPS) continually organizes and exploits previously found solutions to earlier tasks, eciently searching not only the space of domainspecific algorithms, but also the space of search algorithms. Essentially we extend the principles of optimal nonincremental universal search to build an incremental universal learner that is able to improve itself through experience.
The Speed Prior: A New Simplicity Measure Yielding NearOptimal Computable Predictions
 Proceedings of the 15th Annual Conference on Computational Learning Theory (COLT 2002), Lecture Notes in Artificial Intelligence
, 2002
"... Solomonoff's optimal but noncomputable method for inductive inference assumes that observation sequences x are drawn from an recursive prior distribution p(x). Instead of using the unknown p() he predicts using the celebrated universal enumerable prior M() which for all exceeds any recursiv ..."
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Cited by 51 (21 self)
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Solomonoff's optimal but noncomputable method for inductive inference assumes that observation sequences x are drawn from an recursive prior distribution p(x). Instead of using the unknown p() he predicts using the celebrated universal enumerable prior M() which for all exceeds any recursive p(), save for a constant factor independent of x. The simplicity measure M() naturally implements "Occam's razor " and is closely related to the Kolmogorov complexity of . However, M assigns high probability to certain data that are extremely hard to compute. This does not match our intuitive notion of simplicity. Here we suggest a more plausible measure derived from the fastest way of computing data. In absence of contrarian evidence, we assume that the physical world is generated by a computational process, and that any possibly infinite sequence of observations is therefore computable in the limit (this assumption is more radical and stronger than Solomonoff's).
Developmental Robotics, Optimal Artificial Curiosity, Creativity, Music, and the Fine Arts
, 2006
"... Even in absence of external reward, babies and scientists and others explore their world. Using some sort of adaptive predictive world model, they improve their ability to answer questions such as: what happens if I do this or that? They lose interest in both the predictable things and those predict ..."
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Cited by 37 (15 self)
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Even in absence of external reward, babies and scientists and others explore their world. Using some sort of adaptive predictive world model, they improve their ability to answer questions such as: what happens if I do this or that? They lose interest in both the predictable things and those predicted to remain unpredictable despite some effort. One can design curious robots that do the same. The author’s basic idea for doing so (1990, 1991): a reinforcement learning (RL) controller is rewarded for action sequences that improve the predictor. Here this idea is revisited in the context of recent results on optimal predictors and optimal RL machines. Several new variants of the basic principle are proposed. Finally it is pointed out how the fine arts can be formally understood as a consequence of the principle: given some subjective observer, great works of art and music yield observation histories exhibiting more novel, previously unknown compressibility / regularity / predictability (with respect to the observer’s particular learning algorithm) than lesser works, thus deepening the observer’s understanding of the world and what is possible in it.
Formal Theory of Creativity, Fun, and Intrinsic Motivation (19902010)
"... The simple but general formal theory of fun & intrinsic motivation & creativity (1990) is based on the concept of maximizing intrinsic reward for the active creation or discovery of novel, surprising patterns allowing for improved prediction or data compression. It generalizes the traditio ..."
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Cited by 36 (14 self)
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The simple but general formal theory of fun & intrinsic motivation & creativity (1990) is based on the concept of maximizing intrinsic reward for the active creation or discovery of novel, surprising patterns allowing for improved prediction or data compression. It generalizes the traditional field of active learning, and is related to old but less formal ideas in aesthetics theory and developmental psychology. It has been argued that the theory explains many essential aspects of intelligence including autonomous development, science, art, music, humor. This overview first describes theoretically optimal (but not necessarily practical) ways of implementing the basic computational principles on exploratory, intrinsically motivated agents or robots, encouraging them to provoke event sequences exhibiting previously unknown but learnable algorithmic regularities. Emphasis is put on the importance of limited computational resources for online prediction and compression. Discrete and continuous time formulations are given. Previous practical but nonoptimal implementations (1991, 1995, 19972002) are reviewed, as well as several recent variants by others (2005). A simplified typology addresses current confusion concerning the precise nature of intrinsic motivation.
Gödel machines: Fully selfreferential optimal universal selfimprovers
 Goertzel and C. Pennachin, Artificial General Intelligence
, 2006
"... Summary. We present the first class of mathematically rigorous, general, fully selfreferential, selfimproving, optimally efficient problem solvers. Inspired by Kurt Gödel’s celebrated selfreferential formulas (1931), such a problem solver rewrites any part of its own code as soon as it has found ..."
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Cited by 27 (13 self)
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Summary. We present the first class of mathematically rigorous, general, fully selfreferential, selfimproving, optimally efficient problem solvers. Inspired by Kurt Gödel’s celebrated selfreferential formulas (1931), such a problem solver rewrites any part of its own code as soon as it has found a proof that the rewrite is useful, where the problemdependent utility function and the hardware and the entire initial code are described by axioms encoded in an initial proof searcher which is also part of the initial code. The searcher systematically and efficiently tests computable proof techniques (programs whose outputs are proofs) until it finds a provably useful, computable selfrewrite. We show that such a selfrewrite is globally optimal—no local maxima!—since the code first had to prove that it is not useful to continue the proof search for alternative selfrewrites. Unlike previous nonselfreferential methods based on hardwired proof searchers, ours not only boasts an optimal order of complexity but can optimally reduce any slowdowns hidden by the O()notation, provided the utility of such speedups is provable at all. 1
On Universal Prediction and Bayesian Confirmation
 Theoretical Computer Science
, 2007
"... The Bayesian framework is a wellstudied and successful framework for inductive reasoning, which includes hypothesis testing and confirmation, parameter estimation, sequence prediction, classification, and regression. But standard statistical guidelines for choosing the model class and prior are not ..."
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Cited by 23 (12 self)
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The Bayesian framework is a wellstudied and successful framework for inductive reasoning, which includes hypothesis testing and confirmation, parameter estimation, sequence prediction, classification, and regression. But standard statistical guidelines for choosing the model class and prior are not always available or can fail, in particular in complex situations. Solomonoff completed the Bayesian framework by providing a rigorous, unique, formal, and universal choice for the model class and the prior. I discuss in breadth how and in which sense universal (noni.i.d.) sequence prediction solves various (philosophical) problems of traditional Bayesian sequence prediction. I show that Solomonoff’s model possesses many desirable properties: Strong total and future bounds, and weak instantaneous bounds, and in contrast to most classical continuous prior densities has no zero p(oste)rior problem, i.e. can confirm universal hypotheses, is reparametrization and regrouping invariant, and avoids the oldevidence and updating problem. It even performs well
Convergence and Loss Bounds for Bayesian Sequence Prediction
 In
, 2003
"... The probability of observing $x_t$ at time $t$, given past observations $x_1...x_{t1}$ can be computed with Bayes rule if the true generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If $\mu$ is unknown, but known to belong to a class $M$ one can base ones prediction on the Baye ..."
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Cited by 22 (21 self)
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The probability of observing $x_t$ at time $t$, given past observations $x_1...x_{t1}$ can be computed with Bayes rule if the true generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If $\mu$ is unknown, but known to belong to a class $M$ one can base ones prediction on the Bayes mix $\xi$ defined as a weighted sum of distributions $ u\in M$. Various convergence results of the mixture posterior $\xi_t$ to the true posterior $\mu_t$ are presented. In particular a new (elementary) derivation of the convergence $\xi_t/\mu_t\to 1$ is provided, which additionally gives the rate of convergence. A general sequence predictor is allowed to choose an action $y_t$ based on $x_1...x_{t1}$ and receives loss $\ell_{x_t y_t}$ if $x_t$ is the next symbol of the sequence. No assumptions are made on the structure of $\ell$ (apart from being bounded) and $M$. The Bayesoptimal prediction scheme $\Lambda_\xi$ based on mixture $\xi$ and the Bayesoptimal informed prediction scheme $\Lambda_\mu$ are defined and the total loss $L_\xi$ of $\Lambda_\xi$ is bounded in terms of the total loss $L_\mu$ of $\Lambda_\mu$. It is shown that $L_\xi$ is bounded for bounded $L_\mu$ and $L_\xi/L_\mu\to 1$ for $L_\mu\to \infty$. Convergence of the instantaneous losses is also proven.
Gödel Machines: SelfReferential Universal Problem Solvers Making Provably Optimal SelfImprovements
, 2003
"... An old dream of computer scientists is to build an optimally efficient universal problem solver. We show how to solve arbitrary computational problems in an optimal fashion inspired by Kurt Gödel's celebrated selfreferential formulas (1931). Our Gödel machine's initial software includes ..."
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Cited by 17 (8 self)
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An old dream of computer scientists is to build an optimally efficient universal problem solver. We show how to solve arbitrary computational problems in an optimal fashion inspired by Kurt Gödel's celebrated selfreferential formulas (1931). Our Gödel machine's initial software includes an axiomatic description of: the Gödel machine's hardware, the problemspecific utility function (such as the expected future reward of a robot), known aspects of the environment, costs of actions and computations, and the initial software itself (this is possible without introducing circularity). It also includes a typically suboptimal initial problemsolving policy and an asymptotically optimal proof searcher searching the space of computable proof techniques  that is, programs whose outputs are proofs. Unlike previous approaches, the selfreferential Gödel machine will rewrite any part of its software, including axioms and proof searcher, as soon as it has found a proof that this will improve its future performance, given its typically limited computational resources. We show that selfrewrites are globally optimal  no local minima!since provably none of all the alternative rewrites and proofs (those that could be found by continuing the proof search) are worth waiting for.
The New AI: General & Sound & Relevant for Physics
 ARTIFICIAL GENERAL INTELLIGENCE (ACCEPTED 2002)
, 2003
"... Most traditional artificial intelligence (AI) systems of the past 50 years are either very limited, or based on heuristics, or both. The new millennium, however, has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction, search, induct ..."
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Cited by 16 (9 self)
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Most traditional artificial intelligence (AI) systems of the past 50 years are either very limited, or based on heuristics, or both. The new millennium, however, has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction, search, inductive inference based on Occam’s razor, problem solving, decision making, and reinforcement learning in environments of a very general type. Since inductive inference is at the heart of all inductive sciences, some of the results are relevant not only for AI and computer science but also for physics, provoking nontraditional predictions based on Zuse’s thesis of the computergenerated universe.
On the foundations of universal sequence prediction
 In Proc. 3rd Annual Conference on Theory and Applications of Models of Computation (TAMC’06), volume 3959 of LNCS
, 2006
"... Solomonoff completed the Bayesian framework by providing a rigorous, unique, formal, and universal choice for the model class and the prior. We discuss in breadth how and in which sense universal (noni.i.d.) sequence prediction solves various (philosophical) problems of traditional Bayesian sequenc ..."
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Cited by 11 (4 self)
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Solomonoff completed the Bayesian framework by providing a rigorous, unique, formal, and universal choice for the model class and the prior. We discuss in breadth how and in which sense universal (noni.i.d.) sequence prediction solves various (philosophical) problems of traditional Bayesian sequence prediction. We show that Solomonoff’s model possesses many desirable properties: Fast convergence and strong bounds, and in contrast to most classical continuous prior densities has no zero p(oste)rior problem, i.e. can confirm universal hypotheses, is reparametrization and regrouping invariant, and avoids the oldevidence and updating problem. It even performs well (actually better) in noncomputable environments.