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    Abstract. The problem of replicating the flexibility of human common-sensereasoning has captured the imagination of computer scientists since the early

    days of Alan Turings foundational work on computation and the philosophy

    of artificial intelligence. In the intervening years, the idea of cognition ascomputation has emerged as a fundamental tenet of Artificial Intelligence (AI)

    and cognitive science. But what kind of computation is cognition?

    We describe a computational formalism centered around a probabilisticTuring machine called QUERY, which captures the operation of probabilis-tic conditioning via conditional simulation. Through several examples and

    analyses, we demonstrate how the QUERY abstraction can be used to castcommon-sense reasoning as probabilistic inference in a statistical model of our

    observations and the uncertain structure of the world that generated that ex-perience. This formulation is a recent synthesis of several research programs

    in AI and cognitive science, but it also represents a surprising convergence of

    several of Turings pioneering insights in AI, the foundations of computation,and statistics.

    1. Introduction 12. Probabilistic reasoning and QUERY 53. Computable probability theory 124. Conditional independence and compact representations 195. Hierarchical models and learning probabilities from data 256. Random structure 277. Making decisions under uncertainty 338. Towards common-sense reasoning 42Acknowledgements 44References 45

    1. Introduction

    In his landmark paper Computing Machinery and Intelligence [Tur50], AlanTuring predicted that by the end of the twentieth century, general educated opinionwill have altered so much that one will be able to speak of machines thinking withoutexpecting to be contradicted. Even if Turing has not yet been proven right, theidea of cognition as computation has emerged as a fundamental tenet of ArtificialIntelligence (AI) and cognitive science. But what kind of computationwhat kindof computer programis cognition?






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    AI researchers have made impressive progress since the birth of the field over60 years ago. Yet despite this progress, no existing AI system can reproduce anynontrivial fraction of the inferences made regularly by children. Turing himselfappreciated that matching the capability of children, e.g., in language, presented akey challenge for AI:

    We hope that machines will eventually compete with men in allpurely intellectual fields. But which are the best ones to start with?Even this is a difficult decision. Many people think that a veryabstract activity, like the playing of chess, would be best. It canalso be maintained that it is best to provide the machine with thebest sense organs money can buy, and then teach it to understandand speak English. This process could follow the normal teachingof a child. Things would be pointed out and named, etc. Again Ido not know what the right answer is, but I think both approachesshould be tried. [Tur50, p. 460]

    Indeed, many of the problems once considered to be grand AI challenges havefallen prey to essentially brute-force algorithms backed by enormous amounts ofcomputation, often robbing us of the insight we hoped to gain by studying thesechallenges in the first place. Turings presentation of his imitation game (what wenow call the Turing test), and the problem of common-sense reasoning implicitin it, demonstrates that he understood the difficulty inherent in the open-ended, ifcommonplace, tasks involved in conversation. Over a half century later, the Turingtest remains resistant to attack.

    The analogy between minds and computers has spurred incredible scientificprogress in both directions, but there are still fundamental disagreements aboutthe nature of the computation performed by our minds, and how best to narrowthe divide between the capability and flexibility of human and artificial intelligence.The goal of this article is to describe a computational formalism that has proveduseful for building simplified models of common-sense reasoning. The centerpiece ofthe formalism is a universal probabilistic Turing machine called QUERY that per-forms conditional simulation, and thereby captures the operation of conditioningprobability distributions that are themselves represented by probabilistic Turingmachines. We will use QUERY to model the inductive leaps that typify common-sense reasoning. The distributions on which QUERY will operate are models oflatent unobserved processes in the world and the sensory experience and observa-tions they generate. Through a running example of medical diagnosis, we aim toillustrate the flexibility and potential of this approach.

    The QUERY abstraction is a component of several research programs in AI andcognitive science developed jointly with a number of collaborators. This chapterrepresents our own view on a subset of these threads and their relationship withTurings legacy. Our presentation here draws heavily on both the work of VikashMansinghka on natively probabilistic computing [Man09, MJT08, Man11, MR]and the probabilistic language of thought hypothesis proposed and developed byNoah Goodman [KGT08, GTFG08, GG12, GT12]. Their ideas form core aspectsof the picture we present. The Church probabilistic programming language (intro-duced in [GMR+08] by Goodman, Mansinghka, Roy, Bonawitz, and Tenenbaum)


    and various Church-based cognitive science tutorials (in particular, [GTO11], de-veloped by Goodman, Tenenbaum, and ODonnell) have also had a strong influenceon the presentation.

    This approach also draws from work in cognitive science on theory-basedBayesian models of inductive learning and reasoning [TGK06] due to Tenenbaumand various collaborators [GKT08, KT08, TKGG11]. Finally, some of the theoret-ical aspects that we present are based on results in computable probability theoryby Ackerman, Freer, and Roy [Roy11, AFR11].

    While the particular formulation of these ideas is recent, they have antecedentsin much earlier work on the foundations of computation and computable analysis,common-sense reasoning in AI, and Bayesian modeling and statistics. In all of theseareas, Turing had pioneering insights.

    1.1. A convergence of Turings ideas. In addition to Turings well-known con-tributions to the philosophy of AI, many other aspects of his workacross statistics,the foundations of computation, and even morphogenesishave converged in themodern study of AI. In this section, we highlight a few key ideas that will frequentlysurface during our account of common-sense reasoning via conditional simulation.

    An obvious starting point is Turings own proposal for a research program topass his eponymous test. From a modern perspective, Turings focus on learning(and in particular, induction) was especially prescient. For Turing, the idea ofprogramming an intelligent machine entirely by hand was clearly infeasible, and sohe reasoned that it would be necessary to construct a machine with the ability toadapt its own behavior in light of experiencei.e., with the ability to learn:

    Instead of trying to produce a programme to simulate the adultmind, why not rather try to produce one that simulates the childs?If this were then subjected to an appropriate course of educationone would obtain the adult brain. [Tur50, p. 456]

    Turings notion of learning was inductive as well as deductive, in contrast to muchof the work that followed in the first decade of AI. In particular, he was quick toexplain that such a machine would have its flaws (in reasoning, quite apart fromcalculational errors):

    [A machine] might have some method for drawing conclusions byscientific induction. We must expect such a method to lead occa-sionally to erroneous results. [Tur50, p. 449]

    Turing also appreciated that a machine would not only have to learn facts, butwould also need to learn how to learn:

    Important amongst such imperatives will be ones which regulatethe order in which the rules of the logical system concerned are tobe applied. For at each stage when one is using a logical system,there is a very large number of alternative steps, any of which one ispermitted to apply [. . .] These choices make the difference betweena brilliant and a footling reasoner, not the difference between asound and a fallacious one. [. . .] [Some such imperatives] may begiven by authority, but others may be produced by the machineitself, e.g. by scientific induction. [Tur50, p. 458]

    In addition to making these more abstract points, Turing presented a number ofconcrete proposals for how a machine might be programmed to learn. His ideas


    capture the essence of supervised, unsupervised, and reinforcement learning, eachmajor areas in modern AI.1 In Sections 5 and 7 we will return to Turings writingson these matters.

    One major area of Turings contributions, while often overlooked, is statistics.In fact, Turing, along with I. J. Good, made key advances in statistics in thecourse of breaking the Enigma during World War II. Turing and Good developednew techniques for incorporating evidence and new approximations for estimatingparameters in hierarchical models [Goo79, Goo00] (see also [Zab95, 5] and [Zab12]),which were among the most important applications of Bayesian statistics at thetime [Zab12, 3.2]. Given Turings interest in learning machines and his deepunderstanding of statistical methods, it would have been intriguing to see