9 Using the Language of Thought by Eyal Dechter A.B., Harvard University (2009) Submitted to the Department of Brain and Cognitive Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2018 Massachusetts Institute of Technology 2018. All rights reserved. Signature redacted A uthor .. ........................... / Department of Brain and Cognitive Sciences May 23, 2017 Signature redacted Certified by............ Joshua B. Tenenbaum W Professor Thesis Supervisor rSignature redacted Accepted by.>: Matthew Wilson Sherman Flirchild Professor of Neuroscience and Picower Scholar, Director of Graduate Education for Brain and Cognitive Sciences MASSACHUSETS INSTITUTE OF TECHNOLOGY OCIT 1 12 018 LIBRARIES ARCHIVES Using the Language of Thought by Eyal Dechter Submitted to the Department of Brain and Cognitive Sciences on May 23, 2017, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis, I develop and explore two novel models of how humans might be able to acquire high-level conceputal knowledge by performing probabilistic inference over a language of thought (Fodor 1975) - a space of symbolic and compositional mental representations sufficiently expressive to capture the meanings of human thoughts and utterances. These models and their associated learning algorithms are moti- vated by an attempt to provide an understanding of the algorithmic principles that might underlie a child's ability to search the haystack of sentences in her language of thought to find the needle that corresponds to any specific concept. The first model takes advantage of the compositionality inherent to LOT representations, framing concept acquisition as program induction in a functional programming language; the Exploration-Compression algorithm this model motivates iteratively builds a library of useful program fragments that, when composed, restructures the search space, making more useful programs shorter and easier to find. The second model, the Infi- nite Knowledge Base Model (IKM), frames concept learning as probabilistic inference over the space of relational knowledge bases; the algorithm I develop for learning in this model frames this inference problem as a state-space search over abductive proofs of the learner's observed data. This framing allows us to take advantage of powerful techniques from the heuristic search and classical planning literature to guide the learner. In the final part of this thesis, I explore the behavior of the IKM on several case studies of intuitive theories from the concept learning literature, and I discuss evidence for and against it with respect to other approaches to LOT models. Thesis Supervisor: Joshua B. Tenenbaum Title: Professor 3 4 Acknowledgments I would like to thank my advisor, Josh Tenenbaum, for his guidance and support of my work. I am grateful also to the members of my committee, Laura Schulz, Steve Piantadosi and Roger Levy, for their thoughtful feedback. I am incredibly grateful to my collaborators who contributed to this and related work. They include Ryan Adams, Kevin Ellis, Jon Malmaud, Stephen Muggleton, and Dianhuan Lin. In addition, I would like to thank my colleagues in CoCoSci for their guidance, support, and encouragement in this work. I am grateful to the National Science Foundation Graduate Research Fellowship Program for their funding support. This thesis would not exist if not for the love and support of my parents, Rina and Avi Dechter, and it has been improved greatly by my mother's guidance and insight throughout my time at MIT. Lastly, I would like to thank my wife, Amy, for her support arid love; my daughter, Susanna, for being an endless source of inspiration as she learns and develops; and my dog, Ruby, for being a consistent companion in the office and at home. 5 Contents 1 Foreword 17 2 Bootstrap learning via modular program induction 21 2.1 Introduction ....... ............................... 21 2.2 Related Work ............................. 22 2.3 The Bayes LOT framework .... ........... ....... 24 2.4 The Exploration-Compression algorithm ... ........... 25 2.4.1 Generative M odel ............. .......... 25 2.4.2 Inference ............................ 26 2.4.3 Language ............. .............. 27 2.4.4 A stochastic grammar over programs ............ 29 2.4.5 The EXPLORATION step: best-first enumeration of programs 31 2.4.6 The COMPRESSION step ..... .............. 32 2.5 Variations on the EC algorithm ................... 34 2.5.1 Enumerating the frontier . .................. 34 2.5.2 Sequentially Constructing Programs by Sequential EC . 35 2.5.3 Symbolic Dimensionality Reduction ............. 36 2.6 A pplications .......... .................... 37 2.6.1 Applications of Vanilla EC .................. 37 2.6.2 Applications of Sequential EC ................ 46 2.6.3 Applications of Symbolic Dimensionality Reduction ... 50 2.7 Conclusion & Future Directions ................. .. 53 3 The Infinite Knowledge Base Model 55 3.1 Introduction . ........ ........ ... 55 3.2 Related work ............ ....... 56 3.3 Prelim inaries ............ ....... 57 3.4 The Infinite Knowledge Base Model ....... 59 3.4.1 Knowledge base schema ..... .... 59 3.4.2 The probability distribution ... .... 61 3.5 MAP Inference for the IKM ........... 66 3.6 Experiments and case studies .......... 75 3.6.1 Dealing with noisy data ......... 75 3.6.2 Does the heuristic improve learning times? 76 3.6.3 Heuristic graph size ............ 79 7 3.6.4 Comparison with METAGOL ...... ...... ...... 81 3.7 Conclusion ........ ................................. 84 4 Modeling intuitive theories with the IKM 87 4.1 Introduction ........ ................................ 87 4.2 Related Work ........ ............................... 88 4.3 The Infinite Knowledge Base Model ................... 92 4.3.1 Inference ...... ........ ....... ........ 94 4.4 Case studies ...... ......... .......... ....... 95 4.4.1 M agnetism ................ ............. 95 4.4.2 K inship ................. ............. 96 4.4.3 Counting . ............ ............ ..... 99 4.5 D iscussion .. ............. ............ 109 5 Afterword 115 A EC as variational inference in a hierarchical probabilistic model over functional programs 117 A.1 Circuit Distribution ... ...... ...... ....... ...... 124 8 List of Figures 2-1 Graphical model for the multitask reinforcement learning variant of the Bayes LOT framework. The stochastic grammar G is parameterized by hyper-parameters 1, and latent expressions el, . .. , e,, are generated i.i.d from G. Each task tj is parameterized by its associated expression e ...................................... 2 5 2-2 Combinators implement variable-free language via the routing behavior of S, B , and C ........ ....................... 28 2-3 The typed combinator S *I, which computes f(x) = x 2 over real values R, represented as a binary tree. Nodes are annotated with their types, i.e. if node n has left child i : Tand right child r : o- then the type of n is m gu(T, 0- --+ r). ............... ............. 29 2-4 The space of all typed combinators represented as an infinitely deep AND /O R tree. ....... ....................... 29 2-5 Generative model for symbolic dimensionality reduction ....... 37 2-6 Learning curves as a function of frontier size. As frontier size is in- creased, curves plateau closer to 100% performance. A baseline search over 150000 expressions only hits 3% of the tasks. .......... 38 2-7 How do different task sets affect learning curves? Learning curves at frontier size of 10000 for different task sets. ... ........... 39 2-8 Learning curves from Boolean function learning Experiment 1. Learn- ing curves for several frontier sizes on Zsampled, a task set of Boolean functions generated by sampled Boolean circuits. .......... 41 2-9 Comparing the frontier before and after learning on distribution over boolean circuits. Each row represents a distribution over Boolean func- tions {f If :{0, i}d -+ {0, 1}, d E 1, 2, 3}. There are 276 such functions and they are arrayed left-to-right, with d = 1 on the left and d = 3 on the right. Vertical bands indicate the fraction of expressions/circuits that are equivalent to the corresponding function; a darker band indi- cates a larger proportion. For clarity, where the proportion is zero, the color is pink. ... ...... ...... ..... ...... ..... 41 9 2-10 Cached expressions learned in the Boolean circuit experiment. Each ex- pressions is shown with its name, if interpretable as a standard Boolean connective, and a schematic circuit showing its representation as a cir- cuit. a) The three primitive logic gates used in the sampling procedure for the data set. b) Two higher-order learned expressions. These are used in c) to define TRUE, FALSE, and XOR. ........ ..... 42 2-11 Learning curves from Boolean function learning Experiment 2. 10 learning curves for frontier size of 2000 on Zaii, a task set consisting of all Boolean functions of cardinality 3 or smaller. Note how most of the trajectories get stuck in a local minimum around 50%. ...... .. 43 2-12 Learning curves as a function of frontier size. Larger frontier sizes help EC solve more tasks. .......... ............ ..... 44 2-13 Example input/output pairs of learned programs. Each induced pro- gram takes as input an arbitrarily large list of observed nodes and produces an undirected graphical model. Labeled/unlabeled nodes cor- respond to observed/latent variables. ...........