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Scalable Synthesis Brandon Lucia and Todd Mytkowicz Microsoft Research

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Synthesizing Circuits Inference problem is undecidable in general – hard problems to solve! Can we leverage existing work to make this scale? There exists some parameters, w, such that all inputs x implement a specification

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Neural Networks or… F T F T FF F FT T TF F TT T

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Neural Networks 2 layer neural network can approximate any continuous function! or… FF F FT T TF T TT F F T F T

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Learning Abstractions with ML First layer learns low level features Each subsequent layer learns higher level features Unsupervised training, layer by layer

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Duality of Synthesis and ML Specification is implicit in input/output pairs Machine Learning Synthesis Specification is vector of logical formula

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Synthesizing Sudoku Recognizer … … … A0A1A2 B0B1B2 C0C1C2 K0 If A0 is 0 then no other cell in A0’s column 0 no other cell in A0’s row is 0 no other cell in A0’s unit is 0 If A0 is not 0 then one cell in A0’s column must be 0 one cell in A0’s row must be 0 one cell in A0’s unit must be 0

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Learning Abstractions in Sudoku First layer learns local implications Each subsequent level combines prior levels. Learns “factorization” of potentially exponential specification! Unsupervised training, layer by layer

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Future directions & Questions Flesh out duality into formal details Duality goes both ways: can we help ML methods with our understanding of synthesis / formal verification? CEGIS: Program = Data Structure + Algorithm Learn structure (depth) and algorithm (connectivity) Approximation / Probabilistic need not mean incorrect But it may help scale inference

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