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Programming-by-example (PBE) is a synthesis paradigm that allows users to generate functions by simply providing input-output examples. While a promising interaction paradigm, synthesis is still too slow for realtime interaction and more widespread adoption. Existing approaches to PBE synthesis have used automated reasoning tools, such as SMT solvers, as well as works applying machine learning techniques. At its core, the automated reasoning approach relies on highly domain specific knowledge of programming languages. On the other hand, the machine learning approaches utilize the fact that when working with program code, it is possible to generate arbitrarily large training datasets. In this work, we propose a system for using machine learning in tandem with automated reasoning techniques to solve Syntax Guided Synthesis (SyGuS) style PBE problems. By preprocessing SyGuS PBE problems with a neural network, we can use a data driven approach to reduce the size of the search space, then allow automated reasoning-based solvers to more quickly find a solution analytically. Our system is able to run atop existing SyGuS PBE synthesis tools, decreasing the runtime of the winner of the 2019 SyGuS Competition for the PBE Strings track by 47.65% to outperform all of the competing tools.

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We present counterfactual symbolic execution, a new approach that produces counterexamples that localize the causes of failure of static verification. First, we develop a notion of symbolic weak head normal form and use it to define lazy symbolic execution reduction rules for non-strict languages like Haskell. Second, we introduce counterfactual branching, a new method to identify places where verification fails due to imprecise specifications (as opposed to incorrect code). Third, we show how to use counterfactual symbolic execution to localize refinement type errors, by translating refinement types into assertions. We implement our approach in a new Haskell symbolic execution engine, G2, and evaluate it on a corpus of 7550 errors gathered from users of the LiquidHaskell refinement type system. We show that for 97.7% of these errors, G2 is able to quickly find counterexamples that show how the code or specifications must be fixed to enable verification. CCS Concepts • Theory of computation → Abstraction; • Computing methodologies → Symbolic and algebraic manipulation.

We use recent work of Jonah Blasiak (2012) to prove a stability result for the coefficients in the Kronecker product of two Schur functions: one indexed by a hook partition and one indexed by a rectangle partition. We also give bounds for the size of the partition starting with which the Kronecker coefficients are stable. Moreover, we show that once the bound is reached, no new Schur functions appear in the decomposition of Kronecker product, thus allowing one to recover the decomposition from the smallest case in which the stability holds.