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Synthesis from examples enables non-expert users to generate programs byspecifying examples of their behavior. A domain-specific form of such synthesishas been recently deployed in a widely used spreadsheet software product. Inthis paper we contribute to foundations of such techniques and present acomplete algorithm for synthesis of a class of recursive functions defined bystructural recursion over a given algebraic data type definition. The functionswe consider map an algebraic data type to a string; they are useful for, e.g.,pretty printing and serialization of programs and data. We formalize ourproblem as learning deterministic sequential top-down tree-to-stringtransducers with a single state.  The first problem we consider is learning a tree-to-string transducer fromany set of input/output examples provided by the user. We show that thisproblem is NP-complete in general, but can be solved in polynomial time under a(practically useful) closure condition that each subtree of a tree in theinput/output example set is also part of the input/output examples.  Because coming up with relevant input/output examples may be difficult forthe user while creating hard constraint problems for the synthesizer, we alsostudy a more automated active learning scenario in which the algorithm choosesthe inputs for which the user provides the outputs. Our algorithm asks aworst-case linear number of queries as a function of the size of the algebraicdata type definition to determine a unique transducer.

Polynomial-Time Proactive Synthesis of Tree-to-String Functions from Examples