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

## TL;DR

This paper presents a polynomial-time algorithm for synthesizing recursive tree-to-string functions from examples, with applications in pretty printing and serialization, addressing both user-provided and actively queried input/output data.

## Contribution

It introduces a complete algorithm for learning tree-to-string transducers from examples, including an active learning approach that minimizes user input queries.

## Key findings

- Polynomial-time solution under a closure condition
- NP-completeness in the general case
- Active learning algorithm with linear query complexity

## Abstract

Synthesis from examples enables non-expert users to generate programs by specifying examples of their behavior. A domain-specific form of such synthesis has been recently deployed in a widely used spreadsheet software product. In this paper we contribute to foundations of such techniques and present a complete algorithm for synthesis of a class of recursive functions defined by structural recursion over a given algebraic data type definition. The functions we 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 our problem as learning deterministic sequential top-down tree-to-string transducers with a single state. The first problem we consider is learning a tree-to-string transducer from any set of input/output examples provided by the user. We show that this problem 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 the input/output example set is also part of the input/output examples. Because coming up with relevant input/output examples may be difficult for the user while creating hard constraint problems for the synthesizer, we also study a more automated active learning scenario in which the algorithm chooses the inputs for which the user provides the outputs. Our algorithm asks a worst-case linear number of queries as a function of the size of the algebraic data type definition to determine a unique transducer.