Program Synthesis from Polymorphic Refinement Types (1510.08419v3)

Abstract: We present a method for synthesizing recursive functions that provably satisfy a given specification in the form of a polymorphic refinement type. We observe that such specifications are particularly suitable for program synthesis for two reasons. First, they offer a unique combination of expressive power and decidability, which enables automatic verification---and hence synthesis---of nontrivial programs. Second, a type-based specification for a program can often be effectively decomposed into independent specifications for its components, causing the synthesizer to consider fewer component combinations and leading to a combinatorial reduction in the size of the search space. At the core of our synthesis procedure is a new algorithm for refinement type checking, which supports specification decomposition. We have evaluated our prototype implementation on a large set of synthesis problems and found that it exceeds the state of the art in terms of both scalability and usability. The tool was able to synthesize more complex programs than those reported in prior work (several sorting algorithms and operations on balanced search trees), as well as most of the benchmarks tackled by existing synthesizers, often starting from a more concise and intuitive user input.

An Evaluation of Program Synthesis from Polymorphic Refinement Types

The paper "Program Synthesis from Polymorphic Refinement Types" by Polikarpova, Kuraj, and Solar-Lezama presents a method for synthesizing recursive functions that satisfy specifications denoted by polymorphic refinement types. Refinement types provide the expressive power necessary to handle complex program specifications while maintaining a level of decidability that enables automatic synthesis and verification. The paper introduces a framework for efficiently managing these specifications through a new refinement type checking algorithm, which underpins the synthesis process.

Contributions and Results

The authors propose leveraging refinement types to decompose type-based specifications into independent component specifications. This decomposition significantly reduces the search space during synthesis as fewer combinations need to be considered. This approach is notably advantageous in synthesizing recursive functions, which often represent complex problems in program synthesis.

The core synthesis procedure involves solving subtyping constraints expressed as Horn clauses. The authors describe a new approach, termed "local liquid type checking," which merges top-down specification-driven synthesis with bottom-up constraint solving. This method efficiently leverages refinement types using a bidirectional type checking mechanism to guide synthesis.

The effectiveness of the proposed method is demonstrated through the implementation of a prototype synthesizer, Synquid. Evaluation of Synquid on a varied set of benchmarks shows that it exceeds existing tools in terms of both scalability and usability. The synthesizer proves capable of handling intricate synthesis tasks, such as generating sorting algorithms and operations on balanced search trees, with more concise user input compared to previous approaches.

Technical Insights

The supporting technical innovations include enhancements in type checking and constraint solving:

1. Refinement Type Checking: The paper discusses local liquid type checking that supports incremental unification of type variables, allowing for the early detection of type errors. This approach is vital as it addresses issues associated with previous paradigms that relied heavily on Hindley-Milner style type inference, which lacked support for modularity.
2. Condition Abduction: The refinement of branching terms relies on condition abduction, which utilizes liquid abduction to efficiently determine path conditions necessary for the correctness of synthesized programs.
3. MUSFix Solver: To handle the inherent complexity of greatest-fixpoint computations in constraint solving, the authors introduce the MUSFix algorithm, inspired by minimal unsatisfiable subset enumeration. This solver ensures practical performance, especially in the context of abduction during synthesis.

Implications

The paper's findings suggest significant implications for both the theory and practical application of program synthesis in functional programming. By refining and consolidating modular type systems with efficient synthesis algorithms, the framework enables the automatic generation of complex programs with provable correctness guarantees.

The theoretical implications extend into improved type system design, providing a clear route for integrating parametric polymorphism with refinement types in synthesis tasks. Practically, the framework provides a scalable toolset, potentially expanding into domains like software verification and automated programming where correctness is critical.

Future Directions

The authors suggest future exploration into the integration of richer logical structures, such as intersection types and dependent types. There is also an addressed need for optimizing synthesis strategies to further minimize enumeration and improve synthesis times. The proposed abstraction boundaries in synthesis present opportunities for further refinement, potentially leading to enhanced automation in inferring auxiliary function specifications.

Overall, the paper lays a groundwork that extends program synthesis from simple examples to complex recursive tasks, with broad applicability and potential for further advancements in the field of automated software development.