Example-directed synthesis: a type-theoretic interpretation

Abstract

Input-output examples have emerged as a practical and user-friendly specification mechanism for program synthesis in many environments. While example-driven tools have demonstrated tangible impact that has inspired adoption in industry, their underlying semantics are less well-understood: what are "examples" and how do they relate to other kinds of specifications? This paper demonstrates that examples can, in general, be interpreted as refinement types. Seen in this light, program synthesis is the task of finding an inhabitant of such a type. This insight provides an immediate semantic interpretation for examples. Moreover, it enables us to exploit decades of research in type theory as well as its correspondence with intuitionistic logic rather than designing ad hoc theoretical frameworks for synthesis from scratch. We put this observation into practice by formalizing synthesis as proof search in a sequent calculus with intersection and union refinements that we prove to be sound with respect to a conventional type system. In addition, we show how to handle negative examples, which arise from user feedback or counterexample-guided loops. This theory serves as the basis for a prototype implementation that extends our core language to support ML-style algebraic data types and structurally inductive functions. Users can also specify synthesis goals using polymorphic refinements and import monomorphic libraries. The prototype serves as a vehicle for empirically evaluating a number of different strategies for resolving the nondeterminism of the sequent calculus---bottom-up theorem-proving, term enumeration with refinement type checking, and combinations of both---the results of which classify, explain, and validate the design choices of existing synthesis systems. It also provides a platform for measuring the practical value of a specification language that combines "examples" with the more general expressiveness of refinements.

Component-based synthesis of table consolidation and transformation tasks from examples

“…Example 3. A data analyst has the following raw data about the position of vehicles for a driving simulator [3]: Here, Table 1 contains the unique identification number for each vehicle (e.g., 10,15), with 0 indicating the absence of a vehicle. The column labeled "frame" in Table 1 measures the time step, and the columns "X1", "X2", "X3" track which vehicle is closer to the driver.…”

“…Of these approaches, IN-SYNTH [14] is type-directed rather than example-directed. MYTH [22] and its successors [10] cast type-and exampledirected synthesis as type inhabitation in a refinement type system. SYNQUID [24] steps further by taking advantage of recent advances in polymorphic refinement types [27,32].…”

Learning higher-order logic programs

“…The L2 system [16] synthesises recursive functional algorithms. The MYTH [36] and MYTH2 [18] systems use type systems to synthesise programs. Frankle et al [18] show how example-based specifications can be turned into type specifications.…”

Synthesizing symmetric lenses

“…Over the past decade, the programming languages community has explored the synthesis of programs from a wide variety of angles. One of the key ideas is typically to narrow the program search space by focusing on a specific domain, and imposing constraints on syntax [Alur et al 2013], typing [Augustsson 2004Feser et al 2015;Frankle et al 2015;Gvero et al 2013;Scherer and Rèmy 2015], or both.…”