On the enumeration of closures and environments with an application to random generation

Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size $n$. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environemnts and closures

On the enumeration of closures and environments with an application to random generation

Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size $n$. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environemnts and closures

Hoogle?: Constants and ?-abstractions in Petri-net-based Synthesis using Symbolic Execution

Type-directed component-based program synthesis is the task of automatically building a function with applications of available components and whose type matches a given goal type. Existing approaches to component-based synthesis, based on classical proof search, cannot deal with large sets of components. Recently, Hoogle+, a component-based synthesizer for Haskell, overcomes this issue by reducing the search problem to a Petri-net reachability problem. However, Hoogle+ cannot synthesize constants nor ?-abstractions, which limits the problems that it can solve. We present Hoogle?, an extension to Hoogle+ that brings constants and ?-abstractions to the search space, in two independent steps. First, we introduce the notion of wildcard component, a component that matches all types. This enables the algorithm to produce incomplete functions, i.e., functions containing occurrences of the wildcard component. Second, we complete those functions, by replacing each occurrence with constants or custom-defined ?-abstractions. We have chosen to find constants by means of an inference algorithm: we present a new unification algorithm based on symbolic execution that uses the input-output examples supplied by the user to compute substitutions for the occurrences of the wildcard. When compared to Hoogle+, Hoogle? can solve more kinds of problems, especially problems that require the generation of constants and ?-abstractions, without performance degradation