A Saturation-Based Unification Algorithm for Higher-Order Rational Patterns

Higher-order unification has been shown to be undecidable. Miller discovered the pattern fragment and subsequently showed that higher-order pattern unification is decidable and has most general unifiers. We extend the algorithm to higher-order rational terms (a.k.a. regular B\"{o}hm trees, a form of cyclic $\lambda$-terms) and show that pattern unification on higher-order rational terms is decidable and has most general unifiers. We prove the soundness and completeness of the algorithm

Complete Completion using Types and Weights

Developing modern software typically involves composing functionality from existing libraries. This task is difficult because libraries may expose many methods to the developer. To help developers in such scenarios, we present a technique that synthesizes and suggests valid expressions of a given type at a given program point. As the basis of our technique we use type inhabitation for lambda calculus terms in long normal form. We introduce a succinct representation for type judgements that merges types into equivalence classes to reduce the search space, then reconstructs any desired number of solutions on demand. Furthermore, we introduce a method to rank solutions based on weights derived from a corpus of code. We implemented the algorithm and deployed it as a plugin for the Eclipse IDE for Scala. We show that the techniques we incorporated greatly increase the effectiveness of the approach. Our evaluation benchmarks are code examples from programming practice; we make them available for future comparisons

Efficient intuitionistic theorem proving with the polarized inverse method

Abstract. The inverse method is a generic proof search procedure applicable to non-classical logics satisfying cut elimination and the subformula property. In this paper we describe a general architecture and several high-level optimizations that enable its efficient implementation. Some of these rely on logic-specific properties, such as polarization and focusing, which have been shown to hold in a wide range of non-classical logics. Others, such as rule subsumption and recursive backward subsumption apply in general. We empirically evaluate our techniques on first-order intuitionistic logic with our implementation Imogen and demonstrate a substantial improvement over all other existing intuitionistic theorem provers on problems from the ILTP problem library.

Search Techniques for Code Generation

This dissertation explores techniques that synthesize and generate program fragments and test inputs. The main goal of these techniques is to improve and support automation in program synthesis and test input generation. This is important because performing those processes manually is often tedious, time consuming and error prone. The main challenge that these techniques face is exploring the search space in efficient and scalable ways. In the first part of the dissertation, we present tools InSynth and PolySynth that interactively synthesize code fragments. They take as input a partial program and automatically extract type information, the desired type, and set of visible declarations. They use this input to synthesize ranked list of expressions with the desired type. Finally, they present the expressions to a developer in similar manner to code completions in modern IDEs. InSynth is the first tool that uses a complete algorithm to generate expressions with first class functions and higher order functions. We present the theoretical foundation of the InSynth problem, that is based on type inhabitation, and the type-based backward search algorithm that solves it. PolySynth uses type-driven, resolution based algorithm that considers polymorphic types (generics) to generate expressions. Furthermore, the uniqueness of both tools comes from the fact that their algorithms operate using corpus statistics. The statistics are used to steer the algorithms and the search space exploration towards the most relevant solutions. In the second part of the dissertation we present the tool anyCode that uses natural language input to synthesize expressions. As input it accepts English words or Java program language constructs. This allows a developer to encode her intuition about the desired expression using words or the expression that approximates the desired structure. Thanks to this flexibility, anyCode can also repair broken expressions. It uses a pipeline of natural language and related-word tools to analyze the input. This helps anyCode to identify the set of the most relevant components and to reduce the size of search space. To further reduce the size of search space and to create the most relevant expressions, anyCode uses two statistical models: unigram and probabilistic context free grammar. Finally, in the last part of the dissertation we present UDITA, a Java-like language with support for non-determinism, which allows a user to describe test generation programs. Test generation programs are run on a top of Java PathFinder (JPF), a popular explicit-state model checker, that has a built-in backtracking mechanism and supports non-determinism. Using UDITA programs, JPF generates test inputs. The first benefit of UDITA is that non-determinism empowers a user to describe many test inputs as easily as describing a single test input. The second benefit is that it gives a user more flexibility allowing her to describe test generation programs by arbitrarily combining filters and generators. UDITA reduces the size of search space using an algorithm that reduces the number of generated complex isomorphic structures and that delays non-deterministic choices