Representing bound variables in embedded languages is a challenging problem, often requiring painful trade-offs between expressivity and usability. On the one hand, first-order representations using de Bruijn indices have many nice properties, but quickly become difficult to read and write. On the other hand, higher-order representations can piggy-back on the host language's binders to offer a more ergonomic interface, at a variety of costs depending on the technique. The current state-of-the-art is unembedding, i.e. a translation from the higher-order representation to the first-order and back again to get the best of both worlds. Unfortunately, the fact that this translation is type-safe relies on external metatheoretic arguments, holding unembedding back from its true potential. We solve this problem with a new embedding technique that uses instance resolution to define a context-directed isomorphism between an ergonomic higher-order interface and a first-order representation. Unlike previous techniques, this also applies to embedded languages with modal and substructural (e.g. linear) type systems, making unembedding relevant for modern languages.

Cross-category object perception is one of the essential upstream tasks for generelizable robot object interaction and manipulation. Recently, an increasing number of researchers are focusing on investigating visual Generalizable and Actionable Parts understanding at cross-category level perception. However, these works are built upon the RGB-D or point cloud input, that relies on the depth information capture. Under the circumstances of limited depth camera performance, e.g. transparent or light absorbing material, perception algorithms that do not require depth information are urgently needed. In this paper, we propose DFGAP, a novel depth-free framework for RGB-based GAParts segmentation and pose estimation. Specifically, we independently model the ill-pose problems from the absence of depth for GAPart segmentation and pose estimation, by clearly quantifying the pixel-wise segmentation probability and relative depth. We reduce the uncertainty and benefit learning in these two tasks. The experimental results demonstrate the superior performance and robustness of our DFGAP. Our work provides a new research paradigm in GAParts perception. We believe that our work has the enormous potential to be applied in many areas of embodied AI system.

Large language models (LLMs) show promise in code translation due to their ability to generate idiomatic code. However, a significant limitation when using LLMs for code translation is scalability: existing works have shown a drop in translation success rates for code exceeding around 100 lines. We overcome this limitation by developing a modular approach to translation, where we partition the code into small code fragments which can be translated independently and semantically validated (that is, by checking I/O equivalence). When this approach is applied naively, we discover that LLMs are unreliable when translating features of the source language that do not have a direct mapping to the target language, and that the LLM often gets stuck in repair loops when attempting to fix errors. To address these issues, we introduce two key concepts: (1) feature mapping , which integrates predefined translation rules with LLM-based translation to guide the LLM in navigating subtle language differences and producing semantically accurate code; and (2) type-compatibility , which facilitates localized checks at the function signature level to detect errors early, thereby narrowing the scope of potential repairs. We apply our approach to translating real-world Go codebases to Rust, demonstrating that we can consistently generate reliable Rust translations for projects up to 9,700 lines of code and 780 functions, with an average of 73% of functions successfully validated for I/O equivalence, considerably higher than any existing work.

There has been a recent explosion of research on Large Language Models (LLMs) for software engineering tasks, in particular code generation. However, results from LLMs can be highly unstable; non-deterministically returning very different code for the same prompt. Such non-determinism affects the correctness and consistency of the generated code, undermines developers’ trust in LLMs, and yields low reproducibility in LLM-based papers. Nevertheless, there is no work investigating how serious this non-determinism threat is. To fill this gap, this article conducts an empirical study on the non-determinism of ChatGPT in code generation. We chose to study ChatGPT because it is already highly prevalent in the code generation research literature. We report results from a study of 829 code generation problems across three code generation benchmarks (i.e., CodeContests, APPS and HumanEval) with three aspects of code similarities: semantic similarity, syntactic similarity, and structural similarity. Our results reveal that ChatGPT exhibits a high degree of non-determinism under the default setting: the ratio of coding tasks with zero equal test output across different requests is 75.76%, 51.00% and 47.56% for three different code generation datasets (i.e., CodeContests, APPS and HumanEval), respectively. In addition, we find that setting the temperature to 0 does not guarantee determinism in code generation, although it indeed brings less non-determinism than the default configuration ( temperature \(=\) 1). In order to put LLM-based research on firmer scientific foundations, researchers need to take into account non-determinism in drawing their conclusions.

Abstract Invertible programming languages specify transformations to be run in two directions, such as compression/decompression or encryption/decryption. Two key concepts in invertible programming languages are partial invertibility and local invertibility. Partial invertibility lets invertible code be parameterized by the results of non-invertible code, whereas local invertibility requires all code to be invertible. The former allows for more flexible programming, while the latter has connections to domains such as low-energy computing and quantum computing. We find that existing approaches lack a satisfying treatment of partial invertibility, leaving the connection to local invertibility unclear. In this paper, we identify four core constructs for partially invertible programming, and show how to give them a locally invertible interpretation. We show the expressiveness of the constructs by designing the functional invertible language Kalpis, and show how to give them a locally invertible semantics using the novel arrow combinator language $$\textsc {rrArr}$$ R R A R R —the key idea is viewing partial invertibility as an invertible effect. By formalizing the two systems and giving Kalpis semantics by translation to $$\textsc {rrArr}$$ R R A R R , we reconcile partial and local invertibility, solving an open problem in the field. All formal developments are mechanized in Agda.

Abstract Invertibility is a fundamental concept in computer science, with various manifestations in software development (serializer/deserializer, parser/printer, redo/undo, compressor/decompressor, and so on). Full invertibility necessarily requires bijectivity, but the direct approach of composing bijective functions to develop invertible programs is too restrictive to be useful. In this paper, we take a different approach by focusing on partially invertible functions—functions that become invertible if some of their arguments are fixed. The simplest example of such is addition, which becomes invertible when fixing one of the operands. More involved examples include entropy-based compression methods (e.g., Huffman coding), which carry the occurrence frequency of input symbols (in certain formats such as Huffman tree), and fixing this frequency information makes the compression methods invertible. We develop a language Sparcl for programming such functions in a natural way, where partial invertibility is the norm and bijectivity is a special case, hence gaining significant expressiveness without compromising correctness. The challenge in designing such a language is to allow ordinary programming (the “partially” part) to interact with the invertible part freely, and yet guarantee invertibility by construction. The language Sparcl is linear-typed and has a type constructor to distinguish data that are subject to invertible computation and those that are not. We present the syntax, type system, and semantics of the language and prove that Sparcl correctly guarantees invertibility for its programs. We demonstrate the expressiveness of Sparcl with examples including tree rebuilding from preorder and inorder traversals, Huffman coding, arithmetic coding, and LZ77 compression.

Abstract Dubbed a safer C, Rust is a modern programming language that combines memory safety and low-level control. This interesting combination has made Rust very popular among developers and there is a growing trend of migrating legacy codebases (very often in C) to Rust. In this paper, we present a C to Rust translation approach centred around static ownership analysis. We design a suite of analyses that infer ownership models of C pointers and automatically translate the pointers into safe Rust equivalents. The resulting tool, Crown, scales to real-world codebases (half a million lines of code in less than 10 s) and achieves a high conversion rate.

Inference algorithms for probabilistic programming are complex imperative programs with many moving parts. Efficient inference often requires customising an algorithm to a particular probabilistic model or problem, sometimes called inference programming. Most inference frameworks are implemented in languages that lack a disciplined approach to side effects, which can result in monolithic implementations where the structure of the algorithms is obscured and inference programming is hard. Functional programming with typed effects offers a more structured and modular foundation for programmable inference, with monad transformers being the primary structuring mechanism explored to date.

Abstract Bidirectional transformations (BXs) are a mechanism for maintaining consistency between multiple representations of related data. The lens framework, which usually constructs BXs from lens combinators, has become the mainstream approach to BX programming because of its modularity and correctness by construction. However, the involved bidirectional behaviors of lenses make the equational reasoning and optimization of them much harder than unidirectional programs. We propose a novel approach to deriving efficient lenses from clear specifications via program calculation, a correct-by-construction approach to reasoning about functional programs by algebraic laws. To support bidirectional program calculation, we propose contract lenses , which extend conventional lenses with a pair of predicates to enable safe and modular composition of partial lenses. We define several contract-lens combinators capturing common computation patterns including $\textit{fold}, \textit{filter},\textit{map}$ , and $\textit{scan}$ , and develop several bidirectional calculation laws to reason about and optimize contract lenses. We demonstrate the effectiveness of our new calculation framework based on contract lenses with nontrivial examples.

Expert users of property-based testing often labor to craft random generators that encode detailed knowledge about what it means for a test input to be valid and interesting. Fortunately, the fruits of this labor can also be put to other uses. In the bidirectional programming literature, for example, generators have been repurposed as validity checkers, while Python's Hypothesis library uses the same structures for shrinking and mutating test inputs. To unify and generalize these uses and many others, we propose reflective generators, a new foundation for random data generators that can "reflect" on an input value to calculate the random choices that could have been made to produce it. Reflective generators combine ideas from two existing abstractions: free generators and partial monadic profunctors. They can be used to implement and enhance the aforementioned shrinking and mutation algorithms, generalizing them to work for any values that can be produced by the generator, not just ones for which a trace of the generator's execution is available. Beyond shrinking and mutation, reflective generators generalize a published algorithm for example-based generation, and they can also be used as checkers, partial value completers, and other kinds of test data producers.

Embedding is a language development technique that implements the object language as a library in a host language. There are many advantages of the approach, including being lightweight and the ability to inherit features of the host language. A notable example is the technique of HOAS, which makes crucial use of higher-order functions to represent abstract syntax trees with binders. Despite its popularity, HOAS has its limitations. We observe that HOAS struggles with semantic domains that cannot be naturally expressed as functions, particularly when open expressions are involved. Prominent examples of this include incremental computation and reversible/bidirectional languages. In this paper, we pin-point the challenge faced by HOAS as a mismatch between the semantic domain of host and object language functions, and propose a solution. The solution is based on the technique of unembedding , which converts from the finally-tagless representation to de Bruijn-indexed terms with strong correctness guarantees. We show that this approach is able to extend the applicability of HOAS while preserving its elegance. We provide a generic strategy for Embedding by Unembedding, and then demonstrate its effectiveness with two substantial case studies in the domains of incremental computation and bidirectional transformations. The resulting embedded implementations are comparable in features to the state-of-the-art language implementations in the respective areas.

Test case prioritization (TCP) has been widely studied in regression testing, which aims to optimize the execution order of test cases so as to detect more faults earlier. TCP has been divided into white-box test case prioritization (WTCP) and black-box test case prioritization (BTCP) . WTCP can achieve better prioritization effectiveness by utilizing source code information, but is not applicable in many practical scenarios (where source code is unavailable, e.g., outsourced testing). BTCP has the benefit of not relying on source code information, but tends to be less effective than WTCP. That is, both WTCP and BTCP suffer from limitations in the practical use. To improve the practicability of TCP, we aim to explore better BTCP, significantly bridging the effectiveness gap between BTCP and WTCP. In this work, instead of statically analyzing test cases themselves in existing BTCP techniques, we conduct the first study to explore whether this goal can be achieved via log analysis. Specifically, we propose to mine test logs produced during test execution to more sufficiently reflect test behaviors, and design a new BTCP framework (called LogTCP), including log pre-processing, log representation, and test case prioritization components. Based on the LogTCP framework, we instantiate seven log-based BTCP techniques by combining different log representation strategies with different prioritization strategies. We conduct an empirical study to explore the effectiveness of LogTCP. Based on 10 diverse open-source Java projects from GitHub, we compared LogTCP with three representative BTCP techniques and four representative WTCP techniques. Our results show that all of our LogTCP techniques largely perform better than all the BTCP techniques in average fault detection, to the extent that they become competitive to the WTCP techniques. That demonstrates the great potential of logs in practical TCP.

Fault localization is to identify faulty program elements. Among the large number of fault localization approaches in the literature, coverage-based fault localization, especially spectrum-based fault localization has been intensively studied due to its effectiveness and lightweightness. Despite the rich literature, almost all existing fault localization approaches and studies are conducted on imperative programming languages such as Java and C, leaving a gap in other programming paradigms. In this paper, we aim to study fault localization approaches for the functional programming paradigm, using Haskell language as a representation. We build up the first dataset on real Haskell projects including both real and seeded faults, which enables the research of fault localization for functional languages. With this dataset, we explore fault localization techniques for Haskell. In particular, as typically for SBFL approaches, we study methods for coverage collection as well as formulae for suspiciousness scores computation, and carefully adapt these two components to Haskell considering the language features and characteristics, resulting in a series of adaption approaches and a learning-based approach, which are evaluated on the dataset to demonstrate the promises of the direction.

Abstract We propose a technique for synthesizing bidirectional programs from the corresponding unidirectional code plus input/output examples. The core ideas are: (1) constructing a sketch using the given unidirectional program as a specification, and (2) filling the sketch in a modular fashion by exploiting the properties of bidirectional programs. These ideas are enabled by our choice of programming language, HOBiT, which is specifically designed to maintain the unidirectional program structure in bidirectional programming, and keep the parts that control bidirectional behavior modular. To evaluate our approach, we implemented it in a tool called Synbit and used it to generate bidirectional programs for intricate microbenchmarks, as well as for a few larger, more realistic problems. We also compared Synbit to a state-of-the-art unidirectional synthesis tool on the task of synthesizing backward computations. This is an extended version of the paper “Synbit: Synthesizing Bidirectional Programs using Unidirectional Sketches”, published at OOPSLA 2021. In addition to the OOPSLA’21 paper, this journal will contain additional formalization and detailed examples.

Probabilistic programming languages (PPLs) allow programmers to construct statistical models and then simulate data or perform inference over them. Many PPLs restrict models to a particular instance of simulation or inference, limiting their reusability. In other PPLs, models are not readily composable. Using Haskell as the host language, we present an embedded domain specific language based on algebraic effects, where probabilistic models are modular, first-class, and reusable for both simulation and inference. We also demonstrate how simulation and inference can be expressed naturally as composable program transformations using algebraic effect handlers.

We present new language-based dynamic analysis techniques for linking visualisations and other structured outputs to data in a fine-grained way, allowing users to explore how data attributes and visual or other output elements are related by selecting (focusing on) substructures of interest. Our approach builds on bidirectional program slicing techiques based on Galois connections, which provide desirable round-tripping properties. Unlike the prior work, our approach allows selections to be negated, equipping the bidirectional analysis with a De Morgan dual which can be used to link different outputs generated from the same input. This offers a principled language-based foundation for a popular view coordination feature called brushing and linking where selections in one chart automatically select corresponding elements in another related chart.

Multi-stage programming using typed code quotation is an established technique for writing optimizing code generators with strong type-safety guarantees. Unfortunately, quotation in Haskell interacts poorly with type classes, making it difficult to write robust multi-stage programs. We study this unsound interaction and propose a resolution, staged type class constraints, which we formalize in a source calculus λ ⇒ that elaborates into an explicit core calculus F . We show type soundness of both calculi, establishing that well-typed, well-staged source programs always elaborate to well-typed, well-staged core programs, and prove beta and eta rules for code quotations. Our design allows programmers to incorporate type classes into multi-stage programs with confidence. Although motivated by Haskell, it is also suitable as a foundation for other languages that support both overloading and quotation.

We propose a technique for synthesizing bidirectional programs from the corresponding unidirectional code plus a few input/output examples. The core ideas are: (1) constructing a sketch using the given unidirectional program as a specification, and (2) filling the sketch in a modular fashion by exploiting the properties of bidirectional programs. These ideas are enabled by our choice of programming language, HOBiT, which is specifically designed to maintain the unidirectional program structure in bidirectional programming, and keep the parts that control bidirectional behavior modular. To evaluate our approach, we implemented it in a tool called Synbit and used it to generate bidirectional programs for intricate microbenchmarks, as well as for a few larger, more realistic problems. We also compared Synbit to a state-of-the-art unidirectional synthesis tool on the task of synthesizing backward computations.

This paper presents a large-scale study that investigates the bug resolution characteristics among popular Github projects written in different programming languages. We explore correlations but, of course, we cannot infer causation. Specifically, we analyse bug resolution data from approximately 70 million Source Line of Code, drawn from 3 million commits to 600 GitHub projects, primarily written in 10 programming languages. We find notable variations in apparent bug resolution time and patch (fix) size. While interpretation of results from such large-scale empirical studies is inherently difficult, we believe that the differences in medians are sufficiently large to warrant further investigation, replication, re-analysis and follow up research. For example, in our corpus, the median apparent bug resolution time (elapsed time from raise to resolve) for Ruby was 4X that for Go and 2.5X for Java. We also found that patches tend to touch more files for the corpus of strongly typed and for statically typed programs. However, we also found evidence for alowerelapsed resolution time for bug resolution committed to projects constructed from statically typed languages. These findings, if replicated in subsequent follow on studies, may shed further empirical light on the debate about the importance of static typing.

Invertibility is a fundamental concept in computer science, with various manifestations in software development (serializer/deserializer, parser/printer, redo/undo, compressor/decompressor, and so on). Full invertibility necessarily requires bijectivity, but the direct approach of composing bijective functions to develop invertible programs is too restrictive to be useful. In this paper, we take a different approach by focusing on partially-invertible functions—functions that become invertible if some of their arguments are fixed. The simplest example of such is addition, which becomes invertible when fixing one of the operands. More involved examples include entropy-based compression methods (e.g., Huffman coding), which carry the occurrence frequency of input symbols (in certain formats such as Huffman tree), and fixing this frequency information makes the compression methods invertible. We develop a language Sparcl for programming such functions in a natural way, where partial-invertibility is the norm and bijectivity is a special case, hence gaining significant expressiveness without compromising correctness. The challenge in designing such a language is to allow ordinary programming (the “partially” part) to interact with the invertible part freely, and yet guarantee invertibility by construction. The language Sparcl is linear-typed, and has a type constructor to distinguish data that are subject to invertible computation and those that are not. We present the syntax, type system, and semantics of the language, and prove that Sparcl correctly guarantees invertibility for its programs. We demonstrate the expressiveness of Sparcl with examples including tree rebuilding from preorder and inorder traversals and Huffman coding.

Software frequently converts data from one representation to another and vice versa. Naïvely specifying both conversion directions separately is error prone and introduces conceptual duplication. Instead, bidirectional programming techniques allow programs to be written which can be interpreted in both directions. However, these techniques often employ unfamiliar programming idioms via restricted, specialised combinator libraries. Instead, we introduce a framework for composing bidirectional programs monadically, enabling bidirectional programming with familiar abstractions in functional languages such as Haskell. We demonstrate the generality of our approach applied to parsers/printers, lenses, and generators/predicates. We show how to leverage compositionality and equational reasoning for the verification of round-tripping properties for such monadic bidirectional programs.

Code coverage is the most widely adopted criteria for measuring test effectiveness in software quality assurance. The performance of coverage criteria (in indicating test suites' effectiveness) has been widely studied in prior work. Most of the studies use randomly constructed pseudo test suites to facilitate data collection for correlation analysis, yet no previous work has systematically studied whether pseudo test suites would lead to inflated correlation results. This paper focuses on the potentially wide-spread threat with a study over 123 real-world Java projects. Following the typical experimental process of studying coverage criteria, we investigate the correlation between statement/assertion coverage and mutation score using both pseudo and original test suites. Except for direct correlation analysis, we control the number of assertions and the test suite size to conduct partial correlation analysis. The results reveal that 1) the correlation (between coverage criteria and mutation score) derived from pseudo test suites is much higher than from original test suites (from 0.21 to 0.39 higher in Kendall value); 2) contrary to previously reported, statement coverage has a stronger correlation with mutation score than assertion coverage.

We propose HOBiT, a higher-order bidirectional programming language, in which users can write bidirectional programs in the familiar style of conventional functional programming, while enjoying the full expressiveness of lenses. A bidirectional transformation, or a lens, is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws—a pattern that is found in databases, model-driven development, compiler construction, and so on. The most common way of programming lenses is with lens combinators, which are lens-to-lens functions that compose simpler lenses to form more complex ones. Lens combinators preserve the bidirectionality of lenses and are expressive; but they compel programmers to a specialised point-free style—i.e., no naming of intermediate computation results—limiting the scalability of bidirectional programming. To address this issue, we propose a new bidirectional programming language HOBiT, in which lenses are represented as standard functions, and combinators are mapped to language constructs with binders. This design transforms bidirectional programming, enabling programmers to write bidirectional programs in a flexible functional style and at the same time access the full expressiveness of lenses. We formally define the syntax, type system, and the semantics of the language, and then show that programs in HOBiT satisfy bidirectionality. Additionally, we demonstrate HOBiT ’s programmability with examples.

Abstract A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws. One way to reduce the development and maintenance effort of bidirectional transformations is to have specialized languages in which the resulting programs are bidirectional by construction—giving rise to the paradigm of bidirectional programming. In this paper, we develop a framework for applicative-style and higher-order bidirectional programming, in which we can write bidirectional transformations as unidirectional programs in standard functional languages, opening up access to the bundle of language features previously only available to conventional unidirectional languages. Our framework essentially bridges two very different approaches of bidirectional programming, namely the lens framework and Voigtländer's semantic bidirectionalization, creating a new programming style that is able to obtain benefits from both.

Many tools directly change programs, such as bug-fixing tools, program migration tools, etc. We call them program-modification tools. On the other hand, many programming languages use the C preprocessor, such as C, C++, and Objective-C. Because of the complexity of preprocessors, many program-modification tools either fail to produce sound results under the presence of preprocessor directives, or give up completely and deal only with preprocessed code.

The correlation between test coverage and test effectiveness is important to justify the use of coverage in practice. Existing results on imperative programs mostly show that test coverage predicates effectiveness. However, since functional programs are usually structurally different from imperative ones, it is unclear whether the same result may be derived and coverage can be used as a prediction of effectiveness on functional programs. In this paper we report the first empirical study on the correlation between test coverage and test effectiveness on functional programs. We consider four types of coverage: as input coverages, statement/branch coverage and expression coverage, and as oracle coverages, count of assertions and checked coverage. We also consider two types of effectiveness: raw effectiveness and normalized effectiveness. Our results are twofold. (1) In general the findings on imperative programs still hold on functional programs, warranting the use of coverage in practice. (2) On specific coverage criteria, the results may be unexpected or different from the imperative ones, calling for further studies on functional programs.

A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws. One way to reduce the development and maintenance effort of bidirectional transformations is to have specialized languages in which the resulting programs are bidirectional by construction---giving rise to the paradigm of bidirectional programming. In this paper, we develop a framework for applicative-style and higher-order bidirectional programming, in which we can write bidirectional transformations as unidirectional programs in standard functional languages, opening up access to the bundle of language features previously only available to conventional unidirectional languages. Our framework essentially bridges two very different approaches of bidirectional programming, namely the lens framework and Voigtländer's semantic bidirectionalization, creating a new programming style that is able to bag benefits from both.

A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws. Over the years, a lot of effort has been made to offer better language support for programming such transformations, essentially allowing the programmers to construct one mapping of the pair and have the other automatically generated.

A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws. Over the years, a lot of effort has been made to offer better language support for programming such transformations. In particular, a technique known as bidirectionalization is able to analyze and transform unidirectional programs written in general purpose languages, and "bidirectionalize" them.

Abstract Matsuda et al . (Matsuda, K., Hu, Z., Nakano, K., Hamana, M. & Takeichi, M. (2007) Bidirectionalization transformation based on automatic derivation of view complement functions. In Proceedings of the International Conference on Functional Programming . ACM Press, pp. 47–58) and Voigtländer (Voigtländer, J. (2009) Bidirectionalization for free! In Proceedings of Principles of Programming Languages . ACM Press, pp. 165–176) have introduced two techniques that given a source-to-view function provide an update propagation function mapping an original source and an updated view back to an updated source, subject to standard consistency conditions. Previously, we developed a synthesis of the two techniques, based on a separation of shape and content aspects (Voigtländer, J., Hu, Z., Matsuda, K. & Wang, M. (2010) Combining syntactic and semantic bidirectionalization. In Proceedings of the International Conference on Functional Programming . ACM Press, pp. 181–192). Here we carry that idea further, reworking the technique of Voigtländer such that any shape bidirectionalizer (based on the work of Matsuda et al . (2007) or not) can be used as a plug-in, to good effect. We also provide a data-type-generic account, enabling wider reuse, including the use of pluggable bidirectionalization itself as a plug-in.

In mathematics, an enumeration of a set S is a bijective function from (an initial segment of) the natural numbers to S. We define "functional enumerations" as efficiently computable such bijections. This paper describes a theory of functional enumeration and provides an algebra of enumerations closed under sums, products, guarded recursion and bijections. We partition each enumerated set into numbered, finite subsets.

A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly. The key to handling large data objects that are subject to relatively small modifications is to process the updates incrementally. Incrementality has been explored in the semi-structured settings of relational databases and graph transformations; this flexibility in structure makes it relatively easy to divide the data into separate parts that can be transformed and updated independently. The same is not true if the data is to be encoded with more general-purpose algebraic datatypes, with transformations defined as functions: dividing data into well-typed separate parts is tricky, and recursions typically create interdependencies. In this paper, we study transformations that support incremental updates, and devise a constructive process to achieve this incrementality.

Matsuda et al. [2007, ICFP] and Voigtländer [2009, POPL] introduced two techniques that given a source-to-view function provide an update propagation function mapping an original source and an updated view back to an updated source, subject to standard consistency conditions. Being fundamentally different in approach, both techniques have their respective strengths and weaknesses. Here we develop a synthesis of the two techniques to good effect. On the intersection of their applicability domains we achieve more than what a simple union of applying the techniques side by side delivers.

The VISITOR design pattern shows how to separate the structure of an object hierarchy from the behaviour of traversals over that hierarchy. The pattern is very flexible; this very flexibility makes it difficult to capture the pattern as anything more formal than prose, pictures and prototypes.

Aspect-oriented programming (AOP) has been shown to be a useful model for software development. Special care must be taken when we try to adapt AOP to strongly typed functional languages which come with features like a type inference mechanism, polymorphic types, higher-order functions and type-scoped pointcuts. Our main contribution lies in a seamless integration of these two paradigms through a static weaving process which deals with around advices with type-scoped pointcuts in the presence of higher-order functions. We give a source-level type inference system for a higher-order, polymorphic language coupled with type-scoped pointcuts. The type system ensures that base programs are oblivious to the type of around advices. We present a type-directed translation scheme which resolves all advice applications at static time. The translation removes advice declarations from source programs and produces translated code which is typable in the Hindley-Milner system.