Comparing the expressiveness of transducers and simply-typed linear lambda-calculi

比较不同的计算模型和形式主义的表达能力是理论计算机科学的许多领域中的一个重要主题，包括复杂性理论，描述复杂性和自动机理论。这个项目的主要目的是利用来自自动机理论和线性逻辑语义学的技术，对自动机、自动演算和逻辑解释自然定义的函数类进行比较。1996年，自动演算和自动机之间的惊人联系被发现：使用Church编码的从字符串到布尔值的简单类型的函数准确地识别了正则语言的类。这种联系随后被线性逻辑和自动机的专家进一步完善和利用，以解决与高阶模型检查和在无限数据上运行的自动机相关的问题。在此之后，近年来有一种努力，系统地比较线性代数演算和串到串转换中Church编码的表达能力，揭示了一些非平凡的连接，包括正则转换的特征。该技术涉及的语义评估，分析的范式在Alzada演算和非平凡的自动机理论的结果的混合。为了了解所采用的目标和方法，可以浏览以下内容：-概述该计划的摘要：https://cs-web.swan.ac.uk/~cpradic/smp-abstract.pdf-关于该主题的博士论文：https://nguyentito.eu/thesis.pdf这个项目将是该努力的延续。有一些具体问题需要解决，可以作为在这种环境下工作的良好开端，其中包括：- 比较非可交换线性可达演算和一阶转换器的表达性-设计最小类型编程语言，以Bojanczyk关于多正则函数的工作的精神，捕获在具有Church编码的线性可达演算中可实现的内容，但是对于无比较的多正则函数--对无限结构上的换能器和可使用共归纳数据集的Church编码定义的函数进行类似的比较--如果我们允许经典线性逻辑的全部力量而不是直觉主义的片段，检查在表达性上是否有差异，这可以通过对同一领域中更具挑战性的问题或对所涉及的某个领域的更广泛关注的调查来跟进（例如涉及定义树转换的良好分类的研究或线性逻辑中更专业的主题，例如弱指数或线性类型上的量化）虽然有数学逻辑或理论计算机科学的背景是必要的，但成功的申请人在开始之前肯定不会熟悉上述所有工具。

Comparing the expressiveness of different computation models and formalisms is an important theme in many fields of theoretical computer science including complexity theory, descriptive complexity and automata theory. The main aims of this project would be to carry out such comparisons across classes of functions naturally defined by automata, lambda-calculi and logical interpretations, using techniques coming from automata theory and the semantics of linear logic.In 1996, a striking connection between lambda-calculus and automata was uncovered: the simply-typed functions from strings to booleans using Church encodings recognize exactly the class of regular languages. This sort of connection was then refined and further exploited by specialists of linear logic and automata to tackle problems related to higher-order model checking and automata running over infinite data.Following this, there was an effort in recent years to systematically compare the expressive power of Church encodings in linear lambda-calculi and string-to-string transductions, unveiling some non-trivial connections including a characterization of regular transductions. The techniques involve a mix of semantic evaluation, analyses of the normal forms in the lambda-calculus and non-trivial automata-theoretic results. To get a flavor of both the aims and the methods employed, one may skim through the following:- abstract outlining the programme: https://cs-web.swan.ac.uk/~cpradic/smp-abstract.pdf- PhD thesis on that topic: https://nguyentito.eu/thesis.pdf This project would be a continuation of that effort. There are a number of concrete problems to tackle that can serve as a good introduction to working in that setting, including:- comparing the expressiveness of the non-commutative linear lambda-calculus and first-order transducers- designing minimalistic typed programming languages capturing what is achievable in the linear lambda-calculus with Church encodings in the spirit of Bojanczyk's work on polyregular functions, but for comparison-free polyregular functions- carrying out a similar comparison for transducers over infinite structures and functions lambda-definable using the Church encoding of coinductive datatypes- checking whether there is difference in expressiveness if we allow the full power of classical linear logic instead of the intuitionistic fragmentThis could be followed-up by investigations in more challenging problems in the same area or into broader concerns specific to one of the domains involved (such as investigations which involving defining well behaved-class of tree transductions or more specialized topics in linear logic such as weak exponentials or quantification over linear types).While having a background in either mathematical logic or theoretical computer science would be necessary, a successful applicant would certainly not be expected to be familiar with all the tools mentioned above before starting.