24 pages, 7 figures, Extended version of an article in the proceedings of the 24th International Conference on Rewriting Techniques and Applications, Einhoven, the Netherlands, June 24-26, 2013
We address a problem connected to the unfolding semantics of functional programming languages: give a useful characterization of those infinite lambda-terms that are lambda_{letrec}-expressible in the sense that they arise as infinite unfoldings of terms in lambda_{letrec}, the lambda-calculus with letrec. We provide two characterizations, using concepts we introduce for infinite lambda-terms: regularity, strong regularity, and binding-capturing chains. It turns out that lambda_{letrec}-expressible infinite lambda-terms form a proper subclass of the regular infinite lambda-terms. In this paper we establish these characterizations only for expressibility in lambda_{mu}, the lambda-calculus with explicit mu-recursion. We show that for all infinite lambda-terms T the following are equivalent: (i): T is lambda_{mu}-expressible; (ii): T is strongly regular; (iii): T is regular, and it only has finite binding-capturing chains.
We define regularity and strong regularity for infinite lambda-terms as two different generalizations of regularity for infinite first-order terms: as the existence of only finitely many subterms that are defined as the reducts of two rewrite relations for decomposing lambda-terms. These rewrite relations act on infinite lambda-terms furnished with a marked prefix of abstractions for collecting decomposed lambda-abstractions and keeping the terms closed under decomposition. They differ in how vacuous abstractions in the prefix are removed.
This report accompanies the article with the same title for the proceedings of the conference RTA 2013, and mainly differs from that by providing the proof of the characterization of lambda_{mu}-expressibility with binding-capturing chains.
A comparison of Landin's form of lambda calculus with Church's shows that, independently of the lambda calculus, there exists a mechanism for converting functions with arguments indexed by variables to the usual kind of function where the arguments are indexed numerically. We call this the "lambda mechanism" and show how it can be used in other calculi. In first-order predicate logic it can be used to define new functions and new predicates in terms of existing ones. In a purely imperative programming language it can be used to provide an Algol-like procedure facility.
We develop the operational semantics of an untyped probabilistic lambda-calculus with continuous distributions, as a foundation for universal probabilistic programming languages such as Church, Anglican, and Venture. Our first contribution is to adapt the classic operational semantics of lambda-calculus to a continuous setting via creating a measure space on terms and defining step-indexed approximations. We prove equivalence of big-step and small-step formulations of this distribution-based semantics. To move closer to inference techniques, we also define the sampling-based semantics of a ter…
We aim to reason about the correctness of behaviour-preserving transformations of Erlang programs. Behaviour preservation is characterised by semantic equivalence. Based upon our existing formal semantics for Core Erlang, we investigate potential definitions of suitable equivalence relations. In particular we adapt a number of existing approaches of expression equivalence to a simple functional programming language that carries the main features of sequential Core Erlang; we then examine the properties of the equivalence relations and formally establish connections between them. The results pr…
Context-free session types provide a typing discipline for recursive structured communication protocols on bidirectional channels. They overcome the restriction of regular session type systems to tail recursive protocols. This extension enables us to implement serialisation and deserialisation of tree structures in a fully type-safe manner. We present the theory underlying the language FreeST 2, which features context-free session types in an extension of System F with linear types and a kind system to distinguish message types and channel types. The system presents some metatheoretical challe…
Cartesian difference categories are a recent generalisation of Cartesian differential categories which introduce a notion of "infinitesimal" arrows satisfying an analogue of the Kock-Lawvere axiom, with the axioms of a Cartesian differential category being satisfied only "up to an infinitesimal perturbation". In this work, we construct a simply-typed calculus in the spirit of the differential lambda-calculus equipped with syntactic infinitesimals and show how its models correspond to difference lambda-categories, a family of Cartesian difference categories equipped with suitably well-behaved e…
We propose an intersection type system for an imperative lambda-calculus based on a state monad and equipped with algebraic operations to read and write to the store. The system is derived by solving a suitable domain equation in the category of omega-algebraic lattices; the solution consists of a filter-model generalizing the well-known construction for ordinary lambda-calculus. Then the type system is obtained out of the term interpretations into the filter-model itself. The so obtained type system satisfies the "type-semantics" property, and it is sound and complete by construction.
A notion of probabilistic lambda-calculus usually comes with a prescribed reduction strategy, typically call-by-name or call-by-value, as the calculus is non-confluent and these strategies yield different results. This is a break with one of the main advantages of lambda-calculus: confluence, which means results are independent from the choice of strategy. We present a probabilistic lambda-calculus where the probabilistic operator is decomposed into two syntactic constructs: a generator, which represents a probabilistic event; and a consumer, which acts on the term depending on a given event. …
We formally verify an abstract machine for a call-by-value lambda-calculus with de Bruijn terms, simple substitution, and small-step semantics. We follow a stepwise refinement approach starting with a naive stack machine with substitution. We then refine to a machine with closures, and finally to a machine with a heap providing structure sharing for closures. We prove the correctness of the three refinement steps with compositional small-step bottom-up simulations. There is an accompanying Coq development verifying all results.
We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program alw…
LORIA, UMR 7503, Université de Lorraine, CNRS, Inria（洛林大学洛林计算机科学实验室（LORIA），UMR 7503，洛林大学，法国国家科学研究中心，法国国家信息与自动化研究所）
In compositional model-theoretic semantics, researchers assemble truth-conditions or other kinds of denotations using the lambda calculus. It was previously observed that the lambda terms and/or the denotations studied tend to follow the same pattern: they are instances of a monad. In this paper, we present an extension of the simply-typed lambda calculus that exploits this uniformity using the recently discovered technique of effect handlers. We prove that our calculus exhibits some of the key formal properties of the lambda calculus and we use it to construct a modular semantics for a small …
This paper presents Favalon, a functional programming language built on the premise of a lambda calculus for use as an interactive shell replacement. Favalon seamlessly integrates with typed versions of existing libraries and commands using type inference, flexible runtime type metadata, and the same techniques employed by shells to link commands together. Much of Favalon's syntax is customizable via user-defined functions, allowing it to be extended by anyone who is familiar with a command-line shell. Furthermore, Favalon's type inference engine can be separated from its runtime library and e…
We have previously published the Isabelle/HOL formalization of a general theory of syntax with bindings. In this companion paper, we instantiate the general theory to the syntax of lambda-calculus and formalize the development leading to several fundamental constructions and results: sound semantic interpretation, the Church-Rosser and standardization theorems, and higher-order abstract syntax (HOAS) encoding. For Church-Rosser and standardization, our work covers both the call-by-name and call-by-value versions of the calculus, following classic papers by Takahashi and Plotkin. During the for…
We present a novel lambda calculus that casts the categorical approach to the study of quantum protocols into the rich and well established tradition of type theory. Our construction extends the linear typed lambda calculus with a linear negation of "trivialised" De Morgan duality. Reduction is realised through explicit substitution, based on a symmetric notion of binding of global scope, with rules acting on the entire typing judgement instead of on a specific subterm. Proofs of subject reduction, confluence, strong normalisation and consistency are provided, and the language is shown to be a…
We observe that normalization by evaluation for simply-typed lambda-calculus with weak coproducts can be carried out in a weak bi-cartesian closed category of presheaves equipped with a monad that allows us to perform case distinction on neutral terms of sum type. The placement of the monad influences the normal forms we obtain: for instance, placing the monad on coproducts gives us eta-long beta-pi normal forms where pi refers to permutation of case distinctions out of elimination positions. We further observe that placing the monad on every coproduct is rather wasteful, and an optimal placem…
Linear/non-linear (LNL) models, as described by Benton, soundly model a LNL term calculus and LNL logic closely related to intuitionistic linear logic. Every such model induces a canonical enrichment that we show soundly models a LNL lambda calculus for string diagrams, introduced by Rios and Selinger (with primary application in quantum computing). Our abstract treatment of this language leads to simpler concrete models compared to those presented so far. We also extend the language with general recursion and prove soundness. Finally, we present an adequacy result for the diagram-free fragmen…
Generalized metrics, arising from Lawvere's view of metric spaces as enriched categories, have been widely applied in denotational semantics as a way to measure to which extent two programs behave in a similar, although non equivalent, way. However, the application of generalized metrics to higher-order languages like the simply typed lambda calculus has so far proved unsatisfactory. In this paper we investigate a new approach to the construction of cartesian closed categories of generalized metric spaces. Our starting point is a quantitative semantics based on a generalization of usual logica…
The elementary affine lambda-calculus was introduced as a polyvalent setting for implicit computational complexity, allowing for characterizations of polynomial time and hyperexponential time predicates. But these results rely on type fixpoints (a.k.a. recursive types), and it was unknown whether this feature of the type system was really necessary. We give a positive answer by showing that without type fixpoints, we get a characterization of regular languages instead of polynomial time. The proof uses the semantic evaluation method. We also propose an aesthetic improvement on the characteriza…
We investigate the possibility of a semantic account of the execution time (i.e. the number of beta-steps leading to the normal form, if any) for the shuffling calculus, an extension of Plotkin's call-by-value lambda-calculus. For this purpose, we use a linear logic based denotational model that can be seen as a non-idempotent intersection type system: relational semantics. Our investigation is inspired by similar ones for linear logic proof-nets and untyped call-by-name lambda-calculus. We first prove a qualitative result: a (possibly open) term is normalizable for weak reduction (which does …
We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus with the possibility to make arbitrary linear combinations of terms alpha.t + beta.u. We describe how to "execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the field of …
In this paper, we take a pervasively effectful (in the style of ML) typed lambda calculus, and show how to extend it to permit capturing pure expressions with types. Our key observation is that, just as the pure simply-typed lambda calculus can be extended to support effects with a monadic type discipline, an impure typed lambda calculus can be extended to support purity with a comonadic type discipline. We establish the correctness of our type system via a simple denotational model, which we call the capability space model. Our model formalizes the intuition common to systems programmers that…
We introduce a category-theoreticabstraction of a syntax with auxiliary functions, called an admissiblemonad morphism. Relying on an abstract form of structural recursion,we then design generic tools to construct admissible monad morphismsfrom basic data. These tools automate ubiquitous standard patternslike (1) defining auxiliary functions in successive, potentiallydependent layers, and (2) proving properties of auxiliary functions byinduction on syntax. We cover significant examples from theliterature, including the standard lambda-calculus withcapture-avoiding substitution, a lambda-calculu…
The Functional Machine Calculus (FMC), recently introduced by the authors, is a generalization of the lambda-calculus which may faithfully encode the effects of higher-order mutable store, I/O and probabilistic/non-deterministic input. Significantly, it remains confluent and can be simply typed in the presence of these effects. In this paper, we explore the denotational semantics of the FMC. We have three main contributions: first, we argue that its syntax -- in which both effects and lambda-calculus are realised using the same syntactic constructs -- is semantically natural, corresponding clo…
We present a formalization of session types in a multi-threaded lambda-calculus (MTLC) equipped with a linear type system, establishing for the MTLC both type preservation and global progress. The latter (global progress) implies that the evaluation of a well-typed program in the MTLC can never reach a deadlock. As this formulated MTLC can be readily embedded into ATS, a full-fledged language with a functional programming core that supports both dependent types (of DML-style) and linear types, we obtain a direct implementation of session types in ATS. In addition, we gain immediate support for…
Typed Answer Set Programming and Inverse Lambda Algorithms
Arizona State University（亚利桑那州立大学）
我们更广泛的目标是将英语句子自动翻译成适当的知识表示语言中的公式，作为理解和回答英语文本问题的一步。本文的重点是回答集编程语言（ASP）。我们的方法来翻译句子ASP规则的启发蒙塔古的使用λ演算公式的单词和短语的意义。以ASP为目标语言，单词和短语的含义是ASP-lambda公式。在早期的工作中，我们通过手动开发单词字典及其ASP lambda公式来说明我们的方法。然而，这种方法是不可扩展的。在本文中，我们的重点是两个算法，允许一个以逆的方式构建ASP-lambda公式。特别地，这两个算法将两个微积分表达式G和H作为输入，并且计算微积分表达式F，使得输入为G的F（由F@G表示）等于H;并且类似地G@F = H。我们目前的正确性和复杂性的结果，这些算法。为了做到这一点，我们开发了类型化ASP-λ演算理论及其顺序的概念，并将其用于开发完备性结果。(To《逻辑程序设计的理论与实践》（Theory and Practice of Logic Programming）
We study the two Girard's translations of intuitionistic implication into linear logic by exploiting the bang calculus, a paradigmatic functional language with an explicit box-operator that allows both call-by-name and call-by-value lambda-calculi to be encoded in. We investigate how the bang calculus subsumes both call-by-name and call-by-value lambda-calculi from a syntactic and a semantic viewpoint.
