A Theory of Explicit Substitutions with Safe and Full Composition

Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the first part of this paper. Then, very simple technology in named variable-style notation is used to establish a theory of explicit substitutions for the lambda-calculus which enjoys a whole set of useful properties such as full composition, simulation of one-step beta-reduction, preservation of beta-strong normalisation, strong normalisation of typed terms and confluence on metaterms. Normalisation of related calculi is also discussed.Comment: 29 pages Special Issue: Selected Papers of the Conference "International Colloquium on Automata, Languages and Programming 2008" edited by Giuseppe Castagna and Igor Walukiewic

A Case Study on Logical Relations using Contextual Types

Proofs by logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe the completeness proof of algorithmic equality for simply typed lambda-terms by Crary where we reason about logically equivalent terms in the proof environment Beluga. There are three key aspects we rely upon: 1) we encode lambda-terms together with their operational semantics and algorithmic equality using higher-order abstract syntax 2) we directly encode the corresponding logical equivalence of well-typed lambda-terms using recursive types and higher-order functions 3) we exploit Beluga's support for contexts and the equational theory of simultaneous substitutions. This leads to a direct and compact mechanization, demonstrating Beluga's strength at formalizing logical relations proofs.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

Explicit Substitutions for Contextual Type Theory

In this paper, we present an explicit substitution calculus which distinguishes between ordinary bound variables and meta-variables. Its typing discipline is derived from contextual modal type theory. We first present a dependently typed lambda calculus with explicit substitutions for ordinary variables and explicit meta-substitutions for meta-variables. We then present a weak head normalization procedure which performs both substitutions lazily and in a single pass thereby combining substitution walks for the two different classes of variables. Finally, we describe a bidirectional type checking algorithm which uses weak head normalization and prove soundness.Comment: In Proceedings LFMTP 2010, arXiv:1009.218

Comparing Calculi of Explicit Substitutions with Eta-reduction1 1Partially supported by the Brazilian CNPq research council grant number 47488101-6.

AbstractThe past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. Three styles of explicit substitutions are treated in this paper: the λσ and the λse which have proved useful for solving practical problems like higher order unification, and the suspension calculus related to the implementation of the language λ-Prolog. We enlarge the suspension calculus with an adequate eta-reduction which we show to preserve termination and confluence of the associated substitution calculus and to correspond to the eta-reductions of the other two calculi. Additionally, we prove that λσ and λse as well as λσ and the suspension calculus are non comparable while λse is more adequate than the suspension calculus

A Lambda Term Representation Inspired by Linear Ordered Logic

We introduce a new nameless representation of lambda terms inspired by ordered logic. At a lambda abstraction, number and relative position of all occurrences of the bound variable are stored, and application carries the additional information where to cut the variable context into function and argument part. This way, complete information about free variable occurrence is available at each subterm without requiring a traversal, and environments can be kept exact such that they only assign values to variables that actually occur in the associated term. Our approach avoids space leaks in interpreters that build function closures. In this article, we prove correctness of the new representation and present an experimental evaluation of its performance in a proof checker for the Edinburgh Logical Framework. Keywords: representation of binders, explicit substitutions, ordered contexts, space leaks, Logical Framework.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time. Is lambda-calculus a reasonable machine? Is there a way to measure the computational complexity of a lambda-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based over a standard notion in the theory of lambda-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model. Such a theorem cannot be proved by directly relating lambda-calculus with Turing machines or random access machines, because of the size explosion problem: there are terms that in a linear number of steps produce an exponentially long output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modeled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that LSC is invariant with respect to the lambda-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the lambda-calculus. We solve such an impasse by introducing a new form of shared normal form and shared reduction, deemed useful. Useful evaluation avoids those steps that only unshare the output without contributing to beta-redexes, i.e. the steps that cause the blow-up in size. The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties.Comment: arXiv admin note: substantial text overlap with arXiv:1405.331

A correct, precise and efficient integration of set-sharing, freeness and linearity for the analysis of finite and rational tree languages

It is well known that freeness and linearity information positively interact with aliasing information, allowing both the precision and the efficiency of the sharing analysis of logic programs to be improved. In this paper, we present a novel combination of set-sharing with freeness and linearity information, which is characterized by an improved abstract unification operator. We provide a new abstraction function and prove the correctness of the analysis for both the finite tree and the rational tree cases. Moreover, we show that the same notion of redundant information as identified in Bagnara et al. (2000) and Zaffanella et al. (2002) also applies to this abstract domain combination: this allows for the implementation of an abstract unification operator running in polynomial time and achieving the same precision on all the considered observable properties