5,245 research outputs found
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
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
Multi-level Contextual Type Theory
Contextual type theory distinguishes between bound variables and
meta-variables to write potentially incomplete terms in the presence of
binders. It has found good use as a framework for concise explanations of
higher-order unification, characterize holes in proofs, and in developing a
foundation for programming with higher-order abstract syntax, as embodied by
the programming and reasoning environment Beluga. However, to reason about
these applications, we need to introduce meta^2-variables to characterize the
dependency on meta-variables and bound variables. In other words, we must go
beyond a two-level system granting only bound variables and meta-variables.
In this paper we generalize contextual type theory to n levels for arbitrary
n, so as to obtain a formal system offering bound variables, meta-variables and
so on all the way to meta^n-variables. We obtain a uniform account by
collapsing all these different kinds of variables into a single notion of
variabe indexed by some level k. We give a decidable bi-directional type system
which characterizes beta-eta-normal forms together with a generalized
substitution operation.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
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
On generic context lemmas for lambda calculi with sharing
This paper proves several generic variants of context lemmas and thus contributes to improving the tools to develop observational semantics that is based on a reduction semantics for a language. The context lemmas are provided for may- as well as two variants of mustconvergence and a wide class of extended lambda calculi, which satisfy certain abstract conditions. The calculi must have a form of node sharing, e.g. plain beta reduction is not permitted. There are two variants, weakly sharing calculi, where the beta-reduction is only permitted for arguments that are variables, and strongly sharing calculi, which roughly correspond to call-by-need calculi, where beta-reduction is completely replaced by a sharing variant. The calculi must obey three abstract assumptions, which are in general easily recognizable given the syntax and the reduction rules. The generic context lemmas have as instances several context lemmas already proved in the literature for specific lambda calculi with sharing. The scope of the generic context lemmas comprises not only call-by-need calculi, but also call-by-value calculi with a form of built-in sharing. Investigations in other, new variants of extended lambda-calculi with sharing, where the language or the reduction rules and/or strategy varies, will be simplified by our result, since specific context lemmas are immediately derivable from the generic context lemma, provided our abstract conditions are met
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