400 research outputs found
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 Focused Sequent Calculus Framework for Proof Search in Pure Type Systems
Basic proof-search tactics in logic and type theory can be seen as the
root-first applications of rules in an appropriate sequent calculus, preferably
without the redundancies generated by permutation of rules. This paper
addresses the issues of defining such sequent calculi for Pure Type Systems
(PTS, which were originally presented in natural deduction style) and then
organizing their rules for effective proof-search. We introduce the idea of
Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the
syntax of a permutation-free sequent calculus for propositional logic due to
Herbelin, which is strongly related to natural deduction and already well
adapted to proof-search. The operational semantics is adapted from Herbelin's
and is defined by a system of local rewrite rules as in cut-elimination, using
explicit substitutions. We prove confluence for this system. Restricting our
attention to PTSC, a type system for the ground terms of this system, we obtain
the Subject Reduction property and show that each PTSC is logically equivalent
to its corresponding PTS, and the former is strongly normalising iff the latter
is. We show how to make the logical rules of PTSC into a syntax-directed system
PS for proof-search, by incorporating the conversion rules as in
syntax-directed presentations of the PTS rules for type-checking. Finally, we
consider how to use the explicitly scoped meta-variables of PTSCalpha to
represent partial proof-terms, and use them to analyse interactive proof
construction. This sets up a framework PE in which we are able to study
proof-search strategies, type inhabitant enumeration and (higher-order)
unification
Confluence via strong normalisation in an algebraic \lambda-calculus with rewriting
The linear-algebraic lambda-calculus and the algebraic lambda-calculus are
untyped lambda-calculi extended with arbitrary linear combinations of terms.
The former presents the axioms of linear algebra in the form of a rewrite
system, while the latter uses equalities. When given by rewrites, algebraic
lambda-calculi are not confluent unless further restrictions are added. We
provide a type system for the linear-algebraic lambda-calculus enforcing strong
normalisation, which gives back confluence. The type system allows an abstract
interpretation in System F.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Mechanising syntax with binders in Coq
Mechanising binders in general-purpose proof assistants such as Coq is cumbersome and difficult. Yet binders, substitutions, and instantiation of terms with substitutions are a critical ingredient of many programming languages. Any practicable mechanisation of the meta-theory of the latter hence requires a lean formalisation of the former. We investigate the topic from three angles: First, we realise formal systems with binders based on both pure and scoped de Bruijn algebras together with basic syntactic rewriting lemmas and automation. We automate this process in a compiler called Autosubst; our final tool supports many-sorted, variadic, and modular syntax. Second, we justify our choice of realisation and mechanise a proof of convergence of the sigma calculus, a calculus of explicit substitutions that is complete for equality of the de Bruijn algebra corresponding to the lambda calculus. Third, to demonstrate the practical usefulness of our approach, we provide concise, transparent, and accessible mechanised proofs for a variety of case studies refined to de Bruijn substitutions.Die Mechanisierung von Bindern in universellen Beweisassistenten wie Coq ist arbeitsaufwändig und schwierig. Binder, Substitutionen und die Instantiierung von Substitutionen sind jedoch kritischer Bestandteil
vieler Programmiersprachen. Deshalb setzt eine praktikable Mechanisierung der Metatheorie von Programmiersprachen eine elegante Formalisierung von Bindern voraus. Wir nähern uns dem Thema aus drei Richtungen an:
Zuerst realisieren wir formale Systeme mit Bindern mit Hilfe von reinen und indizierten de Bruijn Algebren, zusammen mit grundlegenden syntaktischen Gleichungen und Automatisierung. Wir automatisieren diesen Prozess in
einem Kompilierer namens Autosubst. Unser finaler Kompilierer unterstützt Sortenlogik, variadische Syntax und modulare Syntax. Zweitens rechtfertigen wir unsere Repräsentation und mechanisieren einen Beweis der Konvergenz
des SP-Kalküls, einem Kalkül expliziter Substitutionen der bezüglich der Gleichheit der puren de Bruijn Algebra des -Kalküls vollständig ist. Drittens entwickeln wir kurze, transparente und leicht zugängliche mechanisierte
Beweise für diverse Fallstudien, die wir an de Bruijn Substitutionen angepasst haben. Wir weisen so die praktische Anwendbarkeit unseres Ansatzes nach
On the enumeration of closures and environments with an application to random generation
Environments and closures are two of the main ingredients of evaluation in
lambda-calculus. A closure is a pair consisting of a lambda-term and an
environment, whereas an environment is a list of lambda-terms assigned to free
variables. In this paper we investigate some dynamic aspects of evaluation in
lambda-calculus considering the quantitative, combinatorial properties of
environments and closures. Focusing on two classes of environments and
closures, namely the so-called plain and closed ones, we consider the problem
of their asymptotic counting and effective random generation. We provide an
asymptotic approximation of the number of both plain environments and closures
of size . Using the associated generating functions, we construct effective
samplers for both classes of combinatorial structures. Finally, we discuss the
related problem of asymptotic counting and random generation of closed
environemnts and closures
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