10 research outputs found
Sequence Types for Hereditary Permutators
The invertible terms in Scott\u27s model D_infty are known as the hereditary permutators. Equivalently, they are terms which are invertible up to beta eta-conversion with respect to the composition of the lambda-terms. Finding a type-theoretic characterization to the set of hereditary permutators was problem # 20 of TLCA list of problems. In 2008, Tatsuta proved that this was not possible with an inductive type system. Building on previous work, we use an infinitary intersection type system based on sequences (i.e., families of types indexed by integers) to characterize hereditary permutators with a unique type. This gives a positive answer to the problem in the coinductive case
Normal Form Bisimulations By Value
Normal form bisimilarities are a natural form of program equivalence resting
on open terms, first introduced by Sangiorgi in call-by-name. The literature
contains a normal form bisimilarity for Plotkin's call-by-value
-calculus, Lassen's \emph{enf bisimilarity}, which validates all of
Moggi's monadic laws and can be extended to validate . It does not
validate, however, other relevant principles, such as the identification of
meaningless terms -- validated instead by Sangiorgi's bisimilarity -- or the
commutation of \letexps. These shortcomings are due to issues with open terms
of Plotkin's calculus. We introduce a new call-by-value normal form
bisimilarity, deemed \emph{net bisimilarity}, closer in spirit to Sangiorgi's
and satisfying the additional principles. We develop it on top of an existing
formalism designed for dealing with open terms in call-by-value. It turns out
that enf and net bisimilarities are \emph{incomparable}, as net bisimilarity
does not validate Moggi's laws nor . Moreover, there is no easy way to
merge them. To better understand the situation, we provide an analysis of the
rich range of possible call-by-value normal form bisimilarities, relating them
to Ehrhard's relational model.Comment: Rewritten version (deleted toy similarity and explained proof method
on naive similarity) -- Submitted to POPL2
The infinitary lambda calculus of the infinite eta Böhm trees
In this paper, we introduce a strong form of eta reduction called etabang that we use to construct a confluent and normalising infinitary lambda calculus, of which the normal forms correspond to Barendregt's infinite eta Böhm trees. This new infinitary perspective on the set of infinite eta Böhm trees allows us to prove that the set of infinite eta Böhm trees is a model of the lambda calculus. The model is of interest because it has the same local structure as Scott's D∞-models, i.e. two finite lambda terms are equal in the infinite eta Böhm model if and only if they have the same interpretation in Scott's D∞-models
Nominal Recursors as Epi-Recursors: Extended Technical Report
We study nominal recursors from the literature on syntax with bindings and
compare them with respect to expressiveness. The term "nominal" refers to the
fact that these recursors operate on a syntax representation where the names of
bound variables appear explicitly, as in nominal logic. We argue that nominal
recursors can be viewed as epi-recursors, a concept that captures abstractly
the distinction between the constructors on which one actually recurses, and
other operators and properties that further underpin recursion.We develop an
abstract framework for comparing epi-recursors and instantiate it to the
existing nominal recursors, and also to several recursors obtained from them by
cross-pollination. The resulted expressiveness hierarchies depend on how
strictly we perform this comparison, and bring insight into the relative merits
of different axiomatizations of syntax. We also apply our methodology to
produce an expressiveness hierarchy of nominal corecursors, which are
principles for defining functions targeting infinitary non-well-founded terms
(which underlie lambda-calculus semantics concepts such as B\"ohm trees). Our
results are validated with the Isabelle/HOL theorem prover
Separability of Infinite Lambda Terms
Abstract. Infinite lambda calculi extend finite lambda calculus with infinite terms and transfinite reduction. In this paper we extend some classical results of finite lambda calculus to infinite terms. The first result we extend to infinite terms is Böhm Theorem which states the separability of two finite βη-normal forms. The second result we extend to infinite terms is the equivalence of the prefix relation up to infinite eta expansions and the contextual preorder that observes head normal forms. Finally we prove that the theory given by equality of ∞η-Böhm trees is the largest theory induced by the confluent and normalising infinitary lambda calculi extending the calculus of Böhm trees.