37 research outputs found
Degrees of extensionality in the theory of B\"ohm trees and Sall\'e's conjecture
The main observational equivalences of the untyped lambda-calculus have been
characterized in terms of extensional equalities between B\"ohm trees. It is
well known that the lambda-theory H*, arising by taking as observables the head
normal forms, equates two lambda-terms whenever their B\"ohm trees are equal up
to countably many possibly infinite eta-expansions. Similarly, two lambda-terms
are equal in Morris's original observational theory H+, generated by
considering as observable the beta-normal forms, whenever their B\"ohm trees
are equal up to countably many finite eta-expansions.
The lambda-calculus also possesses a strong notion of extensionality called
"the omega-rule", which has been the subject of many investigations. It is a
longstanding open problem whether the equivalence B-omega obtained by closing
the theory of B\"ohm trees under the omega-rule is strictly included in H+, as
conjectured by Sall\'e in the seventies. In this paper we demonstrate that the
two aforementioned theories actually coincide, thus disproving Sall\'e's
conjecture.
The proof technique we develop for proving the latter inclusion is general
enough to provide as a byproduct a new characterization, based on bounded
eta-expansions, of the least extensional equality between B\"ohm trees.
Together, these results provide a taxonomy of the different degrees of
extensionality in the theory of B\"ohm trees
Efficient Type Checking for Path Polymorphism
A type system combining type application, constants as types, union types (associative, commutative and idempotent) and recursive types has recently been proposed for statically typing path polymorphism, the ability to define functions that can operate uniformly over
recursively specified applicative data structures. A typical pattern such functions resort to is dataterm{x}{y} which decomposes a compound, in other words any applicative tree structure, into its parts. We study type-checking for this type system in two stages. First we propose algorithms for checking type equivalence and subtyping based on coinductive characterizations of those relations. We then formulate a syntax-directed presentation and prove its equivalence with the original one. This yields a type-checking algorithm which unfortunately has exponential time complexity in the worst case. A second algorithm is then proposed, based on automata techniques, which yields a polynomial-time type-checking algorithm
A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice
We present a Kleene realizability semantics for the intensional level of the
Minimalist Foundation, for short mtt, extended with inductively generated
formal topologies, Church's thesis and axiom of choice. This semantics is an
extension of the one used to show consistency of the intensional level of the
Minimalist Foundation with the axiom of choice and formal Church's thesis in
previous work. A main novelty here is that such a semantics is formalized in a
constructive theory represented by Aczel's constructive set theory CZF extended
with the regular extension axiom
Type-Theoretic Constructions of the Final Coalgebra of the Finite Powerset Functor
The finite powerset functor is a construct frequently employed for the specification of nondeterministic transition systems as coalgebras. The final coalgebra of the finite powerset functor, whose elements characterize the dynamical behavior of transition systems, is a well-understood object which enjoys many equivalent presentations in set-theoretic foundations based on classical logic.
In this paper, we discuss various constructions of the final coalgebra of the finite powerset functor in constructive type theory, and we formalize our results in the Cubical Agda proof assistant. Using setoids, the final coalgebra of the finite powerset functor can be defined from the final coalgebra of the list functor. Using types instead of setoids, as it is common in homotopy type theory, one can specify the finite powerset datatype as a higher inductive type and define its final coalgebra as a coinductive type. Another construction is obtained by quotienting the final coalgebra of the list functor, but the proof of finality requires the assumption of the axiom of choice. We conclude the paper with an analysis of a classical construction by James Worrell, and show that its adaptation to our constructive setting requires the presence of classical axioms such as countable choice and the lesser limited principle of omniscience
RPO, Second-order Contexts, and Lambda-calculus
First, we extend Leifer-Milner RPO theory, by giving general conditions to
obtain IPO labelled transition systems (and bisimilarities) with a reduced set
of transitions, and possibly finitely branching. Moreover, we study the weak
variant of Leifer-Milner theory, by giving general conditions under which the
weak bisimilarity is a congruence. Then, we apply such extended RPO technique
to the lambda-calculus, endowed with lazy and call by value reduction
strategies.
We show that, contrary to process calculi, one can deal directly with the
lambda-calculus syntax and apply Leifer-Milner technique to a category of
contexts, provided that we work in the framework of weak bisimilarities.
However, even in the case of the transition system with minimal contexts, the
resulting bisimilarity is infinitely branching, due to the fact that, in
standard context categories, parametric rules such as the beta-rule can be
represented only by infinitely many ground rules.
To overcome this problem, we introduce the general notion of second-order
context category. We show that, by carrying out the RPO construction in this
setting, the lazy observational equivalence can be captured as a weak
bisimilarity equivalence on a finitely branching transition system. This result
is achieved by considering an encoding of lambda-calculus in Combinatory Logic.Comment: 35 pages, published in Logical Methods in Computer Scienc