Extensional Collapse Situations I: non-termination and unrecoverable errors

We consider a simple model of higher order, functional computation over the booleans. Then, we enrich the model in order to encompass non-termination and unrecoverable errors, taken separately or jointly. We show that the models so defined form a lattice when ordered by the extensional collapse situation relation, introduced in order to compare models with respect to the amount of "intensional information" that they provide on computation. The proofs are carried out by exhibiting suitable applied {\lambda}-calculi, and by exploiting the fundamental lemma of logical relations

Inhabitation for Non-idempotent Intersection Types

The inhabitation problem for intersection types in the lambda-calculus is known to be undecidable. We study the problem in the case of non-idempotent intersection, considering several type assignment systems, which characterize the solvable or the strongly normalizing lambda-terms. We prove the decidability of the inhabitation problem for all the systems considered, by providing sound and complete inhabitation algorithms for them

Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion

We study the semantics of a resource-sensitive extension of the lambda calculus in a canonical reflexive object of a category of sets and relations, a relational version of Scott's original model of the pure lambda calculus. This calculus is related to Boudol's resource calculus and is derived from Ehrhard and Regnier's differential extension of Linear Logic and of the lambda calculus. We extend it with new constructions, to be understood as implementing a very simple exception mechanism, and with a "must" parallel composition. These new operations allow to associate a context of this calculus with any point of the model and to prove full abstraction for the finite sub-calculus where ordinary lambda calculus application is not allowed. The result is then extended to the full calculus by means of a Taylor Expansion formula. As an intermediate result we prove that the exception mechanism is not essential in the finite sub-calculus

Graph Lambda Theories

to appear in MSCSInternational audienceA longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly lambda-beta or the the least sensible lambda-theory H (generated by equating all the unsolvable terms). A related question, raised recently by C. Berline, is whether, given a class of lambda models, there are a minimal lambda-theory and a minimal sensible lambda-theory represented by it. In this paper, we give a positive answer to this question for the class of graph models à la Plotkin-Scott-Engeler. In particular, we build two graph models whose theories are respectively the set of equations satisfied in any graph model and in any sensible graph model. We conjecture that the least sensible graph theory, where ''graph theory" means ''lambda-theory of a graph model", is equal to H, while in one of the main results of the paper we show the non-existence of a graph model whose equational theory is exactly the beta-theory. Another related question is whether, given a class of lambda models, there is a maximal sensible lambdatheory represented by it. In the main result of the paper we characterize the greatest sensible graph theory as the lambda-theory B generated by equating lambda-terms with the same Boehm tree. This result is a consequence of the main technical theorem of the paper: all the equations between solvable lambda-terms, which have different Boehm trees, fail in every sensible graph model. A further result of the paper is the existence of a continuum of different sensible graph theories strictly included in B

On Noncommutative Generalisations of Boolean Algebras

Skew Boolean algebras (SBA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed SBAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use SBAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed SBAs in terms of nBAs of n-partitions

Exploring New Topologies for the Theory of Clones

Clones of operations of arity $\omega$ (referred to as $\omega$-operations) have been employed by Neumann to represent varieties of infinitary algebras defined by operations of at most arity $\omega$. More recently, clone algebras have been introduced to study clones of functions, including $\omega$-operations, within the framework of one-sorted universal algebra. Additionally, polymorphisms of arity $\omega$, which are $\omega$-operations preserving the relations of a given first-order structure, have recently been used to establish model theory results with applications in the field of complexity of CSP problems. In this paper, we undertake a topological and algebraic study of polymorphisms of arity $\omega$ and their corresponding invariant relations. Given a Boolean ideal $X$ on the set $A^\omega$, we propose a method to endow the set of $\omega$-operations on $A$ with a topology, which we refer to as $X$-topology. Notably, the topology of pointwise convergence can be retrieved as a special case of this approach. Polymorphisms and invariant relations are then defined parametrically, with respect to the $X$-topology. We characterise the $X$-closed clones of $\omega$-operations in terms of $Pol^\omega$-$Inv^\omega$ and present a method to relate $Inv^\omega$-$Pol^\omega$ to the classical (finitary) $Inv$-$Pol$

Strong Normalization through Intersection Types and Memory

AbstractWe characterize β-strongly normalizing λ-terms by means of a non-idempotent intersection type system. More precisely, we first define a memory calculus K together with a non-idempotent intersection type system K, and we show that a K-term t is typable in K if and only if t is K-strongly normalizing. We then show that β-strong normalization is equivalent to K-strong normalization. We conclude since λ-terms are strictly included in K-terms

Sequent calculi of finite dimension

In recent work, the authors introduced the notion of n-dimensional Boolean algebra and the corresponding propositional logic nCL. In this paper, we introduce a sequent calculus for nCL and we show its soundness and completeness.Comment: arXiv admin note: text overlap with arXiv:1806.0653