Models Of Λ-calculus Research Articles

Why Are Proofs Relevant in Proof-Relevant Models?

Relational models of λ-calculus can be presented as type systems, the relational interpretation of a λ-term being given by the set of its typings. Within a distributors-induced bicategorical semantics generalizing the relational one, we identify the class of ‘categorified’ graph models and show that they can be presented as type systems as well. We prove that all the models living in this class satisfy an Approximation Theorem stating that the interpretation of a program corresponds to the filtered colimit of the denotations of its approximants. As in the relational case, the quantitative nature of our models allows to prove this property via a simple induction, rather than using impredicative techniques. Unlike relational models, our 2-dimensional graph models are also proof-relevant in the sense that the interpretation of a λ-term does not contain only its typings, but the whole type derivations. The additional information carried by a type derivation permits to reconstruct an approximant having the same type in the same environment. From this, we obtain the characterization of the theory induced by the categorified graph models as a simple corollary of the Approximation Theorem: two λ-terms have isomorphic interpretations exactly when their B'ohm trees coincide.

Defining totality in the enumeration degrees

We show that if A and B form a nontrivial K-pair, then there is a semi-computable set C such that A ≤e C and B ≤e C. As a consequence, we obtain a definition of the total enumeration degrees: a nonzero enumeration degree is total if and only if it is the join of a nontrivial maximal K-pair. This answers a longstanding question of Hartley Rogers, Jr. We also obtain a definition of the “c.e. in” relation for total degrees in the enumeration degrees. Two of the most fundamental concepts in computability theory are effective computation and effective enumeration, i.e., computation and enumeration of sets of natural numbers as achieved by computer programs. Each notion induces a natural measure of the complexity of sets of natural numbers. Turing reducibility was defined by Turing in the late 1930’s to capture the relative complexity of computing sets of natural numbers: A ≤T B means that a set A is computable from another set B. We use the word “reducibility” because the problem of computing A is being reduced to the problem of computing B. In the same way, Enumeration reducibility was defined by Friedberg and Rogers in the late 1950’s to capture the relative complexity of enumerating sets: A ≤e B means that there is a way to effectively produce an enumeration of a set A from any enumeration of a set B. Turing reducibility has been studied extensively, but enumeration reducibility also arises naturally in various contexts. Enumeration reducibility restricted to partial functions is equivalent to Kleene’s [12] notion of relative effective computability of partial functions. Scott [20] used enumeration reducibility to give a countable graph model of λ-calculus. C. F. Miller (unpublished manuscript) and M. Ziegler [28] applied enumeration reducibility in group theory, to state and prove an extension of Higman’s embedding theorem for finitely generated groups. J. Miller [15] used enumeration reducibility in his work on computable analysis; he answered a question of Pour El and Lempp by showing that Turing reducibility is not sufficient to measure the complexity of continuous functions on R (but enumeration reducibility is). Similarly, Richter [16] constructed a wide variety of countable structures for which enumeration reducibility, but not Turing reducibility, is sufficient to measure the complexity of their isomorphism type. There are several other occasions in 2010 Mathematics Subject Classification. 03D30.

Essential and relational models

Intersection type assignment systems can be used as a general framework for building logical models of λ-calculus that allow to reason about the denotation of terms in a finitary way. We defineessentialmodels (a new class of logical models) through a parametric type assignment system using non-idempotent intersection types. Under an interpretation of terms based on typings instead than the usual one based on types, every suitable instance of the parameters induces a λ-model, whose theory is sensible. We prove that this type assignment system provides a logical description of a family of λ-models arising from a category of sets and relations.

Relatively terminal coalgebras

Dana Scott’s model of λ-calculus was based on a limit construction which started from an algebra of a suitable endofunctor F and continued by iterating F. We demonstrate that this is a special case of the concept we call coalgebra relatively terminal w.r.t. the given algebra A. This means a coalgebra together with a universal coalgebra-to-algebra morphism into A.We prove that by iterating F countably many times we obtain the relatively terminal coalgebras whenever F preserves limits of ωop-chains. If F is finitary, we need in general ω+ω steps. And for arbitrary accessible (=bounded) set functors we need an ordinal number of steps in general. Scott’s result is captured by the fact that in a CPO-enriched category, assuming that F is locally continuous, ω steps are sufficient for algebras given by projections.

What is a categorical model of the differential and the resource λ-calculi?

The differential λ-calculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows the application of a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitable primitives to handle explicitly the resources used by a program during its execution. The differential operator also allows us to write the full Taylor expansion of a program. Through this expansion, every program can be decomposed into an infinite sum (representing non-deterministic choice) of ‘simpler’ programs that are strictly linear.The aim of this paper is to develop an abstract ‘model theory’ for the untyped differential λ-calculus. In particular, we investigate what form a general categorical definition of a denotational model for this calculus should take. Starting from the work of Blute, Cockett and Seely on differential categories, we develop the notion of a Cartesian closed differential category and prove that linear reflexive objects living in such categories constitute sound and complete models of the untyped differential λ-calculus. We also give sufficient conditions for Cartesian closed differential categories to model the Taylor expansion. This requires that every model living in such categories equates all programs having the same full Taylor expansion.We then provide a concrete example of a Cartesian closed differential category modelling the Taylor expansion, namely the category MRel of sets and relations from finite multisets to sets. We prove that the extensional model of λ-calculus we have recently built in MRel is linear, and is thus also an extensional model of the untyped differential λ-calculus. In the same category, we build a non-extensional model and prove that it is, nevertheless, extensional on its differential part.Finally, we study the relationship between the differential λ-calculus and the resource calculus, which is a functional programming language combining the ideas behind the differential λ-calculus with those behind Boudol's λ-calculus with multiplicities. We define two translation maps between these two calculi and study the properties of these translations. In particular, this analysis shows that the two calculi share the same notion of a model, and thus that the resource calculus can be interpreted by translation into every linear reflexive object living in a Cartesian closed differential category.

Effective λ-models versus recursively enumerable λ-theories

A longstanding open problem is whether there exists a non-syntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for short). We introduce a notion of effective model of λ-calculus, which covers, in particular, all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ or λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. For Scott's semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim–Skolem theorem.

The interpretation of unsolvable λ-terms in models of untyped λ-calculus

AbstractOur goal in this paper is to analyze the interpretation of arbitrary unsolvable λ-terms in a given model of λ-calculus. We focus on graph models and (a special type of) stable models. We introduce the syntactical notion of a decoration and the semantical notion of a critical sequence. We conjecture that any unsolvable term β-reduces to a term admitting a decoration. The main result of this paper concerns the interconnection between those two notions: given a graph model or stable model , we show that any unsolvable term admitting a decoration and having a non-empty interpretation in generates a critical sequence in the model.In the last section, we examine three classical graph models, namely the model of Plotkin and Scott, Engeler's model and Park's model . We show that and do not contain critical sequences whereas does.

Isomorphism and Equational Equivalence of Continuous [lambda]-Models.

We will present several results on two types of continuous models of λ-calculus, namely graph models and extensional models. By introducing a variant of Engeler's model construction, we are able to generalize the results of [7] and to give invariants that determine a large family of graph models up to applicative isomorphism. This covers all graph models considered in the litterature so far. We indicate briefly how these invariants may be modified in order to determine extensional models as well.

Combining type disciplines

We present a type inference system for pure λ-calculus which includes, in addition to arrow types, also universal and existential type quantifiers, intersection and union types, and type recursion. The interest of this system lies in the fact that it offers a possibility to study in a unified framework a wide range of type constructors. We investigate the main syntactical properties of the system, including an analysis of the preservation of types under parallel reduction strategies, leading to a form of the subject-reduction property. We describe a model for this system where types are special subsets of a D∞ model for λ-calculus, without imposing any formal contractiveness constraint on types of the kind considered for a closely related system by MacQueen, Plotkin and Sethi (1986).

An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus

It is well known that a reflexive object in the Cartesian closed category of complete partial orders and Scott-continuous functions is a model of λ-calculus (briefly a topological model). A topological model, through the interpretation function, induces a λ-theory, i.e., a congruence relation on λ-terms closed under α- and β-reduction. It is natural to ask if all possible λ-theories are induced by a topological model, i.e., if topological models are complete w.r.t. λ-calculus. The authors prove an Approximation Theorem, which holds in all topological models. Using this theorem, they analyze some topological models and their induced λ-theories, and they exhibit a λ-theory which cannot be induced by a topological model. So they prove that topological models are not complete w.r.t. λ-calculus.

Classes of Numeration Models of λ‐Calculus

Classes of Numeration Models of λ‐Calculus

Set-theoretical models of λ-calculus: theories, expansions, isomorphisms

On introduit les termes du λ-calcul et les termes de la logique combinatoire d'ordres divers, correspondant aux niveaux de fonctionnalite ou au nombre de λ-abstractions. On donne les algebres PSE. On presente une caracterisation semantique des λ-termes de tout ordre fini. Applications de theorie des modeles principales