Untyped Λ-calculus Research Articles

From semantics to types: The case of the imperative λ-calculus

We study the logical semantics of an untyped λ-calculus equipped with operators representing read and write operations from and to a global store. Such a logic consists of an intersection type assignment system, which we derive from the denotational semantics of the calculus, based on the monadic approach to model computational λ-calculi.The system is obtained by constructing a filter model in the category of ω-algebraic lattices, such that the typing rules can be recovered out of the term interpretation. By construction, the so-obtained type system satisfies the “type-semantics” property and completeness.

The untyped computational λ-calculus and its intersection type discipline

We study a Curry style type assignment system for untyped λ-calculi with effects, based on Moggi's monadic approach. Moving from the abstract definition of monads, we introduce a version of the call-by-value computational λ-calculus based on Wadler's variant, without let, and with unit and bind operators. We define a notion of reduction for the calculus and prove it confluent.We then introduce an intersection type system inspired by Barendregt, Coppo and Dezani system for ordinary untyped λ-calculus, establishing type invariance under conversion.Finally, we introduce a notion of convergence, which is precisely related to reduction, and characterize convergent terms via their types.

Categorical models of the differential λ-calculus

AbstractThe paper shows how the Scott–Koymans theorem for the untyped λ-calculus can be extended to the differential λ-calculus. The main result is that every model of the untyped differential λ-calculus may be viewed as a differential reflexive object in a Cartesian-closed differential category. This extension of the Scott–Koymans theorem depends critically on unraveling the somewhat subtle issue of which idempotents can be split so that differential structure lifts to the idempotent splitting.The paper uses (total) Turing categories with “canonical codes” as the basic categorical semantics for the λ-calculus. It develops the main result in a modular fashion by showing how to add left-additive structure to a Turing category, and then – on top of that – differential structure. For both levels of structure, it is necessary to identify how “canonical codes” must behave with respect to the added structure and, furthermore, how “universal objects” must behave. The latter is closely tied to the question – which is the crux of the paper – of which idempotents can be split while preserving the differential structure of the setting.This paper is the full version of a conference paper and includes the proofs which were omitted from that version due to page-length restrictions.

BUNDER’S PARADOX

AbstractSystems ofillative logicare logical calculi formulated in the untypedλ-calculus supplemented with certain logical constants.1In this short paper, I consider a paradox that arises in illative logic. I note two prima facie attractive ways of resolving the paradox. The first is well known to be consistent, and I briefly outline a now standard construction used by Scott and Aczel that establishes this. The second, however, has been thought to be inconsistent. I show that this isn’t so, by providing a nonempty class of models that establishes its consistency. I then provide an illative logic which is sound and complete for this class of models. I close by briefly noting some attractive features of the second resolution of this paradox.

Execution time of λ-terms via denotational semantics and intersection types

The multiset-based relational model of linear logic induces a semantics of the untyped λ-calculus, which corresponds with a non-idempotent intersection type system, System R. We prove that, in System R, the size of type derivations and the size of types are closely related to the execution time of λ-terms in a particular environment machine, Krivine's machine.

Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness

We give a semantics for the λ-calculus based on a topological duality theorem in nominal sets. A novel interpretation of λ is given in terms of adjoints, and λ-terms are interpreted absolutely as sets (no valuation is necessary).

Categorical Models of the Differential λ-Calculus Revisited

The paper shows that the Scott-Koymans theorem for the untyped λ-calculus extends to the differential λ-calculus. The main result is that every model of the untyped differential λ-calculus may be viewed as a differential reflexive object in a Cartesian closed differential category. This extension of the Scott-Koymans theorem depends critically on unravelling the somewhat subtle issue of which idempotents can be split so that differential structure lifts to the idempotent splitting.The paper uses (total) Turing categories with “canonical codes” as the basic categorical semantics for the λ-calculus. It shows how the main result may be developed in a modular fashion by first adding left-additive structure to a Turing category, and then – on top of that – differential structure. For both levels of structure it is necessary to identify how “canonical codes” behave with respect to the added structure and, furthermore, how “universal objects” behave. The latter is closely tied to the question – which is the crux of the paper – of which idempotents can be split in these more structured settings.

Fully-abstract compilation by approximate back-translation

A compiler is fully-abstract if the compilation from source language programs to target language programs reflects and preserves behavioural equivalence. Such compilers have important security benefits, as they limit the power of an attacker interacting with the program in the target language to that of an attacker interacting with the program in the source language. Proving compiler full-abstraction is, however, rather complicated. A common proof technique is based on the back-translation of target-level program contexts to behaviourally-equivalent source-level contexts. However, constructing such a back-translation is problematic when the source language is not strong enough to embed an encoding of the target language. For instance, when compiling from the simply-typed λ-calculus (λτ) to the untyped λ-calculus (λu), the lack of recursive types in λτ prevents such a back-translation. We propose a general and elegant solution for this problem. The key insight is that it suffices to construct an approximate back-translation. The approximation is only accurate up to a certain number of steps and conservative beyond that, in the sense that the context generated by the back-translation may diverge when the original would not, but not vice versa. Based on this insight, we describe a general technique for proving compiler full-abstraction and demonstrate it on a compiler from λτ to λu . The proof uses asymmetric cross-language logical relations and makes innovative use of step-indexing to express the relation between a context and its approximate back-translation. We believe this proof technique can scale to challenging settings and enable simpler, more scalable proofs of compiler full-abstraction.

Term-generic logic

We introduce term-generic logic (TGL), a first-order logic parameterized with terms defined axiomatically (rather than constructively), by requiring terms to only provide free variable and substitution operators satisfying some reasonable axioms. TGL has a notion of model that generalizes both first-order models and Henkin models of the λ-calculus. The abstract notions of term syntax and model are shown to be sufficient for obtaining the completeness theorem of a Gentzen system generalizing that of first-order logic. Various systems featuring bindings and contextual reasoning, ranging from pure type systems to the π-calculus, are captured as theories inside TGL. For two particular, but rather typical instances—untyped λ-calculus and System F—the general-purpose TGL models are shown to be equivalent with standard ad hoc models.

Böhm theorem and Böhm trees for the [formula omitted]-calculus

Parigot’s λμ-calculus (Parigot, 1992 [1]) is now a standard reference about the computational content of classical logic as well as for the formal study of control operators in functional languages. In addition to the fine-grained Curry–Howard correspondence between minimal classical deductions and simply typed λμ-terms and to the ability to encode many usual control operators such as call/cc in the λμ-calculus (in its historical call-by-name presentation or in call-by-value versions), the success of theλμ-calculus comes from its simplicity, its good meta-theoretical properties both as a typed and an untyped calculus (confluence, strong normalization, etc.) as well as the fact that it naturally extends Church’s λ-calculus. Though, in 2001, David and Py proved [2] that Böhm’s theorem, which is a fundamental result of the untyped λ-calculus, cannot be lifted to Parigot’s calculus.In the present article, we exhibit a natural extension to Parigot’s calculus, theΛμ-calculus, in which Böhm’s property, also known as separation property, can be stated and proved. This is made possible by a careful and detailed analysis of David and Py’s proof of non-separability and of the characteristics of the λμ-calculus which break the property: we identify that the crucial point lies in the design of Parigot’s λμ-calculus with a two-level syntax. In addition, we establish a standardization theorem for the extended calculus, deduce a characterization of solvability, describe Λμ-Böhm trees and connect the calculus with stream computing and delimited control.

Reducibility Proofs in the λ-Calculus

Reducibility, despite being quite mysterious and inflexible, has been used to prove a number of properties of the λ-calculus and is well known to offer general proofs which can be applied to a number of instantiations. In this paper, we look at two related but different results in λ-calculi with intersection types. 1. We show that one such result which aims at giving reducibility proofs of Church-Rosser, standardisation and weak normalisation for the untyped λ-calculus faces serious problems which break the reducibility method. We provide a proposal to partially repair the method. 2. We consider a second result whose purpose is to use reducibility for typed terms in order to show the Church-Rosser of β-developments for the untyped terms and hence the Church-Rosser of β-reduction. In this second result, strong normalisation is not needed. We extend the second result to encompass both βI-and βη-reduction rather than simply β-reduction.

Intuitionistic differential nets and lambda-calculus

We define pure intuitionistic differential proof nets, extending Ehrhard and Regnier’s differential interaction nets with the exponential box of Linear Logic. Normalization of the exponential reduction and confluence of the full one is proved. These results are directed and adjusted to give a translation of Boudol’s untyped λ-calculus with resources extended with a linear–nonlinear reduction à la Ehrhard and Regnier’s differential λ-calculus. Such reduction comes in two flavours: baby-step and giant-step β-reduction. The translation, based on Girard’s encoding A→B∼!A⊸B and as such extending the usual one for λ-calculus into proof nets, enjoys bisimulation for giant-step β-reduction. From this result we also derive confluence of both reductions.

Realisability for Induction and Coinduction with Applications to Constructive Analysis

We prove the correctness of a formalised realisability interpretation of ex- tensions of first-order theories by inductive and coinductive definitions in an untyped λ-calculus with fixed-points. We illustrate the use of this interpretation for program extraction by some simple examples in the area of exact real number computation and hint at further non-trivial applications in computable analysis.

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.

On a measurement-free quantum lambda calculus with classical control

We study a measurement-free, untyped λ-calculus with quantum data and classical control. This work arises from previous proposals by Selinger and Valiron, and Van Tonder. We focus on operational and expressiveness issues, rather than (denotational) semantics. We prove subject reduction and confluence, and a standardisation theorem. Moreover, we prove the computational equivalence of the proposed calculus with a suitable class of quantum circuit families.

Applying Universal Algebra to Lambda Calculus

The aim of this article is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the long-standing open questions concerning the representability of λ-theories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λ-abstraction algebras. In every combinatory and λ-abstraction algebra, there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λ-abstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e. algebras that cannot be decomposed as the Cartesian product of two other non-trivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e. the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λ-theories. In one of the main results of the article we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.

Church-Rosser property and intersection types

We give a proof via reducibility of the Church-Rosser property for the system D of λ-calculus with intersection types. As a consequence we can get the confluence property for developments directly, without making use of the strong normalization property for developments, by using only the typability in D and a suitable embedding of developments in this system. As an application we get a proof of the Church-Rosser theorem for the untyped λ-calculus.

A Category Theoretic Formulation for Engeler-style Models of the Untyped λ-Calculus

We give a category-theoretic formulation of Engeler-style models for the untyped λ-calculus. In order to do so, we exhibit an equivalence between distributive laws and extensions of one monad to the Kleisli category of another and explore the example of an arbitrary commutative monad together with the monad for commutative monoids. On Set as base category, the latter is the finite multiset monad. We exploit the self-duality of the category Rel, i.e., the Kleisli category for the powerset monad, and the category theoretic structures on it that allow us to build models of the untyped λ-calculus, yielding a variant of the Engeler model. We replace the monad for commutative monoids by that for idempotent commutative monoids, which, on Set, is the finite powerset monad. This does not quite yield a distributive law, so requires a little more subtlety, but, subject to that subtlety, it yields exactly the original Engeler construction.

Lazy Strong Normalization

Among all the reduction strategies for the untyped λ-calculus, the so called lazy β-evaluation is of particular interest due to its large applicability to functional programming languages (e.g. Haskell [Bird, R., “Introduction to Functional Programming using Haskell,” Series in Computer Science (2nd edition), Prentice Hall, (1998)]). This strategy reduces only redexes not inside a lambda abstraction.The lazy strongly β- normalizing terms are the λ-terms that don't have infinite lazy β-reduction sequences.This paper presents a logical characterization of lazy strongly β-normalizing terms using intersection types. This characterization, besides being interesting by itself, allows an interesting connection between call-by-name and call-by-value λ-calculus.In fact, it turns out that the class of lazy strongly β-normalizing terms coincides with that of call-by-value potentially valuable terms. This last class is of particular interest since it is a key notion for characterizing solvability in the call-by-value setting.

De Bruijn's syntax and reductional behaviour of λ-terms: the untyped case

In this paper, a notation influenced by de Bruijn's syntax of the λ-calculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than the classical notation. We define reduction modulo equivalence classes of terms up to the permutation of redexes in canonical forms and show that this reduction contains other notions of reductions in the literature including the σ-reduction of Regnier. We establish all the desirable properties of our reduction modulo equivalence classes for the untyped λ-calculus.