Database Query Languages Embedded in the Typed Lambda Calculus

We investigate the expressive power of the typedλ-calculus when expressing computations over finite structures, i.e., databases. We show that the simply typedλ-calculus can express various database query languages such as the relational algebra, fixpoint logic, and the complex object algebra. In our embeddings, inputs and outputs areλ-terms encoding databases, and a program expressing a query is aλ-term which types when applied to an input and reduces to an output. Our embeddings have the additional property that PTIME computable queries are expressible by programs that, when applied to an input, reduce to an output in a PTIME sequence of reduction steps. Under our database input-output conventions, all elementary queries are expressible in the typedλ-calculus and the PTIME queries are expressible in the order-5 (order-4) fragment of the typedλ-calculus (with equality).     

Infinite intersection types

A type theory with infinitary intersection and union types for an extension of the λ-calculus is introduced. Types are viewed as upper closed subsets of a Scott domain and intersection and union type constructors are interpreted as the set-theoretic intersection and union, respectively, even when they are not finite. The assignment of types to λ-terms extends naturally the basic type assignment system. We prove soundness and completeness using a generalization of Abramsky’s finitary logic of domains. Finally, we apply the framework to applicative transition systems, obtaining a sound a complete infinitary intersection type assignment system for the lazy λ-calculus.     

Translating Combinatory Reduction Systems into the Rewriting Calculus

The last few years have seen the development of the rewriting calculus (or rho-calculus, ρCal) that extends first order term rewriting and λ-calculus. The integration of these two latter formalisms has been already handled either by enriching first-order rewriting with higher-order capabilities, like in the Combinatory Reduction Systems, or by adding to λ-calculus algebraic features. The different higher-order rewriting systems and the rewriting calculus share similar concepts and have similar applications, and thus, it seems natural to compare these formalisms. We analyze in this paper the relationship between the Rewriting Calculus and the Combinatory Reduction Systems and we present a translation of CRS-terms and rewrite rules into rho-terms and we show that for any CRS-reduction we have a corresponding rho-reduction.     

Simple Easy Terms

We illustrate the use of intersection types as a semantic tool for proving easiness result on λ-terms. We single out the notion of simple easiness for λ-terms as a useful semantic property for building filter models with special purpose features. Relying on the notion of easy intersection type theory, given λ-terms M and E, with E simple easy, we successfully build a filter model which equates interpretation of M and E, hence proving that simple easiness implies easiness. We finally prove that a class of λ-terms generated by ω₂ω₂ are simple easy, so providing alternative proof of easiness for them.     

Revisiting the notion of function

Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both processes were implemented in Frege’s Begriffschrift, Russell’s Ramified Type Theory, and the λ-calculus (originally introduced by Church) showing that the λ-calculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the λ-calculus and we show that various logical constructs (e.g., let expressions and definitions and the use of parameters in mathematics), can be seen as forms of the missing part of functionalisation. Our study of the function concept leads to: (a) an extension of the Barendregt cube [4] with all of definitions, Π-reduction and explicit substitutions giving all their advantages in one system; and (b) a natural refinement of the cube with parameters. We show that in the refined Barendregt cube, systems like A , LF, and ML, can be described more naturally and accurately than in the original cube.     

A Symmetric Lambda Calculus for Classical Program Extraction

We introduce aλ-calculus with symmetric reduction rules and “classical” types, i.e., types corresponding to formulas of classical propositional logic. The strong normalization property is proved to hold for such a calculus, as well as for its extension to a system equivalent to Peano arithmetic. A theorem on the shape of terms in normal form is also proved, making it possible to get recursive functions out of proofs ofΠ⁰₂formulas, i.e., those corresponding to program specifications.     

Interaction Nets vs. the ρ-calculus: Introducing Bigraphical Nets

The ρ-calculus generalises both term rewriting and the λ-calculus in a uniform framework. Interaction nets are a form of graph rewriting which proved most successful in understanding the dynamics of the λ-calculus, the prime example being the implementation of optimal β-reduction. It is thus natural to study interaction net encodings of the ρ-calculus as a first step towards the definition of efficient reduction strategies. We give two interaction net encodings which bring a new understanding to the operational semantics of the ρ-calculus; however, these encodings have some drawbacks and to overcome them we introduce bigraphical nets—a new paradigm of computation inspired by Lafont's interactions nets and Milner's bigraphs.     

On the Jacopini Technique

The general concern of the Jacopini technique is the question: “Is it consistent to extend a given lambda calculus with certain equations?” The technique was introduced by Jacopini in 1975 in his proof that in the untyped lambda calculusΩis easy, i.e.,Ωcan be assumed equal to any other (closed) term without violating the consistency of the lambda calculus. The presentations of the Jacopini technique that are known from the literature are difficult to understand and hard to generalise. In this paper we generalise the Jacopini technique for arbitrary lambda calculi. We introduce the concept ofproof-replaceabilityby which the structure of the technique is simplified considerably. We illustrate the simplicity and generality of our formulation of the technique with some examples. We apply the Jacopini technique to theλμ-calculus, and we prove a general theorem concerning the consistency of extensions of theλμ-calculus of a certain form. Many well known examples (e.g., the easiness ofΩ) are immediate consequences of this general theorem.     

An Extension of System F with Subtyping

System F is a well-known typed λ-calculus with polymorphic types, which provides a basis for polymorphic programming languages. We study an extension of F, called F_<: (pronounced ef-sub), that combines parametric polymorphism with subtyping. The main focus of the paper is the equational theory of F_<:, which is related to PER models and the notion of parametricity. We study some categorical properties of the theory when restricted to closed terms, including interesting categorical isomorphisms. We also investigate proof-theoretical properties, such as the conservativity of typing judgments with respect to F. We demonstrate by a set of examples how a range of constructs may be encoded in F_<:. These include record operations and subtyping hierarchies that are related to features of object-oriented languages.     

Term Collections in λ and ρ-calculi

The ρ-calculus generalises term rewriting and the λ-calculus by defining abstractions on arbitrary patterns and by using a pattern-matching algorithm which is a parameter of the calculus. In particular, equational theories that do not have unique principal solutions may be used. In the latter case, all the principal solutions of a matching problem are stored in a “structure” that can also be seen as a collection of terms.Motivated by the fact that there are various approaches to the definition of structures in the ρ-calculus, we study in this paper a version of the λ-calculus with term collections.The contributions of this work include a new syntax and operational semantics for a λ-calculus with term collections, which is related to the λ-calculi with strict parallel functions studied by Boudol and Dezani et al. and a proof of the confluence of the β-reduction relation defined for the calculus (which is a suitable extension of the standard rule of β-reduction in the λ-calculus).     

Completeness of Continuation Models for λμ-Calculus

We show that a certain simple call-by-name continuation semantics of Parigot's λ_μ-calculus is complete. More precisely, for every λ_μ-theory we construct a cartesian closed category such that the ensuing continuation-style interpretation of λ_μ, which maps terms to functions sending abstract continuations to responses, is full and faithful. Thus, any λ_μ-category in the sense of L. Ong (1996, in “Proceedings of LICS '96,” IEEE Press, New York) is isomorphic to a continuation model (Y. Lafont, B. Reus, and T. Streicher, “Continuous Semantics or Expressing Implication by Negation,” Technical Report 93-21, University of Munich) derived from a cartesian-closed category of continuations. We also extend this result to a later call-by-value version of λ_μ developed by C.-H. L. Ong and C. A. Stewart (1997, in “Proceedings of ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Paris, January 1997,” Assoc. Comput. Mach. Press, New York).     

Uniform provability realization of intuitionistic logic, modality and λ-terms

The intended meaning of intuitionistic logic is explained by the Brouwer-Heyting-Kolmogorov (BHK) provability semantics which informally defines intuitionistic truth as provability and specifies the intuitionistic connectives via operations on proofs. The problem of finding an adequate formalization of the provability semantics and establishing the completeness of the intuitionistic logic Int was first raised by Gödel in 1933. This question turned out to be a part of the more general problem of the intended realization for Gödel's modal logic of provability S4 which was open since 1933. In this tutorial talk we present a provability realization of Int and S4 that solves both of these problems. We describe the logic of explicit provability (LP) with the atoms “t is a proof of F” and establish that every theorem of S4 admits a reading in LP as the statement about operations on proofs. Moreover, both S4 and Int are shown to be complete with respect to this realization. In addition, LP subsumes the λ-calculus, modal λ-calculus, combinatory logic and provides a uniform provability realization of modality and λ-terms.     

Congruence of Bisimulation in a Non-Deterministic Call-By-Need Lambda Calculus

We present a call-by-need λ-calculus λ_ND with an erratic non-deterministic operator pick and a non-recursive let. A definition of a bisimulation is given, which has to be based on a further calculus named λ_≈, since the naïve bisimulation definition is useless. The main result is that bisimulation in λ_≈ is a congruence and coincides with the contextual equivalence. The proof is a non-trivial extension of Howe's method. This might be a step towards defining useful bisimulation relations and proving them to be congruences in calculi that extend the λ_ND-calculus.     

Package Duplication in Interaction Nets and Weak Head Reduction in the lambda-calculus

We present a simple implementation of weak head reduction in the λ-calculus with interaction nets using package duplication.     

A Rewriting Calculus for Cyclic Higher-order Term Graphs

Introduced at the end of the nineties, the Rewriting Calculus (ρ-calculus, for short) is a simple calculus that fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the obtained structured results are first class objects of the calculus. The evaluation mechanism, generalizing beta-reduction, strongly relies on term matching in various theories.In this paper we propose an extension of the ρ-calculus, handling graph like structures rather than simple terms. The transformations are performed by explicit application of rewrite rules as first class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.The calculus over terms is naturally generalized by using unification constraints in addition to the standard ρ-calculus matching constraints. This therefore provides us with the basics for a natural extension of an explicit substitution calculus to term graphs. Several examples illustrating the introduced concepts are given.     

Discrimination by Parallel Observers: The Algorithm

The main result of the paper is a constructive proof of the following equivalence: two pureλ-terms are observationally equivalent in the lazy concurrentλ-calculusiffthey have the same Lévy–Longo trees. An algorithm which allows to build a context discriminating any two pureλ-terms with different Lévy–Longo trees is described. It follows that contextual equivalence coincides with behavioural equivalence (bisimulation) as considered by Sangiorgi. Another consequence is that the discriminating power of concurrent lambda contexts is the same as that of Boudol–Laneve's contexts with multiplicities.     

On the representation of McCarthy's in the -calculus

We study the encoding of , the call-by-name λ-calculus enriched with McCarthy's amb operator, into the π-calculus. Semantically, amb is a challenging operator, for the fairness constraints that it expresses. We prove that, under a certain interpretation of divergence in the λ-calculus (weak divergence), a faithful encoding is impossible. However, with a different interpretation of divergence (strong divergence), the encoding is possible, and for this case we derive results and coinductive proof methods to reason about that are similar to those for the encoding of pure λ-calculi. We then use these methods to derive the most important laws concerning amb. We take bisimilarity as behavioural equivalence on the π-calculus, which sheds some light on the relationship between fairness and bisimilarity.     

Descendants and Origins in Term Rewriting

In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first-order term rewriting and λ-calculus, their finitary as well as their infinitary variants. A recurrent theme is the parallel moves lemma. Next to the classical notion of descendant, we introduce an extended version, known as origin tracking. Origin tracking has many applications. Here it is employed to give new proofs of three classical theorems: the genericity lemma in λ-calculus, the theorem of Huet and Lévy on needed reductions in first-order term rewriting, and Berry's sequentiality theorem in (infinitary) λ-calculus.     

Developing (Meta)Theory of λ-calculus in the Theory of Contexts

We present a case study on the formal development of a non trivial (meta)theory in the Theory of Contexts using the Coq proof assistant. The methodology underlying the Theory of Contexts for reasoning on systems presented in HOAS is based on an axiomatic syntactic standpoint. We feel that one of the main advantages of this approach, is that it requires a very low logical overhead.The object system we focus on is the lazy, call-by-name λ-calculus (λ_cbn), both untyped and simply typed. We will see that the formal, fully detailed development of the theory of (λ_cbn) in the Theory of Contexts introduces a small, sustainable overhead with respect to the proofs “on the paper”. Moreover, this will allow for comparison with similar case studies developed in other approaches to the metatheoretical reasoning in higher-order abstract syntax.