Control Flow Analysis for SF Combinator Calculus

Programs that transform other programs often require access to the internal structure of the program to be transformed. This is at odds with the usual extensional view of functional programming, as embodied by the lambda calculus and SK combinator calculus. The recently-developed SF combinator calculus offers an alternative, intensional model of computation that may serve as a foundation for developing principled languages in which to express intensional computation, including program transformation. Until now there have been no static analyses for reasoning about or verifying programs written in SF-calculus. We take the first step towards remedying this by developing a formulation of the popular control flow analysis 0CFA for SK-calculus and extending it to support SF-calculus. We prove its correctness and demonstrate that the analysis is invariant under the usual translation from SK-calculus into SF-calculus.Comment: In Proceedings VPT 2015, arXiv:1512.0221

Systems of combinatory logic related to Quine's ‘New Foundations’

AbstractSystems TRC and TRCU of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to Quine's set theory ‘New Foundations’ (NF) and the fragment NFU + Infinity of NF described by Jensen, respectively. Jensen demonstrated the consistency of NFU + Infinity relative to ZFC; the question of the consistency of NF remains open. TRC and TRCU are presented here as classical first-order theories, although they can be presented as equational theories; they are not constructive

Partial Applicative Theories and Explicit Substitutions

Systems based on theories with partial self-application are relevant to the formalization of constructive mathematics and as a logical basis for functional programming languages. In the literature they are either presented in the form of partial combinatory logic or the partial A calculus, and sometimes these two approaches are erroneously considered to be equivalent. In this paper we address some defects of the partial λ calculus as a constructive framework for partial functions. In particular, the partial λ calculus is not embeddable into partial combinatory logic and it lacks the standard recursion-theoretic model. The main reason is a concept of substitution, which is not consistent with a strongly intensional point of view. We design a weakening of the partial λ calculus, which can be embedded into partial combinatory logic. As a consequence, the natural numbers with partial recursive function application are a model of our system. The novel point will be the use of explicit substitutions, which have previously been studied in the literature in connection with the implementation of functional programming language

Effective lambda-models vs recursively enumerable lambda-theories

A longstanding open problem is whether there exists a non syntactical model of the untyped lambda-calculus whose theory is exactly the least lambda-theory (l-beta). In this paper we investigate the more general question of whether the equational/order theory of a model of the (untyped) lambda-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of lambda-calculus 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 l-beta or l-beta-eta. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning 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 minimum among all theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34

Categorical Realizability for Non-symmetric Closed Structures

In categorical realizability, it is common to construct categories of assemblies and categories of modest sets from applicative structures. These categories have structures corresponding to the structures of applicative structures. In the literature, classes of applicative structures inducing categorical structures such as Cartesian closed categories and symmetric monoidal closed categories have been widely studied. In this paper, we expand these correspondences between categories with structure and applicative structures by identifying the classes of applicative structures giving rise to closed multicategories, closed categories, monoidal bi-closed categories as well as (non-symmetric) monoidal closed categories. These applicative structures are planar in that they correspond to appropriate planar lambda calculi by combinatory completeness. These new correspondences are tight: we show that, when a category of assemblies has one of the structures listed above, the based applicative structure is in the corresponding class. In addition, we introduce planar linear combinatory algebras by adopting linear combinatory algebras of Abramsky, Hagjverdi and Scott to our planar setting, that give rise to categorical models of the linear exponential modality and the exchange modality on the non-symmetric multiplicative intuitionistic linear logic