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

A formally exact field theory for classical systems at equilibrium

We propose a formally exact statistical field theory for describing classical fluids with ingredients similar to those introduced in quantum field theory. We consider the following essential and related problems : i) how to find the correct field functional (Hamiltonian) which determines the partition function, ii) how to introduce in a field theory the equivalent of the indiscernibility of particles, iii) how to test the validity of this approach. We can use a simple Hamiltonian in which a local functional transposes, in terms of fields, the equivalent of the indiscernibility of particles. The diagrammatic expansion and the renormalization of this term is presented. This corresponds to a non standard problem in Feynman expansion and requires a careful investigation. Then a non-local term associated with an interaction pair potential is introduced in the Hamiltonian. It has been shown that there exists a mapping between this approach and the standard statistical mechanics given in terms of Mayer function expansion. We show on three properties (the chemical potential, the so-called contact theorem and the interfacial properties) that in the field theory the correlations are shifted on non usual quantities. Some perspectives of the theory are given.Comment: 20 pages, 8 figure

Programs as Data Structures in λSF-Calculus

© 2016 The Author(s) Lambda-SF-calculus can represent programs as closed normal forms. In turn, all closed normal forms are data structures, in the sense that their internal structure is accessible through queries defined in the calculus, even to the point of constructing the Goedel number of a program. Thus, program analysis and optimisation can be performed entirely within the calculus, without requiring any meta-level process of quotation to produce a data structure. Lambda-SF-calculus is a confluent, applicative rewriting system derived from lambda-calculus, and the combinatory SF-calculus. Its superior expressive power relative to lambda-calculus is demonstrated by the ability to decide if two programs are syntactically equal, or to determine if a program uses its input. Indeed, there is no homomorphism of applicative rewriting systems from lambda-SF-calculus to lambda-calculus. Program analysis and optimisation can be illustrated by considering the conversion of a programs to combinators. Traditionally, a program p is interpreted using fixpoint constructions that do not have normal forms, but combinatory techniques can be used to block reduction until the program arguments are given. That is, p is interpreted by a closed normal form M. Then factorisation (by F) adapts the traditional account of lambda-abstraction in combinatory logic to convert M to a combinator N that is equivalent to M in the following two senses. First, N is extensionally equivalent to M where extensional equivalence is defined in terms of eta-reduction. Second, the conversion is an intensional equivalence in that it does not lose any information, and so can be reversed by another definable conversion. Further, the standard optimisations of the conversion process are all definable within lambda-SF-calculus, even those involving free variable analysis. Proofs of all theorems in the paper have been verified using the Coq theorem prover

Net versus combinatory effects of firm and industry antecedents of sales growth

This study examines antecedents of sales growth using a two-step mixed-method approach including analyses of net effects and combinatory effects. Based on a sample of 453 respondents from manufacturing and service firms, this article shows how the combination of structural equation modeling (SEM) and fuzzy set Qualitative Comparative Analysis (fsQCA) provides more detailed insights into the causal patterns of factors to explain sales growth. This article contributes to the extant literature by highlighting fsQCA as a useful method to analyze complex causality (specifically combinatory effects of antecedent conditions) and by discussing options regarding how this approach can be used to complement findings from conventional causal data analysis procedures that analyze net effects

An Intensional Concurrent Faithful Encoding of Turing Machines

The benchmark for computation is typically given as Turing computability; the ability for a computation to be performed by a Turing Machine. Many languages exploit (indirect) encodings of Turing Machines to demonstrate their ability to support arbitrary computation. However, these encodings are usually by simulating the entire Turing Machine within the language, or by encoding a language that does an encoding or simulation itself. This second category is typical for process calculi that show an encoding of lambda-calculus (often with restrictions) that in turn simulates a Turing Machine. Such approaches lead to indirect encodings of Turing Machines that are complex, unclear, and only weakly equivalent after computation. This paper presents an approach to encoding Turing Machines into intensional process calculi that is faithful, reduction preserving, and structurally equivalent. The encoding is demonstrated in a simple asymmetric concurrent pattern calculus before generalised to simplify infinite terms, and to show encodings into Concurrent Pattern Calculus and Psi Calculi.Comment: In Proceedings ICE 2014, arXiv:1410.701

A Frobenius Algebraic Analysis for Parasitic Gaps

The interpretation of parasitic gaps is an ostensible case of non-linearity in natural language composition. Existing categorial analyses, both in the typelogical and in the combinatory traditions, rely on explicit forms of syntactic copying. We identify two types of parasitic gapping where the duplication of semantic content can be confined to the lexicon. Parasitic gaps in adjuncts are analysed as forms of generalized coordination with a polymorphic type schema for the head of the adjunct phrase. For parasitic gaps affecting arguments of the same predicate, the polymorphism is associated with the lexical item that introduces the primary gap. Our analysis is formulated in terms of Lambek calculus extended with structural control modalities. A compositional translation relates syntactic types and derivations to the interpreting compact closed category of finite dimensional vector spaces and linear maps with Frobenius algebras over it. When interpreted over the necessary semantic spaces, the Frobenius algebras provide the tools to model the proposed instances of lexical polymorphism.Comment: SemSpace 2019, to appear in Journal of Applied Logic

Constructive set theory and Brouwerian principles

The paper furnishes realizability models of constructive Zermelo-Fraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous choice (CC), the fan theorem (FT), and monotone bar induction (BIM), and thereby determines the proof-theoretic strength of CZF augmented by these principles. The upshot is that CZF+CC+FT possesses the same strength as CZF, or more precisely, that CZF+CC+FTis conservative over CZF for 02 statements of arithmetic, whereas the addition of a restricted version of bar induction to CZF (called decidable bar induction, BID) leads to greater proof-theoretic strength in that CZF+BID proves the consistency of CZF