First Class Call Stacks: Exploring Head Reduction

Weak-head normalization is inconsistent with functional extensionality in the call-by-name $\lambda$-calculus. We explore this problem from a new angle via the conflict between extensionality and effects. Leveraging ideas from work on the $\lambda$-calculus with control, we derive and justify alternative operational semantics and a sequence of abstract machines for performing head reduction. Head reduction avoids the problems with weak-head reduction and extensionality, while our operational semantics and associated abstract machines show us how to retain weak-head reduction's ease of implementation.Comment: In Proceedings WoC 2015, arXiv:1606.0583

Classical logic, continuation semantics and abstract machines

One of the goals of this paper is to demonstrate that denotational semantics is useful for operational issues like implementation of functional languages by abstract machines. This is exemplified in a tutorial way by studying the case of extensional untyped call-by-name λ-calculus with Felleisen's control operator 𝒞. We derive the transition rules for an abstract machine from a continuation semantics which appears as a generalization of the ¬¬-translation known from logic. The resulting abstract machine appears as an extension of Krivine's machine implementing head reduction. Though the result, namely Krivine's machine, is well known our method of deriving it from continuation semantics is new and applicable to other languages (as e.g. call-by-value variants). Further new results are that Scott's D∞-models are all instances of continuation models. Moreover, we extend our continuation semantics to Parigot's λμ-calculus from which we derive an extension of Krivine's machine for λμ-calculus. The relation between continuation semantics and the abstract machines is made precise by proving computational adequacy results employing an elegant method introduced by Pitts

Formalizing homogeneous language embeddings

The cost of implementing syntactically distinct Domain Specific Languages (DSLs) can be reduced by homogeneously embedding them in a host language in cooperation with its compiler. Current homogeneous embedding approaches either restrict the embedding of multiple DSLs in order to provide safety guarantees, or allow multiple DSLs to be embedded but force the user to deal with the interoperability burden. In this paper we present the m-calculus which allows parameterisable language embeddings to be specified and analysed. By reducing the problem to its core essentials we are able to show how multiple, expressive language embeddings can be defined in a homogeneous embedding context. We further show how variant calculi with safety guarantees can be defined

Infinitary λ\lambda-Calculi from a Linear Perspective (Long Version)

We introduce a linear infinitary $\lambda$-calculus, called $\ell\Lambda_{\infty}$, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applicative $\lambda$-calculus and is universal for computations over infinite strings. What is particularly interesting about $\ell\Lambda_{\infty}$, is that the refinement induced by linear logic allows to restrict both modalities so as to get calculi which are terminating inductively and productive coinductively. We exemplify this idea by analysing a fragment of $\ell\Lambda$ built around the principles of $\mathsf{SLL}$ and $\mathsf{4LL}$. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary $\lambda$-calculi

An Approach to Call-by-Name Delimited Continuations

International audienceWe show that a variant of Parigot's λμ-calculus, originally due to de Groote and proved to satisfy Böhm's theorem by Saurin, is canonically interpretable as a call-by-name calculus of delim- ited control. This observation is expressed using Ariola et al's call-by-value calculus of delimited control, an extension of λμ-calculus with delimited control known to be equationally equivalent to Danvy and Filinski's calculus with shift and reset. Our main result then is that de Groote and Saurin's variant of λμ-calculus is equivalent to a canonical call-by-name variant of Ariola et al's calculus. The rest of the paper is devoted to a comparative study of the call-by-name and call-by-value variants of Ariola et al's calculus, covering in particular the questions of simple typing, operational semantics, and continuation-passing-style semantics. Finally, we discuss the relevance of Ariola et al's calculus as a uniform framework for representing different calculi of delimited continuations, including "lazy" variants such as Sabry's shift and lazy reset calculus

Indexed linear logic and higher-order model checking

In recent work, Kobayashi observed that the acceptance by an alternating tree automaton A of an infinite tree T generated by a higher-order recursion scheme G may be formulated as the typability of the recursion scheme G in an appropriate intersection type system associated to the automaton A. The purpose of this article is to establish a clean connection between this line of work and Bucciarelli and Ehrhard's indexed linear logic. This is achieved in two steps. First, we recast Kobayashi's result in an equivalent infinitary intersection type system where intersection is not idempotent anymore. Then, we show that the resulting type system is a fragment of an infinitary version of Bucciarelli and Ehrhard's indexed linear logic. While this work is very preliminary and does not integrate key ingredients of higher-order model-checking like priorities, it reveals an interesting and promising connection between higher-order model-checking and linear logic.Comment: In Proceedings ITRS 2014, arXiv:1503.0437

Self-Reference, Biologic and the Structure of Reproduction

This paper concentrates on relationships of formal systems with biology. The paper is based on previous papers by the author. We have freely used texts of those papers where the formulations are of use, and we have extended the concepts and discussions herein considerably beyond the earlier work. We concentrate on formal systems not only for the sake of showing how there is a fundamental mathematical structure to biology, but also to consider and reconsider philosophical and phenomenological points of view in relation to natural science and mathematics. The relationship with phenomenology comes about in the questions that arise about the nature of the observer in relation to the observed that arise in philosophy, but also in science in the very act of determining the context and models upon which it shall be based.We examine the schema behind the reproduction of DNA. The DNA molecule consists of two interwound strands, the Watson Strand (W) and the Crick Strand (C). The two strands are bonded to each other via a backbone of base-pairings and these bonds can be broken by certain enzymes present in the cell. In reproduction of DNA the bonds between the two strands are broken and the two strands then acquire the needed complementary base molecules from the cellular environment to reconstitute each a separate copy of the DNA. At this level the situation can be described by a symbolism like this. DNA = -------> --------> = = DNA DNA. Here E stands for the environment of the cell. The first arrow denotes the separation of the DNA into the two strands. The second arrow denotes the action between the bare strands and the environment that leads to the production of the two DNA molecules. The paper considers and compares many formalisms for self-replication, including aspects of quantum formalism and the Temperley-Lieb algebra.Comment: LaTeX document, 71 pages, 33 figures. arXiv admin note: substantial text overlap with arXiv:quant-ph/020400