Henk Barendregt

Term graph rewriting

Graph rewriting (also called reduction) as defined in Wadsworth [1971] was introduced in order to be able to give a more efficient implementation of functional programming languages in the form of lambda calculus or term rewrite systems: identical subterms are shared using pointers.

Greater efficiency in attentional processing related to mindfulness meditation.

In this study, attentional processing in relation to mindfulness meditation was investigated. Since recent studies have suggested that mindfulness meditation may induce improvements in attentional processing, we have tested 20 expert mindfulness meditators in the attention network test. Their performance was compared to that of 20 age- and gender-matched controls. In addition to attentional network analyses, overall attentional processing was analysed by means of efficiency scores (i.e., accuracy controlled for reaction time). Better orienting and executive attention (reflected by smaller differences in either reaction time or error score, respectively) were observed in the mindfulness meditation group. Furthermore, extensive mindfulness meditation appeared to be related to a reduction of the fraction of errors for responses with the same reaction time. These results provide new insights into differences in attentional processing related to mindfulness meditation and suggest the possibility of increasing the efficiency in attentional processing by extensive mental training.

The regulation of brain states by neuroactive substances distributed via the cerebrospinal fluid; a review

The cerebrospinal fluid (CSF) system provides nutrients to and removes waste products from the brain. Recent findings suggest, however, that in addition, the CSF contains message molecules in the form of actively released neuroactive substances. The concentrations of these vary between locations, suggesting they are important for the changes in brain activity that underlie different brain states, and induce different sensory input and behavioral output relationships.The cranial CSF displays a rapid caudally-directed ventricular flow followed by a slower rostrally-directed subarachnoid flow (mainly towards the cribriform plate and from there into the nasal lymphatics). Thus, many brain areas are exposed to and can be influenced by substances contained in the CSF. In this review we discuss the production and flow of the CSF, including the mechanisms involved in the regulation of its composition. In addition, the available evidence for the release of neuropeptides and other neuroactive substances into the CSF is reviewed, with particular attention to the selective effects of these on distant downstream receptive brain areas. As a conclusion we suggest that (1) the flowing CSF is involved in more than just nutrient and waste control, but is also used as a broadcasting system consisting of coordinated messages to a variety of nearby and distant brain areas; (2) this special form of volume transmission underlies changes in behavioral states.

The type free lambda calculus

Publisher Summary The λ-calculus represents a class of (partial) functions (λ-definable functions) on the integers that turns out to be the class of (partial) recursive functions. The equivalence between the Turing computable functions and the general recursive functions was originally proved via the λ-calculus: the general recursive functions are exactly the λ-definable functions as are the Turing computable functions. The equivalence between the λ-definable functions and the recursive functions was one of the arguments used by Church to defend his thesis proposing the identification of the intuitive class of effectively computable functions with the class of recursive functions; in fact one can give arguments for the Churchs superthesis that states that for the functions involved this identification preserves the intensional character—that is, process of computation.

Autarkic Computations in Formal Proofs

Formal proofs in mathematics and computer science are being studied because these objects can be verified by a very simple computer program. An important open problem is whether these formal proofs can be generated with an effort not much greater than writing a mathematical paper in, say, LATEX. Modern systems for proof development make the formalization of reasoning relatively easy. However, formalizing computations in such a manner that the results can be used in formal proofs is not immediate. In this paper we show how to obtain formal proofs of statements such as Prime(61) in the context of Peano arithmetic or (x+1)(x+1)=x2+2x+1 in the context of rings. We hope that the method will help bridge the gap between the efficient systems of computer algebra and the reliable systems of proof development.

Needed reduction and spine strategies for the lambda calculus

Abstract A redex R in a lambda-term M is called needed if in every reduction of M to normal form (some residual of) R is contracted. Among others the following results are proved: 1. R is needed in M iff R is contracted in the leftmost reduction path of M. 2. Let R : M0 →M1 → M2 → … reduce redexes Ri: Mi → Mi+1, and have the property that ∀i.∃j≥i.Rj is needed in Mj. Then R is normalising, i.e., if M0 has a normal form, then R is finite and terminates at that normal form. 3. Neededness is an undecidable property, but has several efficiently decidable approximations, various versions of the so-called spine redexes.

The impact of the lambda calculus in logic and computer science

One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.

Towards an intermediate language based on graph rewriting

Lean is an experimental language for specifying computations in terms of graph rewriting. It is based on an alternative to Term Rewriting Systems (TRS) in which the terms are replaced by graphs. Such a Graph Rewriting System (GRS) consists of a set of graph rewrite rules which specify how a graph may be rewritten. Besides supporting functional programming, Lean also describes imperative constructs and allows the manipulation of cyclic graphs. Programs may exhibit non-determinism as well as parallelism. In particular, Lean can serve as an intermediate language between declarative languages and machine architectures, both sequential and parallel.

Mindfulness meditation associated with alterations in bottom-up processing: Psychophysiological evidence for reduced reactivity

Mental training by meditation has been related to changes in high-level cognitive functions that involve top-down processing. The aim of this study was to investigate whether the practice of meditation is also related to alterations in low-level, bottom-up processing. Therefore, intersensory facilitation (IF) effects in a group of mindfulness meditators (MM) were compared to IF effects in an age- and gender-matched control group. Smaller and even absent IF effects were found in the MM group, which suggests that changes in bottom-up processing are associated with MM. Furthermore, reduced interference of a visual warning stimulus with the IF effects was found, which suggests an improved allocation of attentional resources in mindfulness meditators, even across modalities.

The challenge of computer mathematics

Progress in the foundations of mathematics has made it possible to formulate all thinkable mathematical concepts, algorithms and proofs in one language and in an impeccable way. This is not in spite of, but partially based on the famous results of Gödel and Turing. In this way statements are about mathematical objects and algorithms, proofs show the correctness of statements and computations, and computations are dealing with objects and proofs. Interactive computer systems for a full integration of defining, computing and proving are based on this. The human defines concepts, constructs algorithms and provides proofs, while the machine checks that the definitions are well formed and the proofs and computations are correct. Results formalized so far demonstrate the feasibility of this ‘computer mathematics’. Also there are very good applications. The challenge is to make the systems more mathematician-friendly, by building libraries and tools. The eventual goal is to help humans to learn, develop, communicate, referee and apply mathematics.