Projections for infinitary rewriting

Proof terms in term rewriting are a representation means for reduction sequences, and more in general for contraction activity, allowing to distinguish e.g simultaneous from sequential reduction. Proof terms for finitary, first-order, left-linear term rewriting are described in the Terese book, chapter 8. In a previous work, we defined an extension of the finitary proof-term formalism, that allows to describe contractions in infinitary first-order term rewriting, and gave a characterisation of permutation equivalence. In this work, we discuss how projections of possibly infinite rewrite sequences can be modeled using proof terms. Again, the foundation is a characterisation of projections for finitary rewriting described in Terese, Section 8.7. We extend this characterisation to infinitary rewriting and also refine it, by describing precisely the role that structural equivalence plays in the development of the notion of projection. The characterisation we propose yields a definite expression, i.e. a proof term, that describes the projection of an infinitary reduction over another. To illustrate the working of projections, we show how a common reduct of a (possibly infinite) reduction and a single step that makes part of it can be obtained via their respective projections. We show, by means of several examples, that the proposed definition yields the expected behavior also in cases beyond those covered by this result. Finally, we discuss how the notion of limit is used in our definition of projection for infinite reduction

Unique normal forms for lambda calculus with surjective pairing

AbstractWe consider the equational theory λπ of λ-calculus extended with constants π, π0, π1 and axioms for surjective pairing: π0(πXY) = X, π1(πXY) = Y, π(π0X)(π1X) = X. Two reduction systems yielding the equality of λπ are introduced; the first is not confluent and, for the second, confluence is an open problem. It is shown, however, that in both systems each term possessing a normal form has a unique normal form. Some additional properties and problems in the syntactical analysis of λπ and the corresponding reduction systems are discussed

Local Termination: theory and practice

The characterisation of termination using well-founded monotone algebras has been a milestone on the way to automated termination techniques, of which we have seen an extensive development over the past years. Both the semantic characterisation and most known termination methods are concerned with global termination, uniformly of all the terms of a term rewriting system (TRS). In this paper we consider local termination, of specific sets of terms within a given TRS. The principal goal of this paper is generalising the semantic characterisation of global termination to local termination. This is made possible by admitting the well-founded monotone algebras to be partial. We also extend our approach to local relative termination. The interest in local termination naturally arises in program verification, where one is probably interested only in sensible inputs, or just wants to characterise the set of inputs for which a program terminates. Local termination will be also be of interest when dealing with a specific class of terms within a TRS that is known to be non-terminating, such as combinatory logic (CL) or a TRS encoding recursive program schemes or Turing machines. We show how some of the well-known techniques for proving global termination, such as stepwise removal of rewrite rules and semantic labelling, can be adapted to the local case. We also describe transformations reducing local to global termination problems. The resulting techniques for proving local termination have in some cases already been automated. One of our applications concerns the characterisation of the terminating S-terms in CL as regular language. Previously this language had already been found via a tedious analysis of the reduction behaviour of S-terms. These findings have now been vindicated by a fully automated and verified proof

Effects of Maternal Obesity and Gestational Diabetes Mellitus on the Placenta: Current Knowledge and Targets for Therapeutic Interventions

Obesity and gestational diabetes mellitus (GDM) are becoming more common among pregnant women worldwide and are individually associated with a number of placenta-mediated obstetric complications, including preeclampsia, macrosomia, intrauterine growth restriction and stillbirth. The placenta serves several functions throughout pregnancy and is the main exchange site for the transfer of nutrients and gas from mother to fetus. In pregnancies complicated by maternal obesity or GDM, the placenta is exposed to environmental changes, such as increased inflammation and oxidative stress, dyslipidemia, and altered hormone levels. These changes can affect placental development and function and lead to abnormal fetal growth and development as well as metabolic and cardiovascular abnormalities in the offspring. This review aims to summarize current knowledge on the effects of obesity and GDM on placental development and function. Understanding these processes is key in developing therapeutic interventions with the goal of mitigating these effects and preventing future cardiovascular and metabolic pathology in subsequent generations

Completing partial combinatory algebras with unique head-normal forms

In this note, we prove that having unique head-normal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CL-terms as well as the pca of natural numbers with partial recursive function application can be extended to total combinatory algebras