On streams that are finitely red

Mixing induction and coinduction, we study alternative definitions of streams being finitely red. We organize our definitions into a hierarchy including also some well-known alternatives in intuitionistic analysis. The hierarchy collapses classically, but is intuitionistically of strictly decreasing strength. We characterize the differences in strength in a precise way by weak instances of the Law of Excluded Middle

Completeness of resolution revisited

AbstractBy a novel argument we prove the completeness of (ground) resolution. The argument allows us to give the completeness proofs for various strategies of resolution in a uniform way, thus contributing to the insight into these strategies. For example, our exposition shows how the more efficient strategies can be derived from an analysis of the redundancies in the completeness proofs. Moreover, by using Zorn's Lemma in dealing with infinite sets of ground clauses, we obtain completeness proofs which are completely independent of the cardinality of both the language and the set of clauses. We discuss the set theoretic status of these results

Extensionality of simply typed logic programs

We set up a framework for the study of extensionality in the context of higher-order logic programming. For simply typed logic programs we propose a novel declarative semantics, consisting of a model class with a semi-computable initial model, and a notion of extensionality. We show that the initial model of a simply typed logic program, in case the program is extensional, collapses into a simple, set-theoretic representation. Given the undecidability of extensionality in general, we develop a decidable, syntactic criterion which is sufficient for extensionality. Some typical examples of higher-order logic programs are shown to be extensional

Modeling of many-body interactions between elastic spheres through symmetry functions

Simple models for spherical particles with a soft shell have been shown to self-assemble into numerous crystal phases and even quasicrystals. However, most of these models rely on a simple pairwise interaction, which is usually a valid approximation only in the limit of small deformations, i.e. low densities. In this work, we consider a many-body yet simple model for the evaluation of the elastic energy associated with the deformation of a spherical shell. The resulting energy evaluation, however, is relatively expensive for direct use in simulations. We significantly reduce the associated numerical cost by fitting the potential using a set of symmetry functions. We propose a method for selecting a suitable set of symmetry functions that capture the most relevant features of the particle environment in a systematic manner. The fitted interaction potential is then used in Monte Carlo simulations to draw the phase diagram of the system in two dimensions. The system is found to form both a fluid and a hexagonal crystal phase.Comment: 10 pages, 9 figure