A Tutorial on the Universality and Expressiveness of Fold

In functional programming, fold is a standard operator that encapsulates a simple pattern of recursion for processing lists. This article is a tutorial on two key aspects of the fold operator for lists. First of all, we emphasize the use of the universal property of fold both as a proof principle that avoids the need for inductive proofs, and as a definition principle that guides the transformation of recursive functions into definitions using fold. Secondly, we show that even though the pattern of recursion encapsulated by fold is simple, in a language with tuples and functions as first-class values the fold operator has greater expressive power than might first be expected.

Fold and Unfold for Program Semantics

In this paper we explain how recursion operators can be used to structure and reason about program semantics within a functional language. In particular, we show how the recursion operator fold can be used to structure denotational semantics, how the dual recursion operator unfold can be used to structure operational semantics, and how algebraic properties of these operators can be used to reason about program semantics. The techniques are explained with the aid of two main examples, the first concerning arithmetic expressions, and the second concerning Milner's concurrent language CCS. The aim of the paper is to give functional programmers new insights into recursion operators, program semantics, and the relationships between them.

DnaG interacts with a linker region that joins the N- and C-domains of DnaB and induces the formation of 3-fold symmetric rings

Loading of the replicative ring helicase onto the origin of replication (oriC) is the final outcome of a well coordinated series of events that collectively constitute a primosomal cascade. Once the ring helicase is loaded, it recruits the primase and signals the switch to the polymerization mode. The transient nature of the helicase-primase (DnaB-DnaG) interaction in the Escherichia coli system has hindered our efforts to elucidate its structure and function. Taking advantage of the stable DnaB-DnaG complex in Bacillus stearothermophilus, we have reviewed conflicting mutagenic data from other bacterial systems and shown that DnaG interacts with the flexible linker that connects the N- and C-terminal domains of DnaB. Furthermore, atomic force microscopy (AFM) imaging experiments show that binding of the primase to the helicase induces predominantly a 3-fold symmetric morphology to the hexameric ring. Overall, three DnaG molecules appear to interact with the hexameric ring helicase but a small number of complexes with two and even one DnaG molecule bound to DnaB were also detected. The structural/functional significance of these data is discussed and a speculative structural model for this complex is suggested.