In this paper we describe the extraction of “efficient” recursion schemes from proofs of well-founded induction principles. This is part of a larger methodology; when these well-founded induction principles are used in proofs, the structure of the program extracted from the proof is determined by the recursion scheme inhabiting the induction principle. Our development is based on Paulson’s pape...

We identify the puncture operator in c = 1 Liouville gravity as the discrete state with spin J = 1 2. The correlation functions involving this operator satisfy the recursion relation which is characteristic in topological gravity. We derive the recursion relation involving the puncture operator by the operator product expansion. Multiple point correlation functions are determined recursively fr...

In a sequence of papers, Moschovakis developed a class of languages of recursion as a new approach to the mathematical notion of algorithm and development of computational semantics, e.g., see Moschovakis [7], for FLR, and Moschovakis [8], for Lar. In particular, the language and theory of acyclic recursion Lar is intended for modeling the logical concepts of meaning and synonymy, from the pers...

It is well known that members of families of polynomials, that are orthogonal with respect to an inner product determined by a nonnegative measure on the real axis, satisfy a three-term recursion relation. Analogous recursion formulas are available for orthogonal Laurent polynomials with a pole at the origin. This paper investigates recursion relations for orthogonal rational functions with arb...

We describe various computational models based initially, but not exclusively, on that of the Turing machine, that are generalized to allow for transfinitely many computational steps. Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals rather than numbers. We outline the connections between such models and the older theories of recur...

We show that Spector’s “restricted” form of bar recursion is sufficient (over system T ) to define Spector’s search functional. This new result is then used to show that Spector’s restricted form of bar recursion is in fact as general as the supposedly more general form of bar recursion. Given that these two forms of bar recursion correspond to the (explicitly controlled) iterated products of s...

This is an elementary expository article regarding the application of Kleene’s Recursion Theorems in making definitions by recursion. Whereas the Second Recursion Theorem (SRT) is applicable in a first-order setting, the First Recursion Theorem (FRT) requires a higherorder setting. In some cases both theorems are applicable, but one is stronger than the other: the FRT always produces least fixe...

We study computer virology from an abstract point of view. Viruses and worms are self-replicating programs, whose definitions are based on Kleene’s second recursion theorem. We introduce a notion of delayed recursion that we apply to both Kleene’s second recursion theorem and Smullyan’s double recursion theorem. This leads us to define four classes of viruses, two of them being polymorphic. The...

We explore recursion theory on the reals, the analog counterpart of recursive function theory. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions, such as composition and various forms of differential equations. We define classes of real recursive functions, in a manner similar to the classical approach in re...

Britto, Cachazo and Feng have recently derived a recursion relation for tree-level scattering amplitudes in Yang-Mills. This relation has a bilinear structure inherited from factorisation on multi-particle poles of the scattering amplitudes – a rather generic feature of field theory. Motivated by this, we propose a new recursion relation for scattering amplitudes of gravitons at tree level. Usi...