We propose a progressive approach to simplify teaching recursion. It is introduced by means of three instances of increasing level of difficulty. First, recursion is introduced in a “pure” form as it is found in formal grammars; its accompanying mechanisms are related to rewriting. Later, a functional programming language allows defining recursive functions and recursive data types; its mechani...

Everybody knows what recursion is and how it should be implemented in a procedural language. The first analysis of recursion implementations is due Kent Beck [Beck92], but no formalization was given. The other work on an OOP implementation of the subject is due to Boby Wolf [Wolf97], but his work only deals with one aspect of the problem: Structural Recursion. Structural Recursion is a particul...

For general information on bar recursion the reader should consult the papers of Spector [8], where it was introduced, Howard [2] and Tait [11]. In this note we shall prove that the terms of Godel's theory T (in its extensional version of Spector [8]) are closed under the rule BRo•1 of bar recursion of types 0 and 1. Our method of proof is based on the notion of an infinite term introduced by T...

We have developed an algebraic framework for the study of recursion. For immediate linear recursion, a Horn clause is represented by a relational algebra operator. We show that the set of all such operators forms a closed semiring. In this formalism, query answering corresponds to solving a linear equation. For the first time, we are able to have the query answer expressed in an explicit algebr...

Polarized SILL is a programming language that combines functional with session-typed message-passing concurrency. It features general recursion; code and channel transmission; synchronous asynchronous communication. To reason about programs, we develop the first program equivalence framework based on observable communications. We give meaning to programs using an observed communication semantic...

Novice Computer Science (CS) students struggle learning recursion for reasons such as unfamiliarity with recursive thinking and difficulty in visualizing program execution. Many tasks in CS require a thorough understanding of recursion. We introduce the recursion module of ChiQat-Tutor, an environment for learning CS algorithms and data structures. ChiQat-Tutor uses the pedagogical tool of Recu...

We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy for quantum field theory on noncommutative spaces grasped as complex matrix model, obeys topological recursion of Chekhov, Eynard and Orantin. By introducing two families correlation functions, one corresponding to meromorphic differentials $\omega_{g,n}$ recursion, we obtain Dyson-Schwinger equations eventually ...

This note reexamines Spector’s remarkable computational interpretation of full classical analysis. Spector’s interpretation makes use of a rather abstruse recursion schema, so-called bar recursion, used to interpret the double negation shift DNS. In this note bar recursion is presented as a generalisation of a simpler primitive recursive functional needed for the interpretation of a finite (int...

We give a recursion-theoretic characterization of the complexity classes NC for k ≥ 1. In the spirit of implicit computational complexity, it uses no explicit bounds in the recursion and also no separation of variables is needed. It is based on three recursion schemes, one corresponds to time (time iteration), one to space allocation (explicit structural recursion) and one to internal computati...

Recursion is a simple but powerful programming technique used extensively in functional languages. For a number of reasons, however, it is often desirable to use structured forms of recursion in programming, encapsulated in so-called recursion operators, in preference to unrestricted recursion. In particular, using recursion operators has proved invaluable for transforming and reasoning about f...