Dynamic Recursion Pattern

Proc. SBLP'00, vol. of ENTCS, 2000

Recursive schemes over inductive data structures have been recognized as categorytheoretic universals, yielding a handful of equational laws for program construction and transformation. This paper introduces the implementation of such recursion patterns as type parametric, or polytypic, functionals in the Camila prototyping language. Several examples are discussed.

2002

We introduce a language based upon lambda calculus with products, coproducts and strictly positive inductive types that allows the definition of recursive terms. We present the implementation (foetus) of a syntactical check that ensures that all such terms are structurally recursive, i.e., recursive calls appear only with arguments structurally smaller than the input parameters of terms considered. To ensure the correctness of the termination checker, we show that all structurally recursive terms are normalizing with respect to a given operational semantics. To this end, we define a semantics on all types and a structural ordering on the values in this semantics and prove that all values are accessible with regard to this ordering. Finally, we point out how to do this proof predicatively using set based operators.

Lecture Notes in Computer Science, 2006

XML query languages need to provide some mechanism to inspect and manipulate nodes at all levels of an input tree. In this paper we investigate the expressive power provided in this regard by structural recursion. We show that the combination of vertical recursion down a tree combined with horizontal recursion across a list of trees gives rise to a robust class of transformations: it captures the class of all primitive recursive queries. Since queries are expected to be computable in at most polynomial time for all practical purposes, we next identify a restriction of structural recursion that captures the polynomial time queries. Although this restriction is semantical in nature, and therefore undecidable, we provide an effective syntax. We also give corresponding results for list-based complex objects.

Information Processing Letters, 1976

2015

Ask any instructor of Pascal, Logo, LISP, data structures, or algorithms to name three topics that students find most difficult. Most probably his or her list will include In this brief paper I will propose an hypothesis for explaining students ’ extreme difficulty with recursion, and will justify the importance of recursion’s place in a mathematics curriculum. First, let us ensure a common vocabulary. The term will mean any process that employs itself as a subprocess. The term will mean any object which contains an instance of itself as a component. will mean the class of recursive processes and recursive objects. AN HYPOTHESIS The distinction I have drawn between recursive processes and recursive objects is essential to formulate my hypothesis concerning students ’ difficulties with recursion. The hypothesis is this: to be able to recognize a problem solution as one requiring a recursive process, students must formulate their solution as a recursive object. Conversely, to recogniz...

ACM Sigcse Bulletin, 2000

We propose a gradual approach to teach recursion. Our main assumption is that the difficulty in learning recursion does not come from the recursion concept itself, but from its interaction with other mechanisms of imperative programming. We use this basic idea to propose a new pedagogical approach. On the one hand, recursion is introduced in a gradual way by means of three fields (grammars, functional programming and imperative programming). On the other hand, each instance of recursion is explained so that all of its accompanying mechanisms are clearly identified.

Frontiers in Psychology, 2014

ACM SIGAPL APL Quote Quad, 1985

Recursion is frequently used in functional programming languages. In LISP, it is the primary means of traversing data structures, and, along with COND, is instrumental in controlling program flow. In APL2, the generalization of arrays to include arrays of arrays encourages the use of recursion. This usage is enhanced by the generalization of, and extensions to, operators. In particular, the ability to define operators allows us to create operators which simplify the coding of recursive logic. This paper studies the use of operators to provide program control for solutions to problems which involve recursion. For the most part, iteration on the elements of a list is used to replace recursive function calls. The program control methods used in this paper are introduced in a companion paper, Operators for Program Control (Eusebi[3]). The reader is encouraged to read that paper first.

Philosophical Transactions of the Royal Society B, 2012

Although recursion has been hypothesized to be a necessary capacity for the evolution of language, the multiplicity of definitions being used has undermined the broader interpretation of empirical results. I propose that only a definition focused on representational abilities allows the prediction of specific behavioural traits that enable us to distinguish recursion from non-recursive iteration and from hierarchical embedding: only subjects able to represent recursion, i.e. to represent different hierarchical dependencies (related by parenthood) with the same set of rules, are able to generalize and produce new levels of embedding beyond those specified a priori (in the algorithm or in the input). The ability to use such representations may be advantageous in several domains: action sequencing, problem-solving, spatial navigation, social navigation and for the emergence of conventionalized communication systems. The ability to represent contiguous hierarchical levels with the same rules may lead subjects to expect unknown levels and constituents to behave similarly, and this prior knowledge may bias learning positively. Finally, a new paradigm to test for recursion is presented. Preliminary results suggest that the ability to represent recursion in the spatial domain recruits both visual and verbal resources. Implications regarding language evolution are discussed.

ACM SIGPLAN Notices, 2004

In the interest of designing a recursive module extension to ML that is as simple and general as possible, we propose a novel type system for general recursion over effectful expressions. The presence of effects seems to necessitate a backpatching semantics for recursion based on Scheme's. Our type system ensures statically that recursion is well-founded (that the body of a recursive expression will evaluate without attempting to access the undefined recursive variable), which avoids some unnecessary run-time costs associated with backpatching. To ensure well-founded recursion in the presence of multiple recursive variables and separate compilation, we track the usage of individual recursive variables, represented statically by "names". So that our type system may eventually be integrated smoothly into ML's, reasoning involving names is only required inside code that uses our recursive construct and does not need to infect existing ML code.