Unifying Structured Recursion Schemes

AbstractFolds and unfolds have been understood as fundamental building blocks for total programming, and have been extended to form an entire zoo of specialised structured recursion schemes. A great number of these schemes were unified by the introduction of adjoint folds, but more exotic beasts such as recursion schemes from comonads proved to be elusive. In this paper, we show how the two canonical derivations of adjunctions from (co)monads yield recursion schemes of significant computational importance: monadic catamorphisms come from the Kleisli construction, and more astonishingly, the elusive recursion schemes from comonads come from the Eilenberg–Moore construction. Thus, we demonstrate that adjoint folds are more unifying than previously believed.</jats:p

Detecting sequential structure

Programming by demonstration requires detection and analysis of sequential patterns in a user’s input, and the synthesis of an appropriate structural model that can be used for prediction. This paper describes SEQUITUR, a scheme for inducing a structural description of a sequence from a single example. SEQUITUR integrates several different inference techniques: identification of lexical subsequences or vocabulary elements, hierarchical structuring of such subsequences, identification of elements that have equivalent usage patterns, inference of programming constructs such as looping and branching, generalisation by unifying grammar rules, and the detection of procedural substructure., Although SEQUITUR operates with abstract sequences, a number of concrete illustrations are provided

Strategic polymorphism requires just two combinators!

In previous work, we introduced the notion of functional strategies: first-class generic functions that can traverse terms of any type while mixing uniform and type-specific behaviour. Functional strategies transpose the notion of term rewriting strategies (with coverage of traversal) to the functional programming paradigm. Meanwhile, a number of Haskell-based models and combinator suites were proposed to support generic programming with functional strategies. In the present paper, we provide a compact and matured reconstruction of functional strategies. We capture strategic polymorphism by just two primitive combinators. This is done without commitment to a specific functional language. We analyse the design space for implementational models of functional strategies. For completeness, we also provide an operational reference model for implementing functional strategies (in Haskell). We demonstrate the generality of our approach by reconstructing representative fragments of the Strafunski library for functional strategies.Comment: A preliminary version of this paper was presented at IFL 2002, and included in the informal preproceedings of the worksho

The Phillips Machine, The Analogue Computing Traditoin in Economics and Computability

In this paper I try to argue for the desirability of analog computation in economics from a variety of perspectives, using the example of the Phillips Machine. Ultimately, a case is made for the underpinning of both analog and digital computing theory in constructive mathematics. Some conceptual confusion in the meaning of analog computing and its non-reliance on the theory of numerical analysis is also discussed. Digital computing has its mathematical foundations in (classical) recursion theory and constructive mathematics. The implicit, working, assumption of those who practice the noble art of analog computing may well be that the mathematical foundations of their subject is as sound as the foundations of the real analysis. That, in turn, implies a reliance on the soundness of set theory plus the axiom of choice. This is, surely, seriously disturbing from a computation point of view. Therefore, in this paper, I seek to locate a foundation for analog computing in exhibiting some tentative dualities with results that are analogous to those that are standard in computability theory. The main question, from the point of view of economics, is whether the Phillips Machine, as an analog computer, has universal computing properties. The conjectured answer is in the negative.Phillips Machine, Analogue Computation, Digital Computation, Computability, General Purpose Analogue Computer

Kernel-based Inference of Functions over Graphs

The study of networks has witnessed an explosive growth over the past decades with several ground-breaking methods introduced. A particularly interesting -- and prevalent in several fields of study -- problem is that of inferring a function defined over the nodes of a network. This work presents a versatile kernel-based framework for tackling this inference problem that naturally subsumes and generalizes the reconstruction approaches put forth recently by the signal processing on graphs community. Both the static and the dynamic settings are considered along with effective modeling approaches for addressing real-world problems. The herein analytical discussion is complemented by a set of numerical examples, which showcase the effectiveness of the presented techniques, as well as their merits related to state-of-the-art methods.Comment: To be published as a chapter in `Adaptive Learning Methods for Nonlinear System Modeling', Elsevier Publishing, Eds. D. Comminiello and J.C. Principe (2018). This chapter surveys recent work on kernel-based inference of functions over graphs including arXiv:1612.03615 and arXiv:1605.07174 and arXiv:1711.0930

Beating the Productivity Checker Using Embedded Languages

Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures that programs are productive, i.e. that every finite prefix of an infinite value can be computed in finite time. However, many productive programs are not guarded, and it can be nontrivial to put them in guarded form. This paper gives a method for turning a productive program into a guarded program. The method amounts to defining a problem-specific language as a data type, writing the program in the problem-specific language, and writing a guarded interpreter for this language.Comment: In Proceedings PAR 2010, arXiv:1012.455