Related papers: An Algebraic Treatment of Recursion
This paper discusses the problem of abstracting conditional probabilistic actions. We identify two distinct types of abstraction: intra-action abstraction and inter-action abstraction. We define what it means for the abstraction of an…
Studies of issues related to computability and computational complexity involve the use of a model of computation. Pivotal to such a model are the computational processes considered. Processes of this kind can be described using an…
We propose an approach to image processing related to algebraic operators acting in the space of images. In view of the interest in the applications in optics and computer science, mathematical aspects of the paper have been simplified as…
We begin by defining Temperley-Lieb algebra, in two different ways: as a presented algebra or as a diagrammatic algebra. Next, we look for a basis algorithmically, using rewriting theory. Finally, we introduce a generalization of the…
Convolution is a ubiquitous operation in mathematics and computing. The Kripke semantics for substructural and interval logics motivates its study for quantale-valued functions relative to ternary relations. The resulting notion of…
This paper introduces abstractions that are meaningful for computers and that can be built and used according to computers' own criteria, i.e., computable abstractions. It is analyzed how abstractions can be seen to serve as the building…
The sequential structure of language, and the order of words in a sentence specifically, plays a central role in human language processing. Consequently, in designing computational models of language, the de facto approach is to present…
This paper is devoted to survey composition algebras and some of their applications. After overviewing the classical algebras of quaternions and octonions, both unital composition algebras (or Hurwitz algebras) and symmetric composition…
General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might…
Algorithmic recourse provides explanations that help users overturn an unfavorable decision by a machine learning system. But so far very little attention has been paid to whether providing recourse is beneficial or not. We introduce an…
We study algebraic synchronization trees, i.e., initial solutions of algebraic recursion schemes over the continuous categorical algebra of synchronization trees. In particular, we investigate the relative expressive power of algebraic…
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication…
We present an approach to program reasoning which inserts between a program and its verification conditions an additional layer, the denotation of the program expressed in a declarative form. The program is first translated into its…
The application of automatic transformation processes during the formal development and optimization of programs can introduce encumbrances in the generated code that programmers usually (or presumably) do not write. An example is the…
While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…
In this paper, we first briefly survey automated termination proof methods for higher-order calculi. We then concentrate on the higher-order recursive path ordering, for which we provide an improved definition, the Computability Path…
We advocate a declarative approach to proving properties of logic programs. Total correctness can be separated into correctness, completeness and clean termination; the latter includes non-floundering. Only clean termination depends on the…
Within the program of finding axiomatizations for various parts of computability logic, it was proved earlier that the logic of interactive Turing reduction is exactly the implicative fragment of Heyting's intuitionistic calculus. That sort…
Induction is the process by which we obtain predictive laws or theories or models of the world. We consider the structural aspect of induction. We answer the question as to whether we can find a finite and minmalistic set of operations on…
Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by…