Related papers: Intensional Constructed Numbers: Towards Formalizi…
Many machine learning algorithms represent input data with vector embeddings or discrete codes. When inputs exhibit compositional structure (e.g. objects built from parts or procedures from subroutines), it is natural to ask whether this…
We find that second order quantification is problematic when a quantified concept variable is supposed to function predicatively. This issue is analyzed and it is shown that a constructive interpretation of the falling under relation…
We compare three notions of effectiveness on uncountable structures. The first notion is that of a $\real$-computable structure, based on a model of computation proposed by Blum, Shub, and Smale, which uses full-precision real arithmetic.…
What does it mean to claim that a physical or natural system computes? One answer, endorsed here, is that computing is about programming a system to behave in different ways. This paper offers an account of what it means for a physical…
Recent advances in programming languages study and design have established a standard way of grounding computational systems representation in category theory. These formal results led to a better understanding of issues of control and…
This article defines a complement of a function and conditions for existence of such a complement function and presents few algorithms to construct a complement.
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
When automatically generating programming exercise tasks one often also needs to automatically generate programs. At the very least when providing sample solutions is part of automated feedback. But programs can also be used as part of the…
Measurements are shown to be processes designed to return figures: they are effective. This effectivity allows for a formalization as Turing machines, which can be described employing computation theory. Inspired in the halting problem we…
This paper proposes a new view to algorithms, Algorithms as defining dynamic systems. This view extends the traditional, deterministic view that an algorithm is a step by step procedure with nondeterminism. As a dynamic system can be…
While a great effort has concerned the development of fully integrated modular understanding systems, few researches have focused on the problem of unifying existing linguistic formalisms with cognitive processing models. The Situated…
An abstract machine is a theoretical model designed to perform a rigorous study of computation. Such a model usually consists of configurations, instructions, programs, inputs and outputs for the machine. In this paper we formalize these…
In an era of widespread influence of Natural Language Processing (NLP), there have been multiple research efforts to supplant traditional manual coding techniques with automated systems capable of generating solutions autonomously. With…
Theoretical computer science discusses foundational issues about computations. It asks and answers questions such as "What is a computation?", "What is computable?", "What is efficiently computable?","What is information?", "What is…
Optimizations in a traditional compiler are applied sequentially, with each optimization destructively modifying the program to produce a transformed program that is then passed to the next optimization. We present a new approach for…
The extensive deployment of probabilistic algorithms has radically changed our perspective on several well-established computational notions. Correctness is probably the most basic one. While a typical probabilistic program cannot be said…
How best to quantify the information of an object, whether natural or artifact, is a problem of wide interest. A related problem is the computability of an object. We present practical examples of a new way to address this problem. By…
In this article we study the "norm" of an integer partition, which we define to be the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of…
Shapiro's notations for natural numbers, and the associated desideratum of acceptability - the property of a notation that all recursive functions are computable in it - is well-known in philosophy of computing. Computable structure theory,…