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Neural networks have in recent years shown promise for helping software engineers write programs and even formally verify them. While semantic information plays a crucial part in these processes, it remains unclear to what degree popular…
One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…
We assume some standard choices for the branch cuts of a group of functions and consider the problem of then calculating the branch cuts of expressions involving those functions. Typical examples include the addition formulae for inverse…
Unifying theories distil common features of programming languages and design methods by means of algebraic operators and their laws. Several practical concerns --- e.g., improvement of a program, conformance of code with design, correctness…
Predictive models are fundamental to engineering reliable software systems. However, designing conservative, computable approximations for the behavior of programs (static analyses) remains a difficult and error-prone process for modern…
As dynamic and control systems become more complex, relying purely on numerical computations for systems analysis and design might become extremely expensive or totally infeasible. Computer algebra can act as an enabler for analysis and…
Holomorphic functions are amazing because their values in an ever so small disk in the complex plane completely determine the function values at arbitrary points in their maximum possible domain. The process of extending such a function…
Classical results in computability theory, notably Rice's theorem, focus on the extensional content of programs, namely, on the partial recursive functions that programs compute. Later and more recent work investigated intensional…
Much recent work has shown how cross-linguistic variation is constrained by competing pressures from efficient communication. However, little attention has been paid to the role of the systematicity of forms (regularity), a key property of…
Unsupervised and self-supervised learning approaches have become a crucial tool to learn representations for downstream prediction tasks. While these approaches are widely used in practice and achieve impressive empirical gains, their…
Formal Concept Analysis (FCA) is a mathematical framework for knowledge representation and discovery. It performs a hierarchical clustering over a set of objects described by attributes, resulting in conceptual structures in which objects…
We present a systematic, algebraically based, design methodology for efficient implementation of computer programs optimized over multiple levels of the processor/memory and network hierarchy. Using a common formalism to describe the…
In this paper, we propose to consider various models of pattern recognition. At the same time, it is proposed to consider models in the form of two operators: a recognizing operator and a decision rule. Algebraic operations are introduced…
For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…
Transformers generate valid and diverse chemical structures, but little is known about the mechanisms that enable these models to capture the rules of molecular representation. We present a mechanistic analysis of autoregressive…
In Programming by Example, a system attempts to infer a program from input and output examples, generally by searching for a composition of certain base functions. Performing a naive brute force search is infeasible for even mildly involved…
Understanding and predicting the evolution of across complex systems remains a fundamental challenge due to the absence of unified and computationally testable frameworks. Here we propose the Recursive Hierarchical Network(RHN),…
This article addresses a fundamental problem faced by the ab initio community: the lack of an effective formalism for the rapid exploration and exchange of new methods. To rectify this, we introduce a novel, basis-set independent,…
We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely…
We design various logics for proving hyper properties of iterative programs by application of abstract interpretation principles. In part I, we design a generic, structural, fixpoint abstract interpreter parameterized by an algebraic…