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The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…
Denotational semantics can be based on algebras with additional structure (order, metric, etc.) which makes it possible to interpret recursive specifications. It was the idea of Elgot to base denotational semantics on iterative theories…
Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such…
This survey synthesizes the principal descriptive set-theoretic perspectives on deterministic Cantor sets on the real line and charts directions for future study. After recounting their historical genesis and compiling an up-to-date…
Due to the increased complexity of software development projects more and more systems are described by models. The sheer size makes it impractical to describe these systems by a single model. Instead many models are developed that provide…
Functional reactive programming (FRP) makes it possible to express temporal aspects of computations in a declarative way. Recently we developed two kinds of categorical models of FRP: abstract process categories (APCs) and concrete process…
Herbrand schemes are a method to extract Herband disjunctions directly from sequent calculus proofs, without appealing to cut elimination, using a formal grammar known as a higher-order recursion scheme. In this note, we show that the core…
Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata. The…
The goal of this paper is to design a causal inference method accounting for complex interactions between causal factors. The proposed method relies on a category theoretical reformulation of the definitions of dependent variables,…
We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying approach using operators where we allow the…
We develop a correspondence between the theory of sequential algorithms and classical reasoning, via Kreisel's no-counterexample interpretation. Our framework views realizers of the no-counterexample interpretation as dynamic processes…
Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of…
We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are…
Affine type systems are substructural type systems where copying of information is restricted, but discarding of information is permissible at all types. Such type systems are well-suited for describing quantum programming languages,…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
Rule-based reasoning is an essential part of human intelligence prominently formalized in artificial intelligence research via logic programs. Describing complex objects as the composition of elementary ones is a common strategy in computer…
The purpose of this work is to complete the algebraic foundations of second-order languages from the viewpoint of categorical algebra as developed by Lawvere. To this end, this paper introduces the notion of second-order algebraic theory…
Bialgebrae provide an abstract framework encompassing the semantics of different kinds of computational models. In this paper we propose a bialgebraic approach to the semantics of logic programming. Our methodology is to study logic…
Logic languages based on the theory of rational, possibly infinite, trees have much appeal in that rational trees allow for faster unification (due to the safe omission of the occurs-check) and increased expressivity (cyclic terms can…
A key challenge in example-based program synthesis is the gigantic search space of programs. To address this challenge, various work proposed to use abstract interpretation to prune the search space. However, most of existing approaches…