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This paper introduces the Modular Neural Computer (MNC), a memory-augmented neural architecture for exact algorithmic computation on variable-length inputs. The model combines an external associative memory of scalar cells, explicit read…
We argue that computation is an abstract algebraic concept, and a computer is a result of a morphism (a structure preserving map) from a finite universal semigroup.
Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html ) is a research program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth which it has more traditionally been.…
Given an algebraic theory which can be described by a (possibly symmetric) operad $P$, we propose a definition of the \emph{weakening} (or \emph{categorification}) of the theory, in which equations that hold strictly for $P$-algebras hold…
Semigroup theory is a branch of abstract algebra, and it provides mathematical tools for the theory of computation. Finite semigroups can describe state transition systems and thus they model physically realizable computers. Engineering…
Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation. There are many problems with high complexity that we have to deal, which is especially true for AI. This raises a big…
We present an algebraic semantics for governed execution in which governance is axiomatized, compositional, and coterminous with expressibility. The framework, mechanized in 32 Rocq modules (~12,000 lines, 454 theorems, 0 admitted), is…
The topological interpretation of modal logics provides descriptive languages and proof systems for reasoning about points of topological spaces. Recent work has been devoted to model checking of spatial logics on discrete spatial…
Human cognition operates on a "Global-first" cognitive mechanism, prioritizing information processing based on coarse-grained details. This mechanism inherently possesses an adaptive multi-granularity description capacity, resulting in…
We define a diagrammatic monoidal category, together with a full and essentially surjective monoidal functor from this category to the category of modules over the exceptional Lie algebra of type $F_4$. In this way, we obtain a set of…
Adapting a result of Bazhenov, Kalimullin, and Yamaleev, we show that if a Turing degree $\textbf{d}$ is the degree of categoricity of a computable structure $\mathcal{M}$ and is not the strong degree of categoricity of any computable…
The machinery of the human brain -- analog, probabilistic, embodied -- can be characterized computationally, but what machinery confers what computational powers? Any such system can be abstractly cast in terms of two computational…
The category Set_* of sets and partial functions is well-known to be traced monoidal, meaning that a partial function S+U -/-> T+U can be coherently transformed into a partial function S -/-> T. This transformation is generally described in…
Constructing complex computation from simpler building blocks is a defining problem of computer science. In algebraic automata theory, we represent computing devices as semigroups. Accordingly, we use mathematical tools like products and…
Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a…
Relational properties describe multiple runs of one or more programs. They characterize many useful notions of security, program refinement, and equivalence for programs with diverse computational effects, and they have received much…
Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic…
The paper introduces the notion of a weak bisimulation for coalgebras whose type is a monad satisfying some extra properties. In the first part of the paper we argue that systems with silent moves should be modelled coalgebraically as…
Monadic programming presents a significant challenge for many programmers. In light of category theory, we offer a new perspective on the use of monads in functional programming. This perspective is clarified through numerous examples coded…
This paper investigates some issues arising in categorical models of reversible logic and computation. Our claim is that the structural (coherence) isomorphisms of these categorical models, although generally overlooked, have decidedly…