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We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and…
Recursive coalgebras provide an elegant categorical tool for modelling recursive algorithms and analysing their termination and correctness. By considering coalgebras over categories of suitably indexed families, the correctness of the…
Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
Quantum operations are the most widely used tool in the theory of quantum information processing, representing elementary transformations of quantum states that are composed to form complex quantum circuits. The class of quantum…
To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this…
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…
The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with…
Category computation theory deals with a web-based systemic processing that underlies the morphic webs, which constitute the basis of categorial logical calculus. It is proven that, for these structures, algorithmically incompressible…
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational…
Many recursive functions can be defined elegantly as the unique homomorphisms, between two algebras, two coalgebras, or one each, that are induced by some universal property of a distinguished structure. Besides the well-known applications…
We develop a compositional framework for generalized reversible computing using copy-discard categories and resource theories. We introduce partitioned matrices between partitioned sets as subdistribution matrices which preserve the…
Substructural logics are formal logical systems that omit familiar structural rules of classical and intuitionistic logic such as contraction, weakening, exchange (commutativity), and associativity. This leads to a resource-sensitive…
Coherence phenomena appear in two different situations. In the context of category theory the term `coherence constraints' refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the…
We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…
Modelling compositionality has been a longstanding area of research in the field of vector space semantics. The categorical approach to compositionality maps grammar onto vector spaces in a principled way, but comes under fire for requiring…
I investigate modal group theory for arbitrary homomorphisms. Possibility is interpreted by the existence of a group homomorphism out of the given group, so the semantics is governed by the possibility of collapse: elements may be…
The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a…
We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the $\Sigma_\beta$ hierarchy. We focus on linear orderings. We show that at the $\Sigma_1$ level all linear…
The usual coherence theorem of MacLane for categories with multiplication assumes that a certain pentagonal diagram commutes in order to conclude that associativity isomorphisms are well defined in a certain practical sense. The practical…