Related papers: Domain Theory in Constructive and Predicative Univ…
We introduce judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and natural deduction as special kinds of…
Recently discovered domain-specific formal systems -- specifically homotopy type theory and simplicial type theory -- provide new perspectives on spaces and categories in a natively equivalence-invariant setting. In this note, we expose…
This paper discusses the extension of the Prototype Verification System (PVS) sub-theory for rings, part of the PVS algebra theory, with theorems related to the division algorithm for Euclidean rings and Unique Factorization Domains that…
The Univalent Foundations requires a logic that allows us to define structures on homotopy types, similar to how first-order logic with equality ($\text{FOL}_=$) allows us to define structures on sets. We develop the syntax, semantics and…
This thesis is about the problem of compositionality in distributional semantics. Distributional semantics presupposes that the meanings of words are a function of their occurrences in textual contexts. It models words as distributions over…
Topological domain walls separating 2+1 dimensional topologically ordered phases can be understood in terms of Witt equivalences between the UMTCs describing anyons in the bulk topological orders. However, this picture does not provide a…
Recent breakthroughs in scale have enabled the emergence of powerful generative language models, and the ability to fine-tune these models on various tasks by casting them into prompts or instructions. In this landscape, the problem of…
Theory-guided machine learning has demonstrated that including authentic domain knowledge directly into model design improves performance, sample efficiency and out-of-distribution generalisation. Yet the process by which a formal domain…
Large language models have achieved impressive performance across diverse tasks. However, their tendency to produce overconfident and factually incorrect outputs, known as hallucinations, poses risks in real world applications. Conformal…
Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article, written as lecture notes to accompany a 3-part mini course…
Motivated by the study of meromorphic vector fields, a model theory of "compact complex manifolds equipped with a generic derivation" is here proposed. This is made precise by the notion of a differential CCM-structure. A first-order…
Recursive domain equations have natural solutions. In particular there are domains defined by strictly positive induction. The class of countably based domains gives a computability theory for possibly non-countably based topological…
We give a simple order-theoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order -- the extensional order -- and a bistable coherence, which captures…
With a frame $L$ as the truth value table, we study the topological representations for frame-valued domains. We introduce the notions of locally super-compact $L$-topological space and strong locally super-compact $L$-topological space.…
We establish a computation-substrate-agnostic inference architecture in which domain is an explicit first-class computational parameter. This produces domain-scoped pruning that reduces per-query search space from O(N) to O(N/K),…
Directed spaces are natural topological extensions of dcpos in domain theory and form a cartesian closed category. In order to model nondeterministic semantics, the power structures over directed spaces were defined through the form of free…
Unexpected stimuli induce "error" or "surprise" signals in the brain. The theory of predictive coding promises to explain these observations in terms of Bayesian inference by suggesting that the cortex implements variational inference in a…
Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…
We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued…
With infinitely many high-quality data points, infinite computational power, an infinitely large foundation model with a perfect training algorithm and guaranteed zero generalization error on the pretext task, can the model be used for…