Related papers: Bootstrapping Inductive and Coinductive Types in H…
This paper introduces Isabelle/HoTT, the first development of homotopy type theory in the Isabelle proof assistant. Building on earlier work by Paulson, I use Isabelle's existing logical framework infrastructure to implement essential…
We consider the properties of the bootstrap as a tool for inference concerning the eigenvalues of a sample covariance matrix computed from an $n\times p$ data matrix $X$. We focus on the modern framework where $p/n$ is not close to 0 but…
The concept of $typed$ $topology$ is introduced. In a typed topological space, some open sets are assigned "types", and topological concepts such as closure, connectedness can be defined using types. A finite data set in $R^2$ is a…
We present initial limit Datalog, a new extensible class of constrained Horn clauses for which the satisfiability problem is decidable. The class may be viewed as a generalisation to higher-order logic (with a simple restriction on types)…
Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary for invertible structures to have…
We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it…
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…
The question how neural systems (of humans) can perform reasoning is still far from being solved. We posit that the process of forming Concepts is a fundamental step required for this. We argue that, first, Concepts are formed as closed…
Within dependent type theory, we provide a topological counterpart of well-founded trees (for short, W-types) by using a proof-relevant version of the notion of inductively generated suplattices introduced in the context of formal topology…
This paper introduces an ML / Haskell like programming language with nested inductive and coinductive algebraic datatypes called \chariot. Functions are defined by arbitrary recursive definitions and can thus lead to non-termination and…
Datatype-generic programming increases program abstraction and reuse by making functions operate uniformly across different types. Many approaches to generic programming have been proposed over the years, most of them for Haskell, but…
Linear and substructural types are powerful tools, but adding them to standard functional programming languages often means introducing extra annotations and typing machinery. We propose a lightweight substructural type system design that…
The second-order dependence structure of purely nondeterministic stationary process is described by the coefficients of the famous Wold representation. These coefficients can be obtained by factorizing the spectral density of the process.…
In previous work, categories of algebras of endofunctors were shown to be enriched in categories of coalgebras of the same endofunctor, and the extra structure of that enrichment was used to define a generalization of inductive data types.…
We propose abstract compilation for precise static type analysis of object-oriented languages based on coinductive logic programming. Source code is translated to a logic program, then type-checking and inference problems amount to queries…
Polysynthetic languages have exceptionally large and sparse vocabularies, thanks to the number of morpheme slots and combinations in a word. This complexity, together with a general scarcity of written data, poses a challenge to the…
Integrating deep learning and causal discovery has encouraged us to spot that learning causal structures and representations in dialogue and video is full of challenges. We defined These data forms as "Indefinite Data", characterized by…
In this article, we present data-subsetting algorithms that allow for the approximate and scalable implementation of the Bayesian bootstrap. They are analogous to two existing algorithms in the frequentist literature: the bag of little…
We propose an enhancement to inductive types and records in a dependent type theory, namely (co)conditions. With a primitive interval type, conditions generalize the cubical syntax of higher inductive types in homotopy type theory, while…
We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive…