Related papers: (Deep) Induction Rules for GADTs
It is well-known that GADTs do not admit standard map functions of the kind supported by ADTs and nested types. In addition, standard map functions are insufficient to distribute their data-changing argument functions over all of the…
A well-known problem in the theory of dependent types is how to handle so-called nested data types. These data types are difficult to program and to reason about in total dependently typed languages such as Agda and Coq. In particular, it…
We consider type inference for guarded recursive data types (GRDTs) -- a recent generalization of algebraic data types. We reduce type inference for GRDTs to unification under a mixed prefix. Thus, we obtain efficient type inference.…
We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values),…
We present an approach to develop folds for nested data types using dependent types. We call such folds $\textit{dependently typed folds}$, they have the following properties. (1) Dependently typed folds are defined by well-founded…
Attention networks have proven to be an effective approach for embedding categorical inference within a deep neural network. However, for many tasks we may want to model richer structural dependencies without abandoning end-to-end training.…
Arguably the key reason for the success of deep neural networks is their ability to autonomously form non-linear combinations of the input features, which can be used in subsequent layers of the network. The analogon to this capability in…
Higher inductive types are inductive types that include nontrivial higher-dimensional structure, represented as identifications that are not reflexivity. While work proceeds on type theories with a computational interpretation of univalence…
Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful…
In functional programming languages, generalized algebraic data types (GADTs) are very useful as the unnecessary pattern matching over them can be ruled out by the failure of unification of type arguments. In dependent type systems, this is…
This paper considers parametricity and its consequent free theorems for nested data types. Rather than representing nested types via their Church encodings in a higher-kinded or dependently typed extension of System F, we adopt a functional…
To foster the verifiability and testability of Deep Neural Networks (DNN), an increasing number of methods for test case generation techniques are being developed. When confronted with testing DNN models, the user can apply any existing…
Given a set of Datalog rules, facts, and a query, answers to the query can be inferred bottom-up starting from the facts or top-down starting from the query. For efficiency, top-down evaluation is extended with memoization of inferred…
Theorem provers are tools that help users to write machine readable proofs. Some of this tools are also interactive. The need of such softwares is increasing since they provide proofs that are more certified than the hand written ones. Agda…
Our goal is to provide a review of deep learning methods which provide insight into structured high-dimensional data. Rather than using shallow additive architectures common to most statistical models, deep learning uses layers of…
In this paper, we propose a sensitivity-free and multi-objective structural design methodology called data-driven topology design. It is schemed to obtain high-performance material distributions from initially given material distributions…
Deep neural networks have been proven powerful at processing perceptual data, such as images and audio. However for tabular data, tree-based models are more popular. A nice property of tree-based models is their natural interpretability. In…
In the impredicative type theory of System F ({\lambda}2), it is possible to create inductive data types, such as natural numbers and lists. It is also possible to create coinductive data types such as streams. They work well in the sense…
We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive families within type theory. By elaborating an inductive definition -- a…
Constructing high-quality features is critical to any quantitative data analysis. While feature engineering was historically addressed by carefully hand-crafting data representations based on domain expertise, deep neural networks (DNNs)…