Related papers: Extending Equational Monadic Reasoning with Monad …

Equational reasoning is among the most important tools that functional programming provides us. Curiously, relatively less attention has been paid to reasoning about monadic programs. In this report we derive a backtracking algorithm for…

Due to their numerous advantages, formal proofs and proof assistants, such as Coq, are becoming increasingly popular. However, one disadvantage of using proof assistants is that the resulting proofs can sometimes be hard to read and…

We show how a commutative monad gives rise to a theory of extensive quantities, including (under suitable further conditions) a differential calculus of such. The relationship to Schwartz distributions is dicussed. The paper is a companion…

Modifiable combining functions are a synthesis of two common approaches to combining evidence. They offer many of the advantages of these approaches and avoid some disadvantages. Because they facilitate the acquisition, representation,…

In complex inferential tasks like question answering, machine learning models must confront two challenges: the need to implement a compositional reasoning process, and, in many applications, the need for this reasoning process to be…

We introduce an extension of first-order logic that comes equipped with additional predicates for reasoning about an abstract state. Sequents in the logic comprise a main formula together with pre- and postconditions in the style of Hoare…

We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the…

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…

Monads in category theory are algebraic structures that can be used to model computational effects in programming languages. We show how the notion of "centre", and more generally "centrality", i.e. the property for an effect to commute…

Proof assistants like Coq are increasingly popular to help mathematicians carry out proofs of the results they conjecture. However, formal proofs remain highly technical and are especially difficult to reuse. In this paper, we present a…

We explore end-to-end trained differentiable models that integrate natural logic with neural networks, aiming to keep the backbone of natural language reasoning based on the natural logic formalism while introducing subsymbolic vector…

We propose a modular method for proving termination of general logic programs (i.e., logic programs with negation). It is based on the notion of acceptable programs, but it allows us to prove termination in a truly modular way. We consider…

In functional programming, datatypes a la carte provide a convenient modular representation of recursive datatypes, based on their initial algebra semantics. Unfortunately it is highly challenging to implement this technique in proof…

The substitution lemma is a renowned theorem within the realm of lambda-calculus theory and concerns the interactional behaviour of the metasubstitution operation. In this work, we augment the lambda-calculus's grammar with an uninterpreted…

We describe a new approach to automatically repairing broken proofs in the Coq proof assistant in response to changes in types. Our approach combines a configurable proof term transformation with a decompiler from proof terms to tactic…

Reynold's parametricity theory captures the property that parametrically polymorphic functions behave uniformly: they produce related results on related instantiations. In dependently-typed programming languages, such relations and…

We survey the theory of Hopf monads on monoidal categories, and present new examples and applications. As applications, we utilise this machinery to present a new theory of cross products, as well as analogues of the Fundamental Theorem of…

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the…

Partial Combinatory Algebras (PCAs) provide a foundational model of the untyped $\lambda$-calculus and serve as the basis for many notions of computability, such as realizability theory. However, PCAs support a very limited notion of…

We present a way to apply quantum logic to the study of quantum programs. This is made possible by using an extension of the usual propositional language in order to make transformations performed on the system appear explicitly. This way,…