Related papers: Coaxioms: flexible coinductive definitions by infe…
Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a…
"Interaction trees" (ITrees) are a general-purpose data structure for representing the behaviors of recursive programs that interact with their environments. A coinductive variant of "free monads," ITrees are built out of uninterpreted…
An argument can be seen as a pair consisting of a set of premises and a claim supported by them. Arguments used by humans are often enthymemes, i.e., some premises are implicit. To better understand, evaluate, and compare enthymemes, it is…
In Constructive Type Theory, recursive and corecursive definitions are subject to syntactic restrictions which guarantee termination for recursive functions and productivity for corecursive functions. However, many terminating and…
Categorization axioms have been proposed to axiomatizing clustering results, which offers a hint of bridging the difference between human recognition system and machine learning through an intuitive observation: an object should be assigned…
People care about decision outcomes and how decisions get made, both when making decisions and reflecting on decisions. But formalizing the full range of normative concerns that drive decisions is an open challenge. We introduce Axiomatic…
Traditionally, formal languages are defined as sets of words. More recently, the alternative coalgebraic or coinductive representation as infinite tries, i.e., prefix trees branching over the alphabet, has been used to obtain compact and…
The rapid evolution of artificial intelligence has led to expectations of transformative impact on science, yet current systems remain fundamentally limited in enabling genuine scientific discovery. This perspective contends that progress…
A new theory of data types which allows for the definition of types as initial algebras of certain functors Fam(C) -> Fam(C) is presented. This theory, which we call positive inductive-recursive definitions, is a generalisation of Dybjer…
Enthymemes are defined as arguments where a premise or conclusion is left implicit. We tackle the task of generating the implicit premise in an enthymeme, which requires not only an understanding of the stated conclusion and premise but…
This paper synthesizes a series of formal proofs to construct a unified theory on the logical limits of the Symbol Grounding Problem. We distinguish between internal meaning (sense), which formal systems can possess via axioms, and external…
A fertile field of research in theoretical computer science investigates the representation of general recursive functions in intensional type theories. Among the most successful approaches are: the use of wellfounded relations,…
We study the system IFP of intuitionistic fixed point logic, an extension of intuitionistic first-order logic by strictly positive inductive and coinductive definitions. We define a realizability interpretation of IFP and use it to extract…
We present a simple and clear foundation for finite inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying lattices of logical statements in a way that satisfies general…
This article explores a generic framework of well-typed and well-scoped syntaxes, with a signature-axiom approach resembling traditional abstract algebra. The boilerplate code needed in defining operations on syntaxes is identified and…
An inductive logic can be formulated in which the elements are not propositions or probability distributions, but information systems. The logic is complete for information systems with binary hypotheses, i.e., it applies to all such…
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision…
In this paper we develop cyclic proof systems for the problem of inclusion between the least sets of models of mutually recursive predicates, when the ground constraints in the inductive definitions belong to the quantifier-free fragments…
There are many ways we can not know. Even in systems that we created ourselves, as, for example, systems in mathematical logic, Go\"edel and Tarski's theorems impose limits on what we can know. As we try to speak of the real world, things…
Extending G\"odel's \emph{Dialectica} interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite…