Related papers: Elaborating Inductive Definitions
We present a type theory combining both linearity and dependency by stratifying typing rules into a level for logics and a level for programs. The distinction between logics and programs decouples their semantics, allowing the type system…
Coinduction refers to both a technique for the definition of infinite streams, so-called codata, and a technique for proving the equality of coinductively specified codata. This article first reviews coinduction in declarative programming.…
Abductive reasoning, reasoning for inferring explanations for observations, is often mentioned in scientific, design-related and artistic contexts, but its understanding varies across these domains. This paper reviews how abductive…
Inductions and game semantics are two useful extensions to traditional logic programming. To be specific, inductions can capture a wider class of provable formulas in logic programming. Adopting game semantics can make logic programming…
We present a soundness theorem for a dependent type theory with context constants with respect to an indexed category of (finite, abstract) simplical complexes. The point of interest for computer science is that this category can be seen to…
The success of neural networks builds to a large extent on their ability to create internal knowledge representations from real-world high-dimensional data, such as images, sound, or text. Approaches to extract and present these…
Python's typing system has evolved pragmatically into a powerful but theoretically fragmented system, with scattered specifications. This paper proposes a formalization to address this fragmentation. The central contribution is a formal…
Codifying mathematical theories in a proof assistant or computer algebra system is a challenging task, of which the most difficult part is, counterintuitively, structuring definitions. This results in a steep learning curve for new users…
Measurable cones, with linear and measurable functions as morphisms, are a model of intuitionistic linear logic and of call-by-name probabilistic PCF which accommodates "continuous data types" such as the real line. So far however, they…
This paper presents and extends our type theoretical framework for a compositional treatment of natural language semantics with some lexical features like coercions (e.g. of a town into a football club) and copredication (e.g. on a town as…
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 introduce a new way of formalizing the intensional identity type based on the fact that a entity known as computational paths can be interpreted as terms of the identity type. Our approach enjoys the fact that our elimination rule is…
This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly…
In this paper, we describe an approach that enables an autonomous system to infer the semantics of a command (i.e. a symbol sequence representing an action) in terms of the relations between changes in the observations and the action…
In this thesis, we try to build a connection between the two schools by introducing syntactic inductive biases for deep learning models. We propose two families of inductive biases, one for constituency structure and another one for…
This work introduces the novel concept of kind refinement, which we develop in the context of an explicitly polymorphic ML-like language with type-level computation. Just as type refinements embed rich specifications by means of…
In quantum mechanics, outcomes of measurements on a state have a probabilistic interpretation while the evolution of the state is treated deterministically. Here we show that one can also treat the evolution as being probabilistic in nature…
The form and justification of inductive inference rules depend strongly on the representation of uncertainty. This paper examines one generic representation, namely, incomplete information. The notion can be formalized by presuming that the…
We give in this paper a purely syntactical definition of input and output types of system F. We define the syntactical data types as input and output types. We show that any type with positive quantifiers is a syntactical data type and that…
This work presents a novel systematic methodology to analyse the capabilities and limitations of Large Language Models (LLMs) with feedback from a formal inference engine, on logic theory induction. The analysis is complexity-graded w.r.t.…