Related papers: Syntax Monads for the Working Formal Metatheorist
Formalized $1$-category theory forms a core component of various libraries of mathematical proofs. However, more sophisticated results in fields from algebraic topology to theoretical physics, where objects have "higher structure," rely on…
This paper introduces Choice Trees (ctrees), a monad for modeling nondeterministic, recursive, and impure programs in Coq. Inspired by Xia et al.'s itrees, this novel data structure embeds computations into coinductive trees with three kind…
Pretrained multi-modal large language models (MLLMs) demonstrate strong performance on diverse multimodal tasks, but remain limited in reasoning capabilities for domains where annotations are difficult to collect. In this work, we focus on…
The Bindlib library for OCaml provides a set of tools for the manipulation of data structures with variable binding. It is very well suited for the representation of abstract syntax trees, and has already been used for the implementation of…
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…
Proof nets are a syntax for linear logic proofs which gives a coarser notion of proof equivalence with respect to syntactic equality together with an intuitive geometrical representation of proofs. In this paper we give an alternative…
Terms are a concise representation of tree structures. Since they can be naturally defined by an inductive type, they offer data structures in functional programming and mechanised reasoning with useful principles such as structural…
A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as `machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special…
This chapter explores dynamical structural equation models (DSEMs) and their nonlinear generalizations into sheaves of dynamical systems. It demonstrates these two disciplines on part of the food web in the Bering Sea. The translation from…
The original idea of proof nets can be formulated by means of interaction nets syntax. Additional machinery as switching, jumps and graph connectivity is needed in order to ensure correspondence between a proof structure and a correct proof…
In a Systems Engineering setting, various models are produced using a variety of methods and tools. Focusing on a type of models -- called descriptive models -- which we shall describe, we argue that, while the clarity and precision of…
We give a natural-deduction-style type theory for symmetric monoidal categories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler…
It is known that different categorial grammars have surface representation in a fragment of first order multiplicative linear logic (MLL1). We show that the fragment of interest is equivalent to the recently introduced extended tensor type…
This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known…
This paper provides a geometric characterization of subclasses of the regular languages. We use finite model theory to characterize objects like strings and trees as relational structures. Logical statements meeting certain criteria over…
Formal, automated theorem proving has long been viewed as a challenge to artificial intelligence. We introduce here a new approach to computer theorem proving, one that employs specialized language models for Lean4 proof generation combined…
Properties of morphisms represented by so-called 'string diagrams' of monoidal categories (and their braided and symmetric derivatives), mainly their resistance in value to isotopic deformation, have made the usage of graphical calculi…
Mechanisms are a fundamental concept in many areas of science. Nonetheless, there has been little effort to develop structures to represent mechanisms. We explore the issues in developing a basic semantic modeling framework for describing…
The design and implementation of static analyzers has become increasingly systematic. Yet for a given language or analysis feature, it often requires tedious and error prone work to implement an analyzer and prove it sound. In short, static…
Category theory is famous for its innovative way of thinking of concepts by their descriptions, in particular by establishing universal properties. Concepts that can be characterized in a universal way receive a certain quality seal, which…