Related papers: Syntax Monads for the Working Formal Metatheorist
There are many books designed to introduce category theory to either a mathematical audience or a computer science audience. In this book, our audience is the broader scientific community. We attempt to show that category theory can be…
The overwhelming majority of the attempts in exploring the problems related to quantum logical structures and their interpretation have been based on an underlying set-theoretic syntactic language. We propose a transition in the involved…
One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a "structured cospan" is a…
Ontologies are useful for automatic machine processing of domain knowledge as they represent it in a structured format. Yet, constructing ontologies requires substantial manual effort. To automate part of this process, large language models…
Bounded treewidth and Monadic Second Order (MSO) logic have proved to be key concepts in establishing fixed-parameter tractability results. Indeed, by Courcelle's Theorem we know: Any property of finite structures, which is expressible by…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
Structural proof theory is praised for being a symbolic approach to reasoning and proofs, in which one can define schemas for reasoning steps and manipulate proofs as a mathematical structure. For this to be possible, proof systems must be…
Static single assignment form, or SSA, has been the dominant compiler intermediate representation for decades. In this paper, we give a type theory for a variant of SSA, including its equational theory, which are strong enough to validate a…
The modal logic S4 can be used via a Curry-Howard style correspondence to obtain a lambda-calculus. Modal (boxed) types are intuitively interpreted as `closed syntax of the calculus'. This lambda-calculus is called modal type theory ---…
Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…
Accounts of semantic phenomena often involve extending types of meanings and revising composition rules at the same time. The concept of monads allows many such accounts -- for intensionality, variable binding, quantification and focus --…
We make the case for language models over logical forms (LFLMs), arguing that such models are more data-efficient than their textual counterparts. To that end, we introduce the Graph-based Formal-Logical Distributional Semantics (GFoLDS)…
The present work proposes and discusses the category of supported sets which provides a uniform foundation for nominal sets of various kinds, such as those for equality symmetry, for the order symmetry, and renaming sets. We show that all…
Theory revision integrates inductive learning and background knowledge by combining training examples with a coarse domain theory to produce a more accurate theory. There are two challenges that theory revision and other theory-guided…
Automated theorem proving is essential for the formal verification of safety-critical systems. As the corpus of formal proofs grows, a natural paradigm is to learn from existing proofs. However, current learning-based approaches…
Explanations for description logic (DL) entailments provide important support for the maintenance of large ontologies. The "justifications" usually employed for this purpose in ontology editors pinpoint the parts of the ontology responsible…
We develop layered monoidal theories -- a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams, monoidal theories provide a graphical…
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 connection between languages defined by computational models and logic for languages is well-studied. Monadic second-order logic and finite automata are shown to closely correspond to each-other for the languages of strings, trees, and…
Clearly articulating the assumptions of the execution environment is crucial for the successful application of code-level formal verification. The process of specifying a model for the environment can be both laborious and error-prone,…