Introduction to Categories and Categorical Logic

Abramsky, Samson; Tzevelekos, Nikos

doi:10.1007/978-3-642-12821-9_1

Cited by 56 publications

(70 citation statements)

References 6 publications

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“…Residuals originate in abstract algebra [30,47]; then they were introduced to logic as the central component of the Lambek calculus [35] for syntactic analysis of natural language. Nowadays residuals are viewed as a natural algebraic interpretation of implication in substructural logics [38,18,14,1].…”

Section: Action Lattices and Their Logicsmentioning

confidence: 99%

The Logic of Action Lattices is Undecidable

Kuznetsov

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. This logic involves Kleene star, axiomatized by an induction scheme. For a stronger system which uses an ω-rule instead (infinitary action logic) Buszkowski and Palka (2007) have proved Π 0 1 -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by D. Kozen in 1994. In this article, we show that it is undecidable, more precisely, Σ 0 1 -complete. We also prove the same complexity results for all recursively enumerable logics between action logic and infinitary action logic; for fragments of those only one of the two lattice (additive) connectives; for action logic extended with the law of distributivity.Here the lower bound is due to Buszkowski and the upper one is due to Palka; recently A. Das and D. Pous [11] gave another proof of the Π 0 1 upper bound for ACT ω , based on non-well-founded proofs.

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Section: Action Lattices and Their Logicsmentioning

confidence: 99%

The Logic of Action Lattices is Undecidable

Kuznetsov

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…for all arrows f and g. Building a categorical product from a tensor product is not the most familiar presentation, but it can be shown to be equivalent (see Proposition 13 in [3], for example).…”

Section: Theorem: Lens Does Not Have Productsmentioning

confidence: 97%

Symmetric lenses

2011

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Lenses-bidirectional transformations between pairs of connected structures-have been extensively studied and are beginning to find their way into industrial practice. However, some aspects of their foundations remain poorly understood. In particular, most previous work has focused on the special case of asymmetric lenses, where one of the structures is taken as primary and the other is thought of as a projection, or view. A few studies have considered symmetric variants, where each structure contains information not present in the other, but these all lack the basic operation of composition. Moreover, while many domain-specific languages based on lenses have been designed, lenses have not been thoroughly explored from an algebraic perspective.We offer two contributions to the theory of lenses. First, we present a new symmetric formulation, based on complements, an old idea from the database literature. This formulation generalizes the familiar structure of asymmetric lenses, and it admits a good notion of composition. Second, we explore the algebraic structure of the space of symmetric lenses. We present generalizations of a number of known constructions on asymmetric lenses and settle some longstanding questions about their properties-in particular, we prove the existence of (symmetric monoidal) tensor products and sums and the non-existence of full categorical products or sums in the category of symmetric lenses. We then show how the methods of universal algebra can be applied to build iterator lenses for structured data such as lists and trees, yielding lenses for operations like mapping, filtering, and concatenation from first principles. Finally, we investigate an even more general technique for constructing mapping combinators, based on the theory of containers.

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“…The basic concepts of category theory also include functors, which are mappings between categories, and natural transformations, which are morphisms in functor categories satisfying certain "naturality" conditions, but we will be making only limited use of these concepts in the present paper. For references on category theory and categorical logic, see [22,21,4,1,19].…”

Section: A Logical Languagementioning

confidence: 99%

How to ground a language for legal discourse in a prototypical perceptual semantics

McCarty

2015

Proceedings of the 15th International Conference on Artificial Intelligence and Law

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In a pair of papers from 1995 and 1997, I developed a computational theory of legal argument, but left open a question about the key concept of a "prototype." Contemporary trends in machine learning have now shed new light on the subject. In this paper, I will describe my recent work on "manifold learning," as well as some work in progress on "deep learning." Taken together, this work leads to a logical language grounded in a prototypical perceptual semantics, with implications for legal theory. The main technical contribution of the paper is a categorical logic based on the category of differential manifolds (Man), which is weaker than a logic based on the category of sets (Set) or the category of topological spaces (Top). The paper also shows how this logic can be extended to a full Language for Legal Discourse (LLD), and suggests a solution to the elusive problem of "coherence" in legal argument.

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Introduction to Categories and Categorical Logic

Cited by 56 publications

References 6 publications

The Logic of Action Lattices is Undecidable

The Logic of Action Lattices is Undecidable

Symmetric lenses

How to ground a language for legal discourse in a prototypical perceptual semantics

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