67 research outputs found
Abstract Tensor Systems as Monoidal Categories
The primary contribution of this paper is to give a formal, categorical
treatment to Penrose's abstract tensor notation, in the context of traced
symmetric monoidal categories. To do so, we introduce a typed, sum-free version
of an abstract tensor system and demonstrate the construction of its associated
category. We then show that the associated category of the free abstract tensor
system is in fact the free traced symmetric monoidal category on a monoidal
signature. A notable consequence of this result is a simple proof for the
soundness and completeness of the diagrammatic language for traced symmetric
monoidal categories.Comment: Dedicated to Joachim Lambek on the occasion of his 90th birthda
From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality
We study how to relate well-known hypergraph grammars based on the double
pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL
(called HL-grammars). It turns out that DPO rules can be naturally encoded by
types of HL using methods similar to those used by Kanazawa for
multiplicative-exponential linear logic. In order to generalize his reasonings
we extend the hypergraph Lambek calculus by adding the exponential modality,
which results in a new calculus HMEL0; then we prove that any DPO grammar can
be converted into an equivalent HMEL0-grammar. We also define the conjunctive
Kleene star, which behaves similarly to this exponential modality, and
establish a similar result. If we add neither the exponential modality nor the
conjunctive Kleene star to HL, then we can still use the same encoding and show
that any DPO grammar with a linear restriction on the length of derivations can
be converted into an equivalent HL-grammar.Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421
Towards Theory and Applications of Generalized Categories to Areas of Type Theory and Categorical Logic
Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to higher category theory, we undertake a detailed study of a new mathematical abstraction, the generalized category. It is a partially defined monoid equipped with endomorphism maps defining sources and targets on arbitrary elements, possibly allowing a proximal behavior with respect to composition. We first present a formal introduction to the theory of generalized categories. We describe functors, equivalences, natural transformations, adjoints, and limits in the generalized setting. Next we indicate how the theory of monads extends to generalized categories, and discuss applications to computer science. In particular we discuss implications for the functional programming paradigm, and discuss how to extend categorical semantics to the generalized setting. Next, we present a variant of the calculus of deductive systems developed in the work of Lambek, and give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of cartesian closed categories and exponential-preserving morphisms that leverages the theory of generalized categories. Next, we develop elementary topos theory in the generalized setting of ideal toposes, by building upon the formalism we have previously developed for the extension of the Curry-Howard-Lambek theorem. In particular, we prove that ideal toposes possess the same Heyting algebra structure and squares of adjoints that ordinary toposes do. Finally, we develop generalized sheaves, and show that such categories form ideal toposes. We extend Lawvere and Tierney\u27s theorem relating j-sheaves and sheaves in the sense of Grothendieck to the generalized setting
Hybrid Type-Logical Grammars, First-Order Linear Logic and the Descriptive Inadequacy of Lambda Grammars
In this article we show that hybrid type-logical grammars are a fragment of
first-order linear logic. This embedding result has several important
consequences: it not only provides a simple new proof theory for the calculus,
thereby clarifying the proof-theoretic foundations of hybrid type-logical
grammars, but, since the translation is simple and direct, it also provides
several new parsing strategies for hybrid type-logical grammars. Second,
NP-completeness of hybrid type-logical grammars follows immediately. The main
embedding result also sheds new light on problems with lambda grammars/abstract
categorial grammars and shows lambda grammars/abstract categorial grammars
suffer from problems of over-generation and from problems at the
syntax-semantics interface unlike any other categorial grammar
Geometry of language
Girard (1987) introduced proof nets as a syntax of linear proofs which
eliminates inessential rule ordering manifested by sequent calculus.
Proof nets adapted to the Lambek calculus (Roorda 1991) fulfill a role
in categorial grammar analogous to that of phrase structure trees in
CFG so that categorial proof nets have a central part to play in
computational syntax and semantics; in particular they allow a
reinterpretation of the "problem" of spurious ambiguity as an
opportunity for parallelism. This article aims to make three
contributions: i) provide a tutorial overview of categorial proof
nets, ii) apply and provide motivation for proof nets by showing how
a partial execution eschews the need for semantic evaluation in
language processing, and iii) analyse the intrinsic geometry of
partially commutative proof nets for the kinds of discontinuity
attested in language, offering proof nets for the in situ binder
type-constructor Q(., ., .) of Moortgat (1991/6).Postprint (published version
Proof nets for linguistic analysis
This book investigates the possible linguistic applications of proof nets, redundancy free
representations of proofs, which were introduced by Girard for linear logic.
We will adapt the notion of proof net to allow the formulation of a proof net calculus which is soundand complete for the multimodal Lambek calculus.
Finally, we will investigate the computational and complexity theoretic consequences of this calculus and give an introduction to a practical grammar development tool based on proof nets
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