arXiv subjects

Harley Eades III

Publications and source records attributed to Harley Eades III.

13 recordsLinked to original sources

arXiv2026

A unification of graded and substructural logics

Type systems which account for resource sensitive computations can generally be split into two styles: First, substructural logics such as Linear Logic which seek to restrict weakening and contraction and reintroduce them in a controlled manner; And second, graded systems which allow weakening and contraction by default, but track the use of variables quantitatively in some algebraic structure -- usually a semiring. We present GRASS (Graded and substructural), a type system which incorporates mechanisms from both of these approaches, thus allowing maximally flexible control over variable usage. Furthermore, GRASS allows grades from an arbitrary collection of grade algebras to coexist in the same system, thus allowing different variables to be controlled with respect to different notions of resource within the same program. We develop the categorical semantics of \gyaru{}, and find that, on the level of categorical semantics, it subsumes multiple established systems such as LNL, Adjoint Logic, and mGL.

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arXiv2024

A Mixed Linear and Graded Logic: Proofs, Terms, and Models (with appendices)

Graded modal logics generalise standard modal logics via families of modalities indexed by an algebraic structure whose operations mediate between the different modalities. The graded "of-course" modality $!_r$ captures how many times a proposition is used and has an analogous interpretation to the of-course modality from linear logic; the of-course modality from linear logic can be modelled by a linear exponential comonad and graded of-course can be modelled by a graded linear exponential comonad. Benton showed in his seminal paper on Linear/Non-Linear logic that the of-course modality can be split into two modalities connecting intuitionistic logic with linear logic, forming a symmetric monoidal adjunction. Later, Fujii et al. demonstrated that every graded comonad can be decomposed into an adjunction and a `strict action'. We give a similar result to Benton, leveraging Fujii et al.'s decomposition, showing that graded modalities can be split into two modalities connecting a graded logic with a graded linear logic. We propose a sequent calculus, its proof theory and categorical model, and a natural deduction system which we show is isomorphic to the sequent calculus system. Interestingly, our system can also be understood as Linear/Non-Linear logic composed with an action that adds the grading, further illuminating the shared principles between linear logic and a class of graded modal logics.

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arXiv2023

Combining dependency, grades, and adjoint logic

We propose two new dependent type systems. The first, is a dependent graded/linear type system where a graded dependent type system is connected via modal operators to a linear type system in the style of Linear/Non-linear logic. We then generalize this system to support many graded systems connected by many modal operators through the introduction of modes from Adjoint Logic. Finally, we prove several meta-theoretic properties of these two systems including graded substitution.

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arXiv2022

A Dependent Dependency Calculus (Extended Version)

Over twenty years ago, Abadi et al. established the Dependency Core Calculus (DCC) as a general purpose framework for analyzing dependency in typed programming languages. Since then, dependency analysis has shown many practical benefits to language design: its results can help users and compilers enforce security constraints, eliminate dead code, among other applications. In this work, we present a Dependent Dependency Calculus (DDC), which extends this general idea to the setting of a dependently-typed language. We use this calculus to track both run-time and compile-time irrelevance, enabling faster type-checking and program execution.

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arXiv2021

Graded Modal Dependent Type Theory

Graded type theories are an emerging paradigm for augmenting the reasoning power of types with parameterizable, fine-grained analyses of program properties. There have been many such theories in recent years which equip a type theory with quantitative dataflow tracking, usually via a semiring-like structure which provides analysis on variables (often called `quantitative' or `coeffect' theories). We present Graded Modal Dependent Type Theory (GrTT for short), which equips a dependent type theory with a general, parameterizable analysis of the flow of data, both in and between computational terms and types. In this theory, it is possible to study, restrict, and reason about data use in programs and types, enabling, for example, parametric quantifiers and linearity to be captured in a dependent setting. We propose GrTT, study its metatheory, and explore various case studies of its use in reasoning about programs and studying other type theories. We have implemented the theory and highlight the interesting details, including showing an application of grading to optimising the type checking procedure itself.

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arXiv2021

A graded dependent type system with a usage-aware semantics (extended version)

Graded Type Theory provides a mechanism to track and reason about resource usage in type systems. In this paper, we develop GraD, a novel version of such a graded dependent type system that includes functions, tensor products, additive sums, and a unit type. Since standard operational semantics is resource-agnostic, we develop a heap-based operational semantics and prove a soundness theorem that shows correct accounting of resource usage. Several useful properties, including the standard type soundness theorem, non-interference of irrelevant resources in computation and single pointer property for linear resources, can be derived from this theorem. We hope that our work will provide a base for integrating linearity, irrelevance and dependent types in practical programming languages like Haskell.

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arXiv2020

Grading Adjoint Logic

We introduce a new logic that combines Adjoint Logic with Graded Necessity Modalities. This results in a very expressive system capable of controlling when and how structural rules are used. We give a sequent calculus, natural deduction, and term assignment for Graded Adjoint Logic.

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arXiv2020

Unifying graded and parameterised monads

Monads are a useful tool for structuring effectful features of computation such as state, non-determinism, and continuations. In the last decade, several generalisations of monads have been suggested which provide a more fine-grained model of effects by replacing the single type constructor of a monad with an indexed family of constructors. Most notably, graded monads (indexed by a monoid) model effect systems and parameterised monads (indexed by pairs of pre- and post-conditions) model program logics. This paper studies the relationship between these two generalisations of monads via a third generalisation. This third generalisation, which we call category-graded monads, arises by generalising a view of monads as a particular special case of lax functors. A category-graded monad provides a family of functors T f indexed by morphisms f of some other category. This allows certain compositions of effects to be ruled out (in the style of a program logic) as well as an abstract description of effects (in the style of an effect system). Using this as a basis, we show how graded and parameterised monads can be unified, studying their similarities and differences along the way.

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arXiv2019

On the Lambek Calculus with an Exchange Modality

In this paper we introduce Commutative/Non-Commutative Logic (CNC logic) and two categorical models for CNC logic. This work abstracts Benton's Linear/Non-Linear Logic by removing the existence of the exchange structural rule. One should view this logic as composed of two logics; one sitting to the left of the other. On the left, there is intuitionistic linear logic, and on the right is a mixed commutative/non-commutative formalization of the Lambek calculus. Then both of these logics are connected via a pair of monoidal adjoint functors. An exchange modality is then derivable within the logic using the adjunction between both sides. Thus, the adjoint functors allow one to pull the exchange structural rule from the left side to the right side. We then give a categorical model in terms of a monoidal adjunction, and then a concrete model in terms of dialectica Lambek spaces.

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arXiv2018

An Intuitionistic Linear Logical Semantics of SAND Attack Trees

In this paper we introduce a new logical foundation of SAND attack trees in intuitionistic linear logic. This new foundation is based on a new logic called the Attack Tree Linear Logic (ATLL). Before introducing ATLL we given several new logical models of attack trees, the first, is a very basic model based in truth tables. Then we lift this semantics into a semantics of attack trees based on lineales which introduces implication, but this can be further lifted into a dialectica model which ATLL is based. One important feature of ATLL is that it supports full distributivity of sequential conjunction over choice.

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arXiv2018

Dialectica Categories for the Lambek Calculus

We revisit the old work of de Paiva on the models of the Lambek Calculus in dialectica models making sure that the syntactic details that were sketchy on the first version got completed and verified. We extend the Lambek Calculus with a κmodality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality !, as Girard did, to re-introduce weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present the categorical semantics, proved sound and complete. Finally we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.

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arXiv2017

A Cointuitionistic Adjoint Logic

One leading question with respect to Bi-intuitionistic logic (BINT) is, what does BINT look like across the three arcs -- logic, typed $λ$-calculi, and category theory -- of the Curry-Howard-Lambek correspondence? Categorically, BINT can be seen as a mixing of two worlds: the first being intuitionistic logic (IL), which is modeled by a cartesian closed category, and the second being the dual to intuitionistic logic called cointuitionistic logic (coIL), which is modeled by a cocartesian coclosed category. Crolard showed that combining these two categories into the same category results in it degenerating to a poset. However, this degeneration does not occur when both logics are linear. We propose that IL and coIL need to be separated, and then mixed in a controlled way using the modalities from linear logic. This separation can be ultimately achieved by an adjoint formalization of bi-intuitionistic logic. This formalization consists of three worlds instead of two: the first is intuitionistic logic, the second is linear bi-intuitionistic (Bi-ILL), and the third is cointuitionistic logic. They are then related via two adjunctions. The adjunction between IL and ILL is known as a Linear/Non-linear model (LNL model) of ILL, and is due to Benton. However, the dual to LNL models which would amount to the adjunction between coILL and coIL has yet to appear in the literature. In this paper we fill this gap by studying the dual to LNL models which we call dual LNL models. We show that dual LNL models correspond to dual linear categories, the dual to Bierman's linear categories proposed by Bellin. Then we give the definition of bi-LNL models by combining our model with LNL models to obtain a new model of bi-intuitionistic logic. Finally, we give a corresponding sequent calculus, natural deduction, and term assignment for dual LNL models.

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arXiv2016

Dualized Simple Type Theory

We propose a new bi-intuitionistic type theory called Dualized Type Theory (DTT). It is a simple type theory with perfect intuitionistic duality, and corresponds to a single-sided polarized sequent calculus. We prove DTT strongly normalizing, and prove type preservation. DTT is based on a new propositional bi-intuitionistic logic called Dualized Intuitionistic Logic (DIL) that builds on Pinto and Uustalu's logic L. DIL is a simplification of L by removing several admissible inference rules while maintaining consistency and completeness. Furthermore, DIL is defined using a dualized syntax by labeling formulas and logical connectives with polarities thus reducing the number of inference rules needed to define the logic. We give a direct proof of consistency, but prove completeness by reduction to L.

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