24 research outputs found
Open Higher-Order Logic
We introduce a variation on Barthe et al.ās higher-order logic in which formulas are interpreted as predicates over open rather than closed objects. This way, concepts which have an intrinsically functional nature, like continuity, differentiability, or monotonicity, can be expressed and reasoned about in a very natural way, following the structure of the underlying program. We give open higher-order logic in distinct flavors, and in particular in its relational and local versions, the latter being tailored for situations in which properties hold only in part of the underlying functionās domain of definition
Abstract Clones for Abstract Syntax
We give a formal treatment of simple type theories, such as the simply-typed ?-calculus, using the framework of abstract clones. Abstract clones traditionally describe first-order structures, but by equipping them with additional algebraic structure, one can further axiomatize second-order, variable-binding operators. This provides a syntax-independent representation of simple type theories. We describe multisorted second-order presentations, such as the presentation of the simply-typed ?-calculus, and their clone-theoretic algebras; free algebras on clones abstractly describe the syntax of simple type theories quotiented by equations such as ?- and ?-equality. We give a construction of free algebras and derive a corresponding induction principle, which facilitates syntax-independent proofs of properties such as adequacy and normalization for simple type theories. Working only with clones avoids some of the complexities inherent in presheaf-based frameworks for abstract syntax
Cubical Syntax for Reflection-Free Extensional Equality
We contribute XTT, a cubical reconstruction of Observational Type Theory
which extends Martin-L\"of's intensional type theory with a dependent equality
type that enjoys function extensionality and a judgmental version of the
unicity of identity types principle (UIP): any two elements of the same
equality type are judgmentally equal. Moreover, we conjecture that the typing
relation can be decided in a practical way. In this paper, we establish an
algebraic canonicity theorem using a novel cubical extension (independently
proposed by Awodey) of the logical families or categorical gluing argument
inspired by Coquand and Shulman: every closed element of boolean type is
derivably equal to either 'true' or 'false'.Comment: Extended version; International Conference on Formal Structures for
Computation and Deduction (FSCD), 201
Recursion and Sequentiality in Categories of Sheaves
We present a fully abstract model of a call-by-value language with
higher-order functions, recursion and natural numbers, as an exponential ideal
in a topos. Our model is inspired by the fully abstract models of O'Hearn,
Riecke and Sandholm, and Marz and Streicher. In contrast with semantics based
on cpo's, we treat recursion as just one feature in a model built by combining
a choice of modular components
Lax Logical Relations
Lax logical relations are a categorical generalisation of logical
relations; though they preserve product types, they need not preserve
exponential types. But, like logical relations, they are preserved by the
meanings of all lambda-calculus terms.We show that lax logical relations
coincide with the correspondences of Schoett, the algebraic relations of
Mitchell and the pre-logical relations of Honsell and Sannella on Henkin
models, but also generalise naturally to models in cartesian closed categories
and to richer languages
Fully abstract models for effectful Ī»-calculi via category-theoretic logical relations
We present a construction which, under suitable assumptions, takes a model of Moggiās computational Ī»-calculus with sum types, effect operations and primitives, and yields a model that is adequate and fully abstract. The construction, which uses the theory of fibrations, categorical glueing, ā¤ā¤-lifting, and ā¤ā¤-closure, takes inspiration from OāHearn & Rieckeās fully abstract model for PCF. Our construction can be applied in the category of sets and functions, as well as the category of diffeological spaces and smooth maps and the category of quasi-Borel spaces, which have been studied as semantics for differentiable and probabilistic programming