271 research outputs found
New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized
The definitional equality of an intensional type theory is its test of type
compatibility. Today's systems rely on ordinary evaluation semantics to compare
expressions in types, frustrating users with type errors arising when
evaluation fails to identify two `obviously' equal terms. If only the machine
could decide a richer theory! We propose a way to decide theories which
supplement evaluation with `-rules', rearranging the neutral parts of
normal forms, and report a successful initial experiment.
We study a simple -calculus with primitive fold, map and append operations on
lists and develop in Agda a sound and complete decision procedure for an
equational theory enriched with monoid, functor and fusion laws
Handling Fibred Algebraic Effects
International audienceWe study algebraic computational effects and their handlers in the dependently typed setting. We describecomputational effects using a generalisation of Plotkin and Pretnar’s effect theories, whose dependentlytyped operations allow us to capture precise notions of computation, e.g., state with location-dependent storetypes and dependently typed update monads. Our treatment of handlers is based on an observation that theirconventional term-level definition leads to unsound program equivalences being derivable in languages thatinclude a notion of homomorphism. We solve this problem by giving handlers a novel type-based treatmentvia a new computation type, the user-defined algebra type, which pairs a value type (the carrier) with a set ofvalue terms (the operations), capturing Plotkin and Pretnar’s insight that effect handlers denote algebras. Wethen show that the conventional presentation of handlers can be routinely derived, and demonstrate that thistype-based treatment of handlers provides a useful mechanism for reasoning about effectful computations.We also equip the resulting language with a sound denotational semantics based on families fibrations
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Abstract and concrete type theories
In this thesis, we study abstract and concrete type theories. We introduce an abstract notion of a type theory to obtain general results in the semantics of type theories, but we also provide a syntactic way of presenting a type theory to allow us a further investigation into a concrete type theory to obtain consistency and independence results
Models of Type Theory Based on Moore Paths
This paper introduces a new family of models of intensional Martin-L\"of type
theory. We use constructive ordered algebra in toposes. Identity types in the
models are given by a notion of Moore path. By considering a particular gros
topos, we show that there is such a model that is non-truncated, i.e. contains
non-trivial structure at all dimensions. In other words, in this model a type
in a nested sequence of identity types can contain more than one element, no
matter how great the degree of nesting. Although inspired by existing
non-truncated models of type theory based on simplicial and cubical sets, the
notion of model presented here is notable for avoiding any form of Kan filling
condition in the semantics of types.Comment: This is a revised and expanded version of a paper with the same name
that appeared in the proceedings of the 2nd International Conference on
Formal Structures for Computation and Deduction (FSCD 2017
- …