920 research outputs found
On Isomorphism of "Functional" Intersection and Union Types
Type isomorphism is useful for retrieving library components, since a
function in a library can have a type different from, but isomorphic to, the
one expected by the user. Moreover type isomorphism gives for free the coercion
required to include the function in the user program with the right type. The
present paper faces the problem of type isomorphism in a system with
intersection and union types. In the presence of intersection and union,
isomorphism is not a congruence and cannot be characterised in an equational
way. A characterisation can still be given, quite complicated by the
interference between functional and non functional types. This drawback is
faced in the paper by interpreting each atomic type as the set of functions
mapping any argument into the interpretation of the type itself. This choice
has been suggested by the initial projection of Scott's inverse limit
lambda-model. The main result of this paper is a condition assuring type
isomorphism, based on an isomorphism preserving reduction.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
Toward Isomorphism of Intersection and Union types
This paper investigates type isomorphism in a lambda-calculus with
intersection and union types. It is known that in lambda-calculus, the
isomorphism between two types is realised by a pair of terms inverse one each
other. Notably, invertible terms are linear terms of a particular shape, called
finite hereditary permutators. Typing properties of finite hereditary
permutators are then studied in a relevant type inference system with
intersection and union types for linear terms. In particular, an isomorphism
preserving reduction between types is defined. Type reduction is confluent and
terminating, and induces a notion of normal form of types. The properties of
normal types are a crucial step toward the complete characterisation of type
isomorphism. The main results of this paper are, on one hand, the fact that two
types with the same normal form are isomorphic, on the other hand, the
characterisation of the isomorphism between types in normal form, modulo
isomorphism of arrow types.Comment: In Proceedings ITRS 2012, arXiv:1307.784
An Intuitionistic Formula Hierarchy Based on High-School Identities
We revisit the notion of intuitionistic equivalence and formal proof
representations by adopting the view of formulas as exponential polynomials.
After observing that most of the invertible proof rules of intuitionistic
(minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms
corresponding to the high-school identities, we show that one can obtain a more
compact variant of a proof system, consisting of non-invertible proof rules
only, and where the invertible proof rules have been replaced by a formula
normalisation procedure.
Moreover, for certain proof systems such as the G4ip sequent calculus of
Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the
non-invertible proof rules as strict inequalities between exponential
polynomials; a careful combinatorial treatment is given in order to establish
this fact.
Finally, we extend the exponential polynomial analogy to the first-order
quantifiers, showing that it gives rise to an intuitionistic hierarchy of
formulas, resembling the classical arithmetical hierarchy, and the first one
that classifies formulas while preserving isomorphism
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Conversion of HOL Light proofs into Metamath
We present an algorithm for converting proofs from the OpenTheory interchange
format, which can be translated to and from any of the HOL family of proof
languages (HOL4, HOL Light, ProofPower, and Isabelle), into the ZFC-based
Metamath language. This task is divided into two steps: the translation of an
OpenTheory proof into a Metamath HOL formalization, ,
followed by the embedding of the HOL formalization into the main ZFC
foundations of the main Metamath library, . This
process provides a means to link the simplicity of the Metamath foundations to
the intense automation efforts which have borne fruit in HOL Light, allowing
the production of complete Metamath proofs of theorems in HOL Light, while also
proving that HOL Light is consistent, relative to Metamath's ZFC
axiomatization.Comment: 14 pages, 2 figures, accepted to Journal of Formalized Reasonin
Diagrammatic Inference
Diagrammatic logics were introduced in 2002, with emphasis on the notions of
specifications and models. In this paper we improve the description of the
inference process, which is seen as a Yoneda functor on a bicategory of
fractions. A diagrammatic logic is defined from a morphism of limit sketches
(called a propagator) which gives rise to an adjunction, which in turn
determines a bicategory of fractions. The propagator, the adjunction and the
bicategory provide respectively the syntax, the models and the inference
process for the logic. Then diagrammatic logics and their morphisms are applied
to the semantics of side effects in computer languages.Comment: 16 page
Theories of analytic monads
We characterize the equational theories and Lawvere theories that correspond
to the categories of analytic and polynomial monads on Set, and hence also the
categories of the symmetric and rigid operads in Set. We show that the category
of analytic monads is equivalent to the category of regular-linear theories.
The category of polynomial monads is equivalent to the category of rigid
theories, i.e. regular-linear theories satisfying an additional global
condition. This solves a problem A. Carboni and P. T. Johnstone. The Lawvere
theories corresponding to these monads are identified via some factorization
systems.Comment: 29 pages. v2: minor correction
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