729 research outputs found
Relational Parametricity and Control
We study the equational theory of Parigot's second-order
λμ-calculus in connection with a call-by-name continuation-passing
style (CPS) translation into a fragment of the second-order λ-calculus.
It is observed that the relational parametricity on the target calculus induces
a natural notion of equivalence on the λμ-terms. On the other hand,
the unconstrained relational parametricity on the λμ-calculus turns
out to be inconsistent with this CPS semantics. Following these facts, we
propose to formulate the relational parametricity on the λμ-calculus
in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc
Second-Order Type Isomorphisms Through Game Semantics
The characterization of second-order type isomorphisms is a purely
syntactical problem that we propose to study under the enlightenment of game
semantics. We study this question in the case of second-order
λ-calculus, which can be seen as an extension of system F to
classical logic, and for which we define a categorical framework: control
hyperdoctrines. Our game model of λ-calculus is based on polymorphic
arenas (closely related to Hughes' hyperforests) which evolve during the play
(following the ideas of Murawski-Ong). We show that type isomorphisms coincide
with the "equality" on arenas associated with types. Finally we deduce the
equational characterization of type isomorphisms from this equality. We also
recover from the same model Roberto Di Cosmo's characterization of type
isomorphisms for system F. This approach leads to a geometrical comprehension
on the question of second order type isomorphisms, which can be easily extended
to some other polymorphic calculi including additional programming features.Comment: accepted by Annals of Pure and Applied Logic, Special Issue on Game
Semantic
Polarizing Double Negation Translations
Double-negation translations are used to encode and decode classical proofs
in intuitionistic logic. We show that, in the cut-free fragment, we can
simplify the translations and introduce fewer negations. To achieve this, we
consider the polarization of the formul{\ae}{} and adapt those translation to
the different connectives and quantifiers. We show that the embedding results
still hold, using a customized version of the focused classical sequent
calculus. We also prove the latter equivalent to more usual versions of the
sequent calculus. This polarization process allows lighter embeddings, and
sheds some light on the relationship between intuitionistic and classical
connectives
Denotational Semantics of the Simplified Lambda-Mu Calculus and a New Deduction System of Classical Type Theory
Classical (or Boolean) type theory is the type theory that allows the type
inference (the type counterpart of
double-negation elimination), where is any type and is
absurdity type. This paper first presents a denotational semantics for a
simplified version of Parigot's lambda-mu calculus, a premier example of
classical type theory. In this semantics the domain of each type is divided
into infinitely many ranks and contains not only the usual members of the type
at rank 0 but also their negative, conjunctive, and disjunctive shadows in the
higher ranks, which form an infinitely nested Boolean structure. Absurdity type
is identified as the type of truth values. The paper then presents a new
deduction system of classical type theory, a sequent calculus called the
classical type system (CTS), which involves the standard logical operators such
as negation, conjunction, and disjunction and thus reflects the discussed
semantic structure in a more straightforward fashion.Comment: In Proceedings CL&C 2016, arXiv:1606.0582
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