12 research outputs found
Curry-style type Isomorphisms and Game Semantics
Curry-style system F, ie. system F with no explicit types in terms, can be
seen as a core presentation of polymorphism from the point of view of
programming languages. This paper gives a characterisation of type isomorphisms
for this language, by using a game model whose intuitions come both from the
syntax and from the game semantics universe. The model is composed of: an
untyped part to interpret terms, a notion of game to interpret types, and a
typed part to express the fact that an untyped strategy plays on a game. By
analysing isomorphisms in the model, we prove that the equational system
corresponding to type isomorphisms for Curry-style system F is the extension of
the equational system for Church-style isomorphisms with a new, non-trivial
equation: forall X.A = A[forall Y.Y/X] if X appears only positively in A.Comment: Accept\'e \`a Mathematical Structures for Computer Science, Special
Issue on Type Isomorphism
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
Quantification du second ordre en sémentique des jeux (application aux isomorphismes de types)
PARIS7-Bibliothèque centrale (751132105) / SudocSudocFranceF
Anomalous price impact and the critical nature of liquidity in financial markets
We propose a dynamical theory of market liquidity that predicts that the average supply/demand profile is V-shaped and {\it vanishes} around the current price. This result is generic, and only relies on mild assumptions about the order flow and on the fact that prices are (to a first approximation) diffusive. This naturally accounts for two striking stylized facts: first, large metaorders have to be fragmented in order to be digested by the liquidity funnel, leading to long-memory in the sign of the order flow. Second, the anomalously small local liquidity induces a breakdown of linear response and a diverging impact of small orders, explaining the "square-root" impact law, for which we provide additional empirical support. Finally, we test our arguments quantitatively using a numerical model of order flow based on the same minimal ingredients.