74 research outputs found
Isomorphisms of types in the presence of higher-order references
We investigate the problem of type isomorphisms in a programming language
with higher-order references. We first recall the game-theoretic model of
higher-order references by Abramsky, Honda and McCusker. Solving an open
problem by Laurent, we show that two finitely branching arenas are isomorphic
if and only if they are geometrically the same, up to renaming of moves
(Laurent's forest isomorphism). We deduce from this an equational theory
characterizing isomorphisms of types in a finitary language with higher order
references. We show however that Laurent's conjecture does not hold on
infinitely branching arenas, yielding a non-trivial type isomorphism in the
extension of this language with natural numbers.Comment: Twenty-Sixth Annual IEEE Symposium on Logic In Computer Science (LICS
2011), Toronto : Canada (2011
Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended version)
We show that a version of Martin-L\"of type theory with an extensional
identity type former I, a unit type N1 , Sigma-types, Pi-types, and a base type
is a free category with families (supporting these type formers) both in a 1-
and a 2-categorical sense. It follows that the underlying category of contexts
is a free locally cartesian closed category in a 2-categorical sense because of
a previously proved biequivalence. We show that equality in this category is
undecidable by reducing it to the undecidability of convertibility in
combinatory logic. Essentially the same construction also shows a slightly
strengthened form of the result that equality in extensional Martin-L\"of type
theory with one universe is undecidable
Thin Games with Symmetry and Concurrent Hyland-Ong Games
We build a cartesian closed category, called Cho, based on event structures.
It allows an interpretation of higher-order stateful concurrent programs that
is refined and precise: on the one hand it is conservative with respect to
standard Hyland-Ong games when interpreting purely functional programs as
innocent strategies, while on the other hand it is much more expressive. The
interpretation of programs constructs compositionally a representation of their
execution that exhibits causal dependencies and remembers the points of
non-deterministic branching.The construction is in two stages. First, we build
a compact closed category Tcg. It is a variant of Rideau and Winskel's category
CG, with the difference that games and strategies in Tcg are equipped with
symmetry to express that certain events are essentially the same. This is
analogous to the underlying category of AJM games enriching simple games with
an equivalence relations on plays. Building on this category, we construct the
cartesian closed category Cho as having as objects the standard arenas of
Hyland-Ong games, with strategies, represented by certain events structures,
playing on games with symmetry obtained as expanded forms of these arenas.To
illustrate and give an operational light on these constructions, we interpret
(a close variant of) Idealized Parallel Algol in Cho
An Analysis of Symmetry in Quantitative Semantics
In this paper, we build on a recent bicategorical model called thin spans of
groupoids, introduced by Clairambault and Forest. Notably, thin spans feature a
decomposition of symmetry into two sub-groupoids of polarized -- positive and
negative -- symmetries. We first construct a variation of the original
exponential of thin spans, based on sequences rather than families. Then we
give a syntactic characterisation of the interpretation of simply-typed
lambda-terms in thin spans, in terms of rigid intersection types and rigid
resource terms. Finally, we formally relate thin spans with the weighted
relational model and generalized species of structure. This allows us to show
how some quantities in those models reflect polarized symmetries: in particular
we show that the weighted relational model counts witnesses from generalized
species of structure, divided by the cardinal of a group of positive
symmetries
Observably Deterministic Concurrent Strategies and Intensional Full Abstraction for Parallel-or
International audienceAlthough Plotkin's parallel-or is inherently deterministic, it has a non-deterministic interpretation in games based on (prime) event structures-in which an event has a unique causal history-because they do not directly support disjunctive causality. General event structures can express disjunctive causality and have a more permissive notion of determinism, but do not support hiding. We show that (structures equivalent to) deterministic general event structures do support hiding, and construct a new category of games based on them with a deterministic interpretation of aPCFpor, an affine variant of PCF extended with parallel-or. We then exploit this deterministic interpretation to give a relaxed notion of determinism (observable determinism) on the plain event structures model. Putting this together with our previously introduced concurrent notions of well-bracketing and innocence, we obtain an intensionally fully abstract model of aPCFpor
Extensional Taylor Expansion
We introduce a calculus of extensional resource terms. These are resource
terms \`a la Ehrhard-Regnier, but in infinite -long form, while retaining
a finite syntax and dynamics: in particular, we prove strong confluence and
normalization.
Then we define an extensional version of Taylor expansion, mapping ordinary
-terms to sets (or infinite linear combinations) of extensional
resource terms: just like for ordinary Taylor expansion, the dynamics of our
resource calculus allows to simulate the -reduction of -terms;
the extensional nature of expansion shows in that we are also able to simulate
-reduction.
In a sense, extensional resource terms form a language of (non-necessarily
normal) finite approximants of Nakajima trees, much like ordinary resource
terms are approximants of B\"ohm-trees. Indeed, we show that the equivalence
induced on -terms by the normalization of extensional Taylor-expansion
is nothing but , the greatest consistent sensible -theory.
Taylor expansion has profoundly renewed the approximation theory of the
-calculus by providing a quantitative alternative to order-based
approximation techniques, such as Scott continuity and B\"ohm trees.
Extensional Taylor expansion enjoys similar advantages: e.g., to exhibit models
of , it is now sufficient to provide a model of the extensional resource
calculus. We apply this strategy to give a new, elementary proof of a result by
Manzonetto: is the -theory induced by a well-chosen reflexive
object in the relational model of the -calculus
Games and Strategies as Event Structures.
In 2011, Rideau and Winskel introduced concurrent games and strategies as
event structures, generalizing prior work on causal formulations of games. In this paper we give a detailed, self-contained and slightly-updated account of the results of Rideau and Winskel: a notion of pre-strategy based on event structures; a characterisation of those pre-strategies (deemed strategies) which are preserved by composition with a copycat strategy; and the construction of a bicategory of these strategies. Furthermore, we prove
that the corresponding category has a compact closed structure, and hence forms the basis for the semantics of concurrent higher-order computation
The Biequivalence of Locally Cartesian Closed Categories and Martin-L\"of Type Theories
Seely's paper "Locally cartesian closed categories and type theory" contains
a well-known result in categorical type theory: that the category of locally
cartesian closed categories is equivalent to the category of Martin-L\"of type
theories with Pi-types, Sigma-types and extensional identity types. However,
Seely's proof relies on the problematic assumption that substitution in types
can be interpreted by pullbacks. Here we prove a corrected version of Seely's
theorem: that the B\'enabou-Hofmann interpretation of Martin-L\"of type theory
in locally cartesian closed categories yields a biequivalence of 2-categories.
To facilitate the technical development we employ categories with families as a
substitute for syntactic Martin-L\"of type theories. As a second result we
prove that if we remove Pi-types the resulting categories with families are
biequivalent to left exact categories.Comment: TLCA 2011 - 10th Typed Lambda Calculi and Applications, Novi Sad :
Serbia (2011
Strategies as Resource Terms, and their Categorical Semantics
As shown by Tsukada and Ong, simply-typed, normal and eta-long resource terms
correspond to plays in Hyland-Ong games, quotiented by Melli\`es' homotopy
equivalence. The original proof of this inspiring result is indirect, relying
on the injectivity of the relational model w.r.t. both sides of the
correspondence -- in particular, the dynamics of the resource calculus is taken
into account only via the compatibility of the relational model with the
composition of normal terms defined by normalization.
In the present paper, we revisit and extend these results. Our first
contribution is to restate the correspondence by considering causal structures
we call augmentations, which are canonical representatives of Hyland-Ong plays
up to homotopy. This allows us to give a direct and explicit account of the
connection with normal resource terms. As a second contribution, we extend this
account to the reduction of resource terms: building on a notion of strategies
as weighted sums of augmentations, we provide a denotational model of the
resource calculus, invariant under reduction. A key step -- and our third
contribution -- is a categorical model we call a resource category, which is to
the resource calculus what differential categories are to the differential
lambda-calculus.Comment: extended versio
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