13 research outputs found
Non uniform (hyper/multi)coherence spaces
In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs.
Intuitively, vertices represent results of computations and the edge relation
witnesses the ability of being assembled into a same piece of data or a same
(strongly) stable function, at arrow types. In (hyper)coherence semantics, the
argument of a (strongly) stable functional is always a (strongly) stable
function. As a consequence, comparatively to the relational semantics, where
there is no edge relation, some vertices are missing. Recovering these vertices
is essential for the purpose of reconstructing proofs/terms from their
interpretations. It shall also be useful for the comparison with other
semantics, like game semantics. In [BE01], Bucciarelli and Ehrhard introduced a
so called non uniform coherence space semantics where no vertex is missing. By
constructing the co-free exponential we set a new version of this last
semantics, together with non uniform versions of hypercoherences and
multicoherences, a new semantics where an edge is a finite multiset. Thanks to
the co-free construction, these non uniform semantics are deterministic in the
sense that the intersection of a clique and of an anti-clique contains at most
one vertex, a result of interaction, and extensionally collapse onto the
corresponding uniform semantics.Comment: 32 page
Lifting Coalgebra Modalities and IMELL Model Structure to Eilenberg-Moore Categories
A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic (IMELL), known as a linear category, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by R. Blute and P. Scott\u27s work on categories of modules of Hopf algebras as models of linear logic, we study Eilenberg-Moore categories of monads as models of IMELL. We define an IMELL lifting monad on a linear category as a Hopf monad - in the Bruguieres, Lack, and Virelizier sense - with a mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to Eilenberg-Moore categories of IMELL lifting monads. We explain how monoids in the Eilenberg-Moore category of the monoidal coalgebra modality can induce IMELL lifting monads and provide sources for such monoids. Along the way, we also define mixed distributive laws of bimonads over coalgebra modalities and lifting differential category structure to Eilenberg-Moore categories of exponential lifting monads
Lifting Coalgebra Modalities and Model Structure to Eilenberg-Moore Categories
A categorical model of the multiplicative and exponential fragments of
intuitionistic linear logic (), known as a \emph{linear
category}, is a symmetric monoidal closed category with a monoidal coalgebra
modality (also known as a linear exponential comonad). Inspired by Blute and
Scott's work on categories of modules of Hopf algebras as models of linear
logic, we study categories of algebras of monads (also known as Eilenberg-Moore
categories) as models of . We define a lifting
monad on a linear category as a Hopf monad -- in the Brugui{\`e}res, Lack, and
Virelizier sense -- with a special kind of mixed distributive law over the
monoidal coalgebra modality. As our main result, we show that the linear
category structure lifts to the category of algebras of lifting
monads. We explain how groups in the category of coalgebras of the monoidal
coalgebra modality induce lifting monads and provide a source
for such groups from enrichment over abelian groups. Along the way we also
define mixed distributive laws of symmetric comonoidal monads over symmetric
monoidal comonads and lifting differential category structure.Comment: An extend abstract version of this paper appears in the conference
proceedings of the 3rd International Conference on Formal Structures for
Computation and Deduction (FSCD 2018), under the title "Lifting Coalgebra
Modalities and Model Structure to Eilenberg-Moore Categories
On dialogue games and coherent strategies
We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended with a necessity modality. The translation is done in such a way that every innocent strategy defines a clique or a configuration in the resulting space of positions. This leads us to study the notion of configuration designed by Curien, Plotkin and Winskel for general bistructures in the particular case of a bistructure associated to a dialogue game. We show that every such configuration may be seen as an interactive strategy equipped with a backward as well as a forward dynamics based on the interplay between the stable order and the extensional order. In that way, the category of bistructures is shown to include a full subcategory of games and coherent strategies of an interesting nature
Some Programming Languages Suggested by Game Models (Extended Abstract)
AbstractWe consider a simple and well-known category of alternating games (also known as sequential data structures) and several categories derived from it. In each case, we present an extension of Plotkin's language FPC (or a suitable linearization thereof) which defines all computable strategies of appropriate types. The quest for such languages results in a novel selection of language primitives for state encapsulation, coroutining and backtracking
Monoidal-Closed Categories of Tree Automata
We propose a realizability semantics for automata on infinite trees, based on categories of games built on usual simple games, and generalizing usual acceptance games of tree automata. Our approach can be summarized with the slogan " automata as objects, strategies as morphisms ". We show that the operations on tree automata used in the translations of MSO-formulae to automata (underlying Rabin's Theorem, that is the decidability of MSO on infinite trees) can be organized in a deduction system based on the multiplica-tive fragment of intuitionistic linear logic (ILL). Namely, we equip a variant of usual alternating tree automata (that we call uniform tree automata) with a fi-bred monoidal closed structure which in particular, via game determinacy handles a linear complementation of alternating automata, as well as deduction rules for exis-tential and universal quantifications. This monoidal structure is actually Cartesian on non-deterministic automata. Moreover, an adaptation of a usual construction for the simulation of alternating automata by non-deterministic ones satisfies the deduction rules of the !(−) ILL-exponential modality. Our realizability semantics satisfies an expected property of witness extraction from proofs of existential statements. Moreover, it allows to combine realizers produced as interpretations of proofs with strategies witnessing (non-)emptiness of tree automata, possibly obtained using external algorithms
Game semantics for an object-oriented language
This thesis investigates the relationship between object-oriented programming
languages and game models of computation. These are intuitively well matched:
an object encapsulates some internal state and presents some behaviour to the
world via its publicly visible methods, while a strategy for some game represents
the possible interactions of a program with its environment.
We work with a simple and well-understood game model. Rather than tailoring
our model to match some existing programming language, we view the
simplicity of our semantic setting as a virtue, and try to find the appropriate
language corresponding to the model.
We define a class-based, stateful object-oriented language, and give a heapbased
operational semantics and an interpretation in our game model. At the
heart of this interpretation lies a novel semantic treatment of the phenomenon
of data abstraction. The model closely guides the design of our language, which
enjoys an intermediate level of expressivity between that of first-order and general
higher-order store.
The agreement between the operational and game interpretations is verified
by a soundness proof. This involves the development of specialised techniques
and a detailed analysis of the relationship between the concrete and abstract
views. We also show that definability and full abstraction hold at certain types
of arbitrary rank, but are problematic at other types.
We conclude by briefly discussing an extended language with a control operator,
along with other extensions leading to a possible core for a more realistic
programming language