10 research outputs found
Indexed linear logic and higher-order model checking
In recent work, Kobayashi observed that the acceptance by an alternating tree
automaton A of an infinite tree T generated by a higher-order recursion scheme
G may be formulated as the typability of the recursion scheme G in an
appropriate intersection type system associated to the automaton A. The purpose
of this article is to establish a clean connection between this line of work
and Bucciarelli and Ehrhard's indexed linear logic. This is achieved in two
steps. First, we recast Kobayashi's result in an equivalent infinitary
intersection type system where intersection is not idempotent anymore. Then, we
show that the resulting type system is a fragment of an infinitary version of
Bucciarelli and Ehrhard's indexed linear logic. While this work is very
preliminary and does not integrate key ingredients of higher-order
model-checking like priorities, it reveals an interesting and promising
connection between higher-order model-checking and linear logic.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
Relational semantics of linear logic and higher-order model-checking
In this article, we develop a new and somewhat unexpected connection between
higher-order model-checking and linear logic. Our starting point is the
observation that once embedded in the relational semantics of linear logic, the
Church encoding of any higher-order recursion scheme (HORS) comes together with
a dual Church encoding of an alternating tree automata (ATA) of the same
signature. Moreover, the interaction between the relational interpretations of
the HORS and of the ATA identifies the set of accepting states of the tree
automaton against the infinite tree generated by the recursion scheme. We show
how to extend this result to alternating parity automata (APT) by introducing a
parametric version of the exponential modality of linear logic, capturing the
formal properties of colors (or priorities) in higher-order model-checking. We
show in particular how to reunderstand in this way the type-theoretic approach
to higher-order model-checking developed by Kobayashi and Ong. We briefly
explain in the end of the paper how his analysis driven by linear logic results
in a new and purely semantic proof of decidability of the formulas of the
monadic second-order logic for higher-order recursion schemes.Comment: 24 pages. Submitte
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
Profinite lambda-terms and parametricity
Combining ideas coming from Stone duality and Reynolds parametricity, we
formulate in a clean and principled way a notion of profinite lambda-term
which, we show, generalizes at every type the traditional notion of profinite
word coming from automata theory. We start by defining the Stone space of
profinite lambda-terms as a projective limit of finite sets of usual
lambda-terms, considered modulo a notion of equivalence based on the finite
standard model. One main contribution of the paper is to establish that,
somewhat surprisingly, the resulting notion of profinite lambda-term coming
from Stone duality lives in perfect harmony with the principles of Reynolds
parametricity. In addition, we show that the notion of profinite lambda-term is
compositional by constructing a cartesian closed category of profinite
lambda-terms, and we establish that the embedding from lambda-terms modulo
beta-eta-conversion to profinite lambda-terms is faithful using Statman's
finite completeness theorem. Finally, we prove that the traditional Church
encoding of finite words into lambda-terms can be extended to profinite words,
and leads to a homeomorphism between the space of profinite words and the space
of profinite lambda-terms of the corresponding Church type
An Indexed Linear Logic for Idempotent Intersection Types (Long version)
Indexed Linear Logic has been introduced by Ehrhard and Bucciarelli, it can
be seen as a logical presentation of non-idempotent intersection types extended
through the relational semantics to the full linear logic. We introduce an
idempotent variant of Indexed Linear Logic. We give a fine-grained
reformulation of the syntax by exposing implicit parameters and by unifying
several operations on formulae via the notion of base change. Idempotency is
achieved by means of an appropriate subtyping relation. We carry on an in-depth
study of indLL as a logic, showing how it determines a refinement of classical
linear logic and establishing a terminating cut-elimination procedure.
Cut-elimination is proved to be confluent up to an appropriate congruence
induced by the subtyping relation
On Phase Semantics and Denotational Semantics in Multiplicative Additive Linear Logic
International audienc
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.