8,317 research outputs found
A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4
We propose a modal linear logic to reformulate intuitionistic modal logic S4 (IS4) in terms of linear logic, establishing an S4-version of Girard translation from IS4 to it. While the Girard translation from intuitionistic logic to linear logic is well-known, its extension to modal logic is non-trivial since a naive combination of the S4 modality and the exponential modality causes an undesirable interaction between the two modalities. To solve the problem, we introduce an extension of intuitionistic multiplicative exponential linear logic with a modality combining the S4 modality and the exponential modality, and show that it admits a sound translation from IS4. Through the Curry-Howard correspondence we further obtain a Geometry of Interaction Machine semantics of the modal lambda-calculus by Pfenning and Davies for staged computation
Type-driven semantic interpretation and feature dependencies in R-LFG
Once one has enriched LFG's formal machinery with the linear logic mechanisms
needed for semantic interpretation as proposed by Dalrymple et. al., it is
natural to ask whether these make any existing components of LFG redundant. As
Dalrymple and her colleagues note, LFG's f-structure completeness and coherence
constraints fall out as a by-product of the linear logic machinery they propose
for semantic interpretation, thus making those f-structure mechanisms
redundant. Given that linear logic machinery or something like it is
independently needed for semantic interpretation, it seems reasonable to
explore the extent to which it is capable of handling feature structure
constraints as well.
R-LFG represents the extreme position that all linguistically required
feature structure dependencies can be captured by the resource-accounting
machinery of a linear or similiar logic independently needed for semantic
interpretation, making LFG's unification machinery redundant. The goal is to
show that LFG linguistic analyses can be expressed as clearly and perspicuously
using the smaller set of mechanisms of R-LFG as they can using the much larger
set of unification-based mechanisms in LFG: if this is the case then we will
have shown that positing these extra f-structure mechanisms is not
linguistically warranted.Comment: 30 pages, to appear in the the ``Glue Language'' volume edited by
Dalrymple, uses tree-dvips, ipa, epic, eepic, fullnam
Density Matrices with Metric for Derivational Ambiguity
Recent work on vector-based compositional natural language semantics has
proposed the use of density matrices to model lexical ambiguity and (graded)
entailment (e.g. Piedeleu et al 2015, Bankova et al 2019, Sadrzadeh et al
2018). Ambiguous word meanings, in this work, are represented as mixed states,
and the compositional interpretation of phrases out of their constituent parts
takes the form of a strongly monoidal functor sending the derivational
morphisms of a pregroup syntax to linear maps in FdHilb. Our aims in this paper
are threefold. Firstly, we replace the pregroup front end by a Lambek
categorial grammar with directional implications expressing a word's
selectional requirements. By the Curry-Howard correspondence, the derivations
of the grammar's type logic are associated with terms of the (ordered) linear
lambda calculus; these terms can be read as programs for compositional meaning
assembly with density matrices as the target semantic spaces. Secondly, we
extend on the existing literature and introduce a symmetric, nondegenerate
bilinear form called a "metric" that defines a canonical isomorphism between a
vector space and its dual, allowing us to keep a distinction between left and
right implication. Thirdly, we use this metric to define density matrix spaces
in a directional form, modeling the ubiquitous derivational ambiguity of
natural language syntax, and show how this alows an integrated treatment of
lexical and derivational forms of ambiguity controlled at the level of the
interpretation.Comment: 24 pages, 10 figures. SemSpace 2019, to appear in J. of Applied
Logic
Dual-Context Calculi for Modal Logic
We present natural deduction systems and associated modal lambda calculi for
the necessity fragments of the normal modal logics K, T, K4, GL and S4. These
systems are in the dual-context style: they feature two distinct zones of
assumptions, one of which can be thought as modal, and the other as
intuitionistic. We show that these calculi have their roots in in sequent
calculi. We then investigate their metatheory, equip them with a confluent and
strongly normalizing notion of reduction, and show that they coincide with the
usual Hilbert systems up to provability. Finally, we investigate a categorical
semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see
arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089
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