5,125 research outputs found
Translating and Evolving: Towards a Model of Language Change in DisCoCat
The categorical compositional distributional (DisCoCat) model of meaning
developed by Coecke et al. (2010) has been successful in modeling various
aspects of meaning. However, it fails to model the fact that language can
change. We give an approach to DisCoCat that allows us to represent language
models and translations between them, enabling us to describe translations from
one language to another, or changes within the same language. We unify the
product space representation given in (Coecke et al., 2010) and the functorial
description in (Kartsaklis et al., 2013), in a way that allows us to view a
language as a catalogue of meanings. We formalize the notion of a lexicon in
DisCoCat, and define a dictionary of meanings between two lexicons. All this is
done within the framework of monoidal categories. We give examples of how to
apply our methods, and give a concrete suggestion for compositional translation
in corpora.Comment: In Proceedings CAPNS 2018, arXiv:1811.0270
Fixed-point elimination in the intuitionistic propositional calculus
It is a consequence of existing literature that least and greatest
fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic
models of the Intuitionistic Propositional Calculus-always exist, even when
these algebras are not complete as lattices. The reason is that these extremal
fixed-points are definable by formulas of the IPC. Consequently, the
-calculus based on intuitionistic logic is trivial, every -formula
being equivalent to a fixed-point free formula. We give in this paper an
axiomatization of least and greatest fixed-points of formulas, and an algorithm
to compute a fixed-point free formula equivalent to a given -formula. The
axiomatization of the greatest fixed-point is simple. The axiomatization of the
least fixed-point is more complex, in particular every monotone formula
converges to its least fixed-point by Kleene's iteration in a finite number of
steps, but there is no uniform upper bound on the number of iterations. We
extract, out of the algorithm, upper bounds for such n, depending on the size
of the formula. For some formulas, we show that these upper bounds are
polynomial and optimal
The Structure of First-Order Causality
Game semantics describe the interactive behavior of proofs by interpreting
formulas as games on which proofs induce strategies. Such a semantics is
introduced here for capturing dependencies induced by quantifications in
first-order propositional logic. One of the main difficulties that has to be
faced during the elaboration of this kind of semantics is to characterize
definable strategies, that is strategies which actually behave like a proof.
This is usually done by restricting the model to strategies satisfying subtle
combinatorial conditions, whose preservation under composition is often
difficult to show. Here, we present an original methodology to achieve this
task, which requires to combine advanced tools from game semantics, rewriting
theory and categorical algebra. We introduce a diagrammatic presentation of the
monoidal category of definable strategies of our model, by the means of
generators and relations: those strategies can be generated from a finite set
of atomic strategies and the equality between strategies admits a finite
axiomatization, this equational structure corresponding to a polarized
variation of the notion of bialgebra. This work thus bridges algebra and
denotational semantics in order to reveal the structure of dependencies induced
by first-order quantifiers, and lays the foundations for a mechanized analysis
of causality in programming languages
Towards Functorial Language-Games
In categorical compositional semantics of natural language one studies
functors from a category of grammatical derivations (such as a Lambek pregroup)
to a semantic category (such as real vector spaces). We compositionally build
game-theoretic semantics of sentences by taking the semantic category to be the
category whose morphisms are open games. This requires some modifications to
the grammar category to compensate for the failure of open games to form a
compact closed category. We illustrate the theory using simple examples of
Wittgenstein's language-games.Comment: In Proceedings CAPNS 2018, arXiv:1811.0270
Equational Characterization of Covariant-Contravariant Simulation and Conformance Simulation Semantics
Covariant-contravariant simulation and conformance simulation generalize
plain simulation and try to capture the fact that it is not always the case
that "the larger the number of behaviors, the better". We have previously
studied their logical characterizations and in this paper we present the
axiomatizations of the preorders defined by the new simulation relations and
their induced equivalences. The interest of our results lies in the fact that
the axiomatizations help us to know the new simulations better, understanding
in particular the role of the contravariant characteristics and their interplay
with the covariant ones; moreover, the axiomatizations provide us with a
powerful tool to (algebraically) prove results of the corresponding semantics.
But we also consider our results interesting from a metatheoretical point of
view: the fact that the covariant-contravariant simulation equivalence is
indeed ground axiomatizable when there is no action that exhibits both a
covariant and a contravariant behaviour, but becomes non-axiomatizable whenever
we have together actions of that kind and either covariant or contravariant
actions, offers us a new subtle example of the narrow border separating
axiomatizable and non-axiomatizable semantics. We expect that by studying these
examples we will be able to develop a general theory separating axiomatizable
and non-axiomatizable semantics.Comment: In Proceedings SOS 2010, arXiv:1008.190
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
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