7,203 research outputs found
Complexity of Grammar Induction for Quantum Types
Most categorical models of meaning use a functor from the syntactic category
to the semantic category. When semantic information is available, the problem
of grammar induction can therefore be defined as finding preimages of the
semantic types under this forgetful functor, lifting the information flow from
the semantic level to a valid reduction at the syntactic level. We study the
complexity of grammar induction, and show that for a variety of type systems,
including pivotal and compact closed categories, the grammar induction problem
is NP-complete. Our approach could be extended to linguistic type systems such
as autonomous or bi-closed categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Equational reasoning with context-free families of string diagrams
String diagrams provide an intuitive language for expressing networks of
interacting processes graphically. A discrete representation of string
diagrams, called string graphs, allows for mechanised equational reasoning by
double-pushout rewriting. However, one often wishes to express not just single
equations, but entire families of equations between diagrams of arbitrary size.
To do this we define a class of context-free grammars, called B-ESG grammars,
that are suitable for defining entire families of string graphs, and crucially,
of string graph rewrite rules. We show that the language-membership and
match-enumeration problems are decidable for these grammars, and hence that
there is an algorithm for rewriting string graphs according to B-ESG rewrite
patterns. We also show that it is possible to reason at the level of grammars
by providing a simple method for transforming a grammar by string graph
rewriting, and showing admissibility of the induced B-ESG rewrite pattern.Comment: International Conference on Graph Transformation, ICGT 2015. The
final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-21145-9_
Two linearities for quantum computing in the lambda calculus
We propose a way to unify two approaches of non-cloning in quantum
lambda-calculi: logical and algebraic linearities. The first approach is to
forbid duplicating variables, while the second is to consider all lambda-terms
as algebraic-linear functions. We illustrate this idea by defining a quantum
extension of first-order simply-typed lambda-calculus, where the type is linear
on superposition, while allows cloning base vectors. In addition, we provide an
interpretation of the calculus where superposed types are interpreted as vector
spaces and non-superposed types as their basis.Comment: Long journal version of TPNC'17 paper
(doi:10.1007/978-3-319-71069-3_22) extended with third author's
"Licenciatura"'s thesi
Exploring the N-th Dimension of Language
This paper is aimed at exploring the hidden fundamental\ud
computational property of natural language that has been so elusive that it has made all attempts to characterize its real computational property ultimately fail. Earlier natural language was thought to be context-free. However, it was gradually realized that this does not hold much water given that a range of natural language phenomena have been found as being of non-context-free character that they have almost scuttled plans to brand natural language contextfree. So it has been suggested that natural language is mildly context-sensitive and to some extent context-free. In all, it seems that the issue over the exact computational property has not yet been solved. Against this background it will be proposed that this exact computational property of natural language is perhaps the N-th dimension of language, if what we mean by dimension is\ud
nothing but universal (computational) property of natural language
!-Graphs with Trivial Overlap are Context-Free
String diagrams are a powerful tool for reasoning about composite structures
in symmetric monoidal categories. By representing string diagrams as graphs,
equational reasoning can be done automatically by double-pushout rewriting.
!-graphs give us the means of expressing and proving properties about whole
families of these graphs simultaneously. While !-graphs provide elegant proofs
of surprisingly powerful theorems, little is known about the formal properties
of the graph languages they define. This paper takes the first step in
characterising these languages by showing that an important subclass of
!-graphs--those whose repeated structures only overlap trivially--can be
encoded using a (context-free) vertex replacement grammar.Comment: In Proceedings GaM 2015, arXiv:1504.0244
Quantum Programming Made Easy
We present IQu, namely a quantum programming language that extends Reynold's
Idealized Algol, the paradigmatic core of Algol-like languages. IQu combines
imperative programming with high-order features, mediated by a simple type
theory. IQu mildly merges its quantum features with the classical programming
style that we can experiment through Idealized Algol, the aim being to ease a
transition towards the quantum programming world. The proposed extension is
done along two main directions. First, IQu makes the access to quantum
co-processors by means of quantum stores. Second, IQu includes some support for
the direct manipulation of quantum circuits, in accordance with recent trends
in the development of quantum programming languages. Finally, we show that IQu
is quite effective in expressing well-known quantum algorithms.Comment: In Proceedings Linearity-TLLA 2018, arXiv:1904.0615
A System F accounting for scalars
The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend
the lambda-calculus with the possibility of making arbitrary linear
combinations of terms. In this paper we provide a fine-grained, System F-like
type system for the linear-algebraic lambda-calculus. We show that this
"scalar" type system enjoys both the subject-reduction property and the
strong-normalisation property, our main technical results. The latter yields a
significant simplification of the linear-algebraic lambda-calculus itself, by
removing the need for some restrictions in its reduction rules. But the more
important, original feature of this scalar type system is that it keeps track
of 'the amount of a type' that is present in each term. As an example of its
use, we shown that it can serve as a guarantee that the normal form of a term
is barycentric, i.e that its scalars are summing to one
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