17 research outputs found
Vector Space Semantics for Lambek Calculus with Soft Subexponentials
We develop a vector space semantics for Lambek Calculus with Soft
Subexponentials, apply the calculus to construct compositional vector
interpretations for parasitic gap noun phrases and discourse units with
anaphora and ellipsis, and experiment with the constructions in a
distributional sentence similarity task. As opposed to previous work, which
used Lambek Calculus with a Relevant Modality the calculus used in this paper
uses a bounded version of the modality and is decidable. The vector space
semantics of this new modality allows us to meaningfully define contraction as
projection and provide a linear theory behind what we could previously only
achieve via nonlinear maps
Colored props for large scale graphical reasoning
The prop formalism allows representation of processes withstring diagrams and
has been successfully applied in various areas such as quantum computing,
electric circuits and control flow graphs. However, these graphical approaches
suffer from scalability problems when it comes to writing large diagrams. A
proposal to tackle this issue has been investigated for ZX-calculus using
colored props. This paper extends the approach to any prop, making it a general
tool for graphical languages manipulation
SZX-Calculus: Scalable Graphical Quantum Reasoning
We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and elaborate quantum evolutions. We extend the ZX-calculus to registers of qubits and allow compact representation of sub-diagrams via binary matrices. We show soundness and completeness of the SZX-calculus and provide two examples of applications, for graph states and error correcting codes
A Bestiary of Sets and Relations
Building on established literature and recent developments in the
graph-theoretic characterisation of its CPM category, we provide a treatment of
pure state and mixed state quantum mechanics in the category fRel of finite
sets and relations. On the way, we highlight the wealth of exotic beasts that
hide amongst the extensive operational and structural similarities that the
theory shares with more traditional arenas of categorical quantum mechanics,
such as the category fdHilb. We conclude our journey by proving that fRel is
local, but not without some unexpected twists.Comment: In Proceedings QPL 2015, arXiv:1511.0118
From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality
We study how to relate well-known hypergraph grammars based on the double
pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL
(called HL-grammars). It turns out that DPO rules can be naturally encoded by
types of HL using methods similar to those used by Kanazawa for
multiplicative-exponential linear logic. In order to generalize his reasonings
we extend the hypergraph Lambek calculus by adding the exponential modality,
which results in a new calculus HMEL0; then we prove that any DPO grammar can
be converted into an equivalent HMEL0-grammar. We also define the conjunctive
Kleene star, which behaves similarly to this exponential modality, and
establish a similar result. If we add neither the exponential modality nor the
conjunctive Kleene star to HL, then we can still use the same encoding and show
that any DPO grammar with a linear restriction on the length of derivations can
be converted into an equivalent HL-grammar.Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421
One-dimensional Cellular Automata in Quantum and Fermionic Theories
The thesis deals with quantum cellular automata (QCAs) and Fermionic quantum cellular automata (FQCAs) on one-dimensional lattices. With the term cellular automaton, we refer to a class of algorithms that can process information distributed on a regular grid in a local fashion. Its quantum counterpart—where at each site of the grid we can find a quantum system—represents a model for massive parallel quantum computation on finitely generated grids. The model is particularly well-suited for describing and simulating a vast class of physical phenomena.
The work presented in the thesis is threefold. We first introduce a new definition of QCA in terms of super maps, namely functions from quantum operations to quantum operations, that preserves locality and composition of transformations. Thereby, we define the so-called T-operator, i.e. a local operator that incorporates all the necessary information for univocally defining a QCA. The T-operator plays here the role of the Choi operator of the automaton.
Secondly, we classify all nearest-neighbor FQCAs over the one-dimensional lattice where each site contains one local Fermionic mode. We observe that the Fermionic automata are divided into two classes. In the first one, we find some FQCAs that are equivalent to a subset of quantum cellular automata. On the other hand, the second class of FQCAs has been found to have no quantum counterparts.
Finally, we report the experimental realization of a photonic platform to simulate the evolution of a one-dimensional quantum walk, i.e. a quantum cellular automaton whose action is linear in the field operators. Specifically, we observe the Zitterbewegung of a particle satisfying the Dirac dispersion relation. The theoretical background, numerical simulation, and optimization of the parameter space are discussed with special attention
The syntax and semantics of quantitative type theory
We present Quantitative Type Theory, a Type Theory that records usage information for each variable in a judgement, based on a previous system by McBride. The usage information is used to give a realizability semantics using a variant of Linear Combinatory Algebras, refining the usual realizability semantics of Type Theory by accurately tracking resource behaviour. We define the semantics in terms of Quantitative Categories with Families, a novel extension of Categories with Families for modelling resource sensitive type theories
Aeronautical engineering: A continuing bibliography with indexes (supplement 291)
This bibliography lists 757 reports, articles, and other documents introduced into the NASA scientific and technical information system in May. 1993. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics
Reversible Computation: Extending Horizons of Computing
This open access State-of-the-Art Survey presents the main recent scientific outcomes in the area of reversible computation, focusing on those that have emerged during COST Action IC1405 "Reversible Computation - Extending Horizons of Computing", a European research network that operated from May 2015 to April 2019. Reversible computation is a new paradigm that extends the traditional forwards-only mode of computation with the ability to execute in reverse, so that computation can run backwards as easily and naturally as forwards. It aims to deliver novel computing devices and software, and to enhance existing systems by equipping them with reversibility. There are many potential applications of reversible computation, including languages and software tools for reliable and recovery-oriented distributed systems and revolutionary reversible logic gates and circuits, but they can only be realized and have lasting effect if conceptual and firm theoretical foundations are established first