4,203 research outputs found
On Linear Information Systems
Scott's information systems provide a categorically equivalent, intensional
description of Scott domains and continuous functions. Following a well
established pattern in denotational semantics, we define a linear version of
information systems, providing a model of intuitionistic linear logic (a
new-Seely category), with a "set-theoretic" interpretation of exponentials that
recovers Scott continuous functions via the co-Kleisli construction. From a
domain theoretic point of view, linear information systems are equivalent to
prime algebraic Scott domains, which in turn generalize prime algebraic
lattices, already known to provide a model of classical linear logic
Linear-algebraic lambda-calculus
With a view towards models of quantum computation and/or the interpretation
of linear logic, we define a functional language where all functions are linear
operators by construction. A small step operational semantic (and hence an
interpreter/simulator) is provided for this language in the form of a term
rewrite system. The linear-algebraic lambda-calculus hereby constructed is
linear in a different (yet related) sense to that, say, of the linear
lambda-calculus. These various notions of linearity are discussed in the
context of quantum programming languages. KEYWORDS: quantum lambda-calculus,
linear lambda-calculus, -calculus, quantum logics.Comment: LaTeX, 23 pages, 10 figures and the LINEAL language
interpreter/simulator file (see "other formats"). See the more recent
arXiv:quant-ph/061219
The Vectorial -Calculus
We describe a type system for the linear-algebraic -calculus. The
type system accounts for the linear-algebraic aspects of this extension of
-calculus: it is able to statically describe the linear combinations
of terms that will be obtained when reducing the programs. This gives rise to
an original type theory where types, in the same way as terms, can be
superposed into linear combinations. We prove that the resulting typed
-calculus is strongly normalising and features weak subject reduction.
Finally, we show how to naturally encode matrices and vectors in this typed
calculus.Comment: Long and corrected version of arXiv:1012.4032 (EPTCS 88:1-15), to
appear in Information and Computatio
Towards an embedding of Graph Transformation in Intuitionistic Linear Logic
Linear logics have been shown to be able to embed both rewriting-based
approaches and process calculi in a single, declarative framework. In this
paper we are exploring the embedding of double-pushout graph transformations
into quantified linear logic, leading to a Curry-Howard style isomorphism
between graphs and transformations on one hand, formulas and proof terms on the
other. With linear implication representing rules and reachability of graphs,
and the tensor modelling parallel composition of graphs and transformations, we
obtain a language able to encode graph transformation systems and their
computations as well as reason about their properties
An algebraic basis for specifying and enforcing access control in security systems
Security services in a multi-user environment are often based on access control mechanisms. Static aspects of an access control policy can be formalised using abstract algebraic models. We integrate these static aspects into a dynamic framework considering requesting access to resources as a process aiming at the prevention of access control violations when a program is executed. We use another algebraic technique, monads, as a meta-language to integrate access control operations into a functional
programming language. The integration of monads and concepts from a denotational model for process algebras provides a framework for programming of access control in security systems
Relational Graph Models at Work
We study the relational graph models that constitute a natural subclass of
relational models of lambda-calculus. We prove that among the lambda-theories
induced by such models there exists a minimal one, and that the corresponding
relational graph model is very natural and easy to construct. We then study
relational graph models that are fully abstract, in the sense that they capture
some observational equivalence between lambda-terms. We focus on the two main
observational equivalences in the lambda-calculus, the theory H+ generated by
taking as observables the beta-normal forms, and H* generated by considering as
observables the head normal forms. On the one hand we introduce a notion of
lambda-K\"onig model and prove that a relational graph model is fully abstract
for H+ if and only if it is extensional and lambda-K\"onig. On the other hand
we show that the dual notion of hyperimmune model, together with
extensionality, captures the full abstraction for H*
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
Wave-Style Token Machines and Quantum Lambda Calculi
Particle-style token machines are a way to interpret proofs and programs,
when the latter are written following the principles of linear logic. In this
paper, we show that token machines also make sense when the programs at hand
are those of a simple quantum lambda-calculus with implicit qubits. This,
however, requires generalising the concept of a token machine to one in which
more than one particle travel around the term at the same time. The presence of
multiple tokens is intimately related to entanglement and allows us to give a
simple operational semantics to the calculus, coherently with the principles of
quantum computation.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441
Resource-Bound Quantification for Graph Transformation
Graph transformation has been used to model concurrent systems in software
engineering, as well as in biochemistry and life sciences. The application of a
transformation rule can be characterised algebraically as construction of a
double-pushout (DPO) diagram in the category of graphs. We show how
intuitionistic linear logic can be extended with resource-bound quantification,
allowing for an implicit handling of the DPO conditions, and how resource logic
can be used to reason about graph transformation systems
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