4,121 research outputs found
Proof Diagrams for Multiplicative Linear Logic
The original idea of proof nets can be formulated by means of interaction
nets syntax. Additional machinery as switching, jumps and graph connectivity is
needed in order to ensure correspondence between a proof structure and a
correct proof in sequent calculus.
In this paper we give an interpretation of proof nets in the syntax of string
diagrams. Even though we lose standard proof equivalence, our construction
allows to define a framework where soundness and well-typeness of a diagram can
be verified in linear time.Comment: In Proceedings LINEARITY 2016, arXiv:1701.0452
Towards 3-Dimensional Rewriting Theory
String rewriting systems have proved very useful to study monoids. In good
cases, they give finite presentations of monoids, allowing computations on
those and their manipulation by a computer. Even better, when the presentation
is confluent and terminating, they provide one with a notion of canonical
representative of the elements of the presented monoid. Polygraphs are a
higher-dimensional generalization of this notion of presentation, from the
setting of monoids to the much more general setting of n-categories. One of the
main purposes of this article is to give a progressive introduction to the
notion of higher-dimensional rewriting system provided by polygraphs, and
describe its links with classical rewriting theory, string and term rewriting
systems in particular. After introducing the general setting, we will be
interested in proving local confluence for polygraphs presenting 2-categories
and introduce a framework in which a finite 3-dimensional rewriting system
admits a finite number of critical pairs
Synthesising Graphical Theories
In recent years, diagrammatic languages have been shown to be a powerful and
expressive tool for reasoning about physical, logical, and semantic processes
represented as morphisms in a monoidal category. In particular, categorical
quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of
quantum theory into abstract structural properties, expressed in the form of
diagrammatic identities. One way we search for these properties is to start
with a concrete model (e.g. a set of linear maps or finite relations) and start
composing generators into diagrams and looking for graphical identities.
Naively, we could automate this procedure by enumerating all diagrams up to a
given size and check for equalities, but this is intractable in practice
because it produces far too many equations. Luckily, many of these identities
are not primitive, but rather derivable from simpler ones. In 2010, Johansson,
Dixon, and Bundy developed a technique called conjecture synthesis for
automatically generating conjectured term equations to feed into an inductive
theorem prover. In this extended abstract, we adapt this technique to
diagrammatic theories, expressed as graph rewrite systems, and demonstrate its
application by synthesising a graphical theory for studying entangled quantum
states.Comment: 10 pages, 22 figures. Shortened and one theorem adde
Linear lambda terms as invariants of rooted trivalent maps
The main aim of the article is to give a simple and conceptual account for
the correspondence (originally described by Bodini, Gardy, and Jacquot) between
-equivalence classes of closed linear lambda terms and isomorphism
classes of rooted trivalent maps on compact oriented surfaces without boundary,
as an instance of a more general correspondence between linear lambda terms
with a context of free variables and rooted trivalent maps with a boundary of
free edges. We begin by recalling a familiar diagrammatic representation for
linear lambda terms, while at the same time explaining how such diagrams may be
read formally as a notation for endomorphisms of a reflexive object in a
symmetric monoidal closed (bi)category. From there, the "easy" direction of the
correspondence is a simple forgetful operation which erases annotations on the
diagram of a linear lambda term to produce a rooted trivalent map. The other
direction views linear lambda terms as complete invariants of their underlying
rooted trivalent maps, reconstructing the missing information through a
Tutte-style topological recurrence on maps with free edges. As an application
in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent
maps as linear lambda terms containing no closed proper subterms, and conclude
by giving a natural reformulation of the Four Color Theorem as a statement
about typing in lambda calculus.Comment: accepted author manuscript, posted six months after publicatio
Encoding !-tensors as !-graphs with neighbourhood orders
Diagrammatic reasoning using string diagrams provides an intuitive language
for reasoning about morphisms in a symmetric monoidal category. To allow
working with infinite families of string diagrams, !-graphs were introduced as
a method to mark repeated structure inside a diagram. This led to !-graphs
being implemented in the diagrammatic proof assistant Quantomatic. Having a
partially automated program for rewriting diagrams has proven very useful, but
being based on !-graphs, only commutative theories are allowed. An enriched
abstract tensor notation, called !-tensors, has been used to formalise the
notion of !-boxes in non-commutative structures. This work-in-progress paper
presents a method to encode !-tensors as !-graphs with some additional
structure. This will allow us to leverage the existing code from Quantomatic
and quickly provide various tools for non-commutative diagrammatic reasoning.Comment: In Proceedings QPL 2015, arXiv:1511.0118
A first-order logic for string diagrams
Equational reasoning with string diagrams provides an intuitive means of
proving equations between morphisms in a symmetric monoidal category. This can
be extended to proofs of infinite families of equations using a simple
graphical syntax called !-box notation. While this does greatly increase the
proving power of string diagrams, previous attempts to go beyond equational
reasoning have been largely ad hoc, owing to the lack of a suitable logical
framework for diagrammatic proofs involving !-boxes. In this paper, we extend
equational reasoning with !-boxes to a fully-fledged first order logic called
with conjunction, implication, and universal quantification over !-boxes. This
logic, called !L, is then rich enough to properly formalise an induction
principle for !-boxes. We then build a standard model for !L and give an
example proof of a theorem for non-commutative bialgebras using !L, which is
unobtainable by equational reasoning alone.Comment: 15 pages + appendi
Innocent strategies as presheaves and interactive equivalences for CCS
Seeking a general framework for reasoning about and comparing programming
languages, we derive a new view of Milner's CCS. We construct a category E of
plays, and a subcategory V of views. We argue that presheaves on V adequately
represent innocent strategies, in the sense of game semantics. We then equip
innocent strategies with a simple notion of interaction. This results in an
interpretation of CCS.
Based on this, we propose a notion of interactive equivalence for innocent
strategies, which is close in spirit to Beffara's interpretation of testing
equivalences in concurrency theory. In this framework we prove that the
analogues of fair and must testing equivalences coincide, while they differ in
the standard setting.Comment: In Proceedings ICE 2011, arXiv:1108.014
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