10 research outputs found
Preservation Theorems Through the Lens of Topology
In this paper, we introduce a family of topological spaces that captures the existence of preservation theorems. The structure of those spaces allows us to study the relativisation of preservation theorems under suitable definitions of surjective morphisms, subclasses, sums, products, topological closures, and projective limits. Throughout the paper, we also integrate already known results into this new framework and show how it captures the essence of their proofs
Fixed Points and Noetherian Topologies
This paper provides a canonical construction of a Noetherian least fixed
point topology. While such least fixed point are not Noetherian in general, we
prove that under a mild assumption, one can use a topological minimal bad
sequence argument to prove that they are. We then apply this fixed point
theorem to rebuild known Noetherian topologies with a uniform proof.
In the case of spaces that are defined inductively (such as finite words and
finite trees), we provide a uniform definition of a divisibility topology using
our fixed point theorem. We then prove that the divisibility topology is a
generalisation of the divisibility preorder introduced by Hasegawa in the case
of well-quasi-orders.Comment: 18 pages, 2 figure
Basic Operational Preorders for Algebraic Effects in General, and for Combined Probability and Nondeterminism in Particular
Z-polyregular functions
Given an MSO formula with free variables , one can
define the function mapping a word to the number of valuations
satisfying in . In this paper, we introduce the class of
-linear combinations of such functions, that we call
-polyregular functions. Indeed, it turns out to be closely related
to the well-studied class of polyregular functions.
The main results of this paper solve two natural decision problems for
-polyregular functions. First, we show that one can minimise the
number of free variables which are needed to describe a function.
Then, we show how to decide if a function can be defined using first-order
formulas, by extending the notion of residual automaton and providing an
original semantic characterisation based on aperiodicity. We also connect this
class of functions to -rational series.Comment: 23 pages, 2 figure
Diagrammatic Semantics for Digital Circuits
We introduce a general diagrammatic theory of digital circuits, based on connections between monoidal categories and graph rewriting. The main achievement of the paper is conceptual, filling a foundational gap in reasoning syntactically and symbolically about a large class of digital circuits (discrete values, discrete delays, feedback). This complements the dominant approach to circuit modelling, which relies on simulation. The main advantage of our symbolic approach is the enabling of automated reasoning about parametrised circuits, with a potentially interesting new application to partial evaluation of digital circuits. Relative to the recent interest and activity in categorical and diagrammatic methods, our work makes several new contributions. The most important is establishing that categories of digital circuits are Cartesian and admit, in the presence of feedback expressive iteration axioms. The second is producing a general yet simple graph-rewrite framework for reasoning about such categories in which the rewrite rules are computationally efficient, opening the way for practical applications
A structural and nominal syntax for diagrams
The correspondence between monoidal categories and graphical languages of
diagrams has been studied extensively, leading to applications in quantum
computing and communication, systems theory, circuit design and more. From the
categorical perspective, diagrams can be specified using (name-free)
combinators which enjoy elegant equational properties. However, conventional
notations for diagrammatic structures, such as hardware description languages
(VHDL, Verilog) or graph languages (Dot), use a different style, which is flat,
relational, and reliant on extensive use of names (labels). Such languages are
not known to enjoy nice syntactic equational properties. However, since they
make it relatively easy to specify (and modify) arbitrary diagrammatic
structures they are more popular than the combinator style. In this paper we
show how the two approaches to diagram syntax can be reconciled and unified in
a way that does not change the semantics and the existing equational theory.
Additionally, we give sound and complete equational theories for the combined
syntax.Comment: In Proceedings QPL 2017, arXiv:1802.0973
Localité et théorÚmes de préservation en logique du premier ordre
International audienceThis paper investigates the expressiveness of a fragment of firstorder sentences in Gaifman normal form, namely the positive Boolean combinations of basic local sentences. We show that they match exactly the first-order sentences preserved under local elementary embeddings, thus providing a new general preservation theorem and extending the ĆĂłs-Tarski Theorem. This full preservation result fails as usual in the finite, and we show furthermore that the naturally related decision problems are undecidable. In the more restricted case of preservation under extensions, it nevertheless yields new well-behaved classes of finite structures: we show that preservation under extensions holds if and only if it holds locally
Basic Operational Preorders for Algebraic Effects in General, and for Combined Probability and Nondeterminism in Particular
International audienceThe "generic operational metatheory" of Johann, Simpson and VoigtlÀnder (LiCS 2010) defines contextual equivalence, in the presence of algebraic effects, in terms of a basic operational preorder on ground-type effect trees. We propose three general approaches to specifying such preorders: (i) operational (ii) denotational, and (iii) axiomatic; coinciding with the three major styles of program semantics. We illustrate these via a nontrivial case study: the combination of probabilistic choice with nondeterminism, for which we show that natural instantiations of the three specification methods (operational in terms of Markov decision processes, denotational using a powerdomain, and axiomatic) all determine the same canonical preorder. We do this in the case of both angelic and demonic nondeterminism
Infinitary Noetherian Constructions II. Transfinite Words and the Regular Subword Topology
International audienceWe show that the spaces of transfinite words, namely ordinalindexed words, over a Noetherian space, is also Noetherian, under a natural topology which we call the regular subword topology. We characterize its sobrification and its specialization ordering, and we give an upper bound on its sobrification rank and on its stature