104 research outputs found

### Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra

In many instances in first order logic or computable algebra, classical
theorems show that many problems are undecidable for general structures, but
become decidable if some rigidity is imposed on the structure. For example, the
set of theorems in many finitely axiomatisable theories is nonrecursive, but
the set of theorems for any finitely axiomatisable complete theory is
recursive. Finitely presented groups might have an nonrecursive word problem,
but finitely presented simple groups have a recursive word problem. In this
article we introduce a topological framework based on closure spaces to show
that many of these proofs can be obtained in a similar setting. We will show in
particular that these statements can be generalized to cover arbitrary
structures, with no finite or recursive presentation/axiomatization. This
generalizes in particular work by Kuznetsov and others. Examples from first
order logic and symbolic dynamics will be discussed at length

### The rational fragment of the ZX-calculus

We introduce here a new axiomatisation of the rational fragment of the
ZX-calculus, a diagrammatic language for quantum mechanics. Compared to the
previous axiomatisation introduced in [8], our axiomatisation does not use any
metarule , but relies instead on a more natural rule, called the cyclotomic
supplementarity rule, that was introduced previously in the literature. Our
axiomatisation is only complete for diagrams using rational angles , and is not
complete in the general case. Using results on diophantine geometry, we
characterize precisely which diagram equality involving arbitrary angles are
provable in our framework without any new axioms, and we show that our
axiomatisation is continuous, in the sense that a diagram equality involving
arbitrary angles is provable iff it is a limit of diagram equalities involving
rational angles. We use this result to give a complete characterization of all
Euler equations that are provable in this axiomatisation

### Aperiodic Subshifts of Finite Type on Groups

In this note we prove the following results:
$\bullet$ If a finitely presented group $G$ admits a strongly aperiodic SFT,
then $G$ has decidable word problem. More generally, for f.g. groups that are
not recursively presented, there exists a computable obstruction for them to
admit strongly aperiodic SFTs.
$\bullet$ On the positive side, we build strongly aperiodic SFTs on some new
classes of groups. We show in particular that some particular monster groups
admits strongly aperiodic SFTs for trivial reasons. Then, for a large class of
group $G$, we show how to build strongly aperiodic SFTs over $\mathbb{Z}\times
G$. In particular, this is true for the free group with 2 generators,
Thompson's groups $T$ and $V$, $PSL_2(\mathbb{Z})$ and any f.g. group of
rational matrices which is bounded.Comment: New version. Adding results about monster group

### Infinite Communication Complexity

Suppose that Alice and Bob are given each an infinite string, and they want
to decide whether their two strings are in a given relation. How much
communication do they need? How can communication be even defined and measured
for infinite strings? In this article, we propose a formalism for a notion of
infinite communication complexity, prove that it satisfies some natural
properties and coincides, for relevant applications, with the classical notion
of amortized communication complexity. More-over, an application is given for
tackling some conjecture about tilings and multidimensional sofic shifts.Comment: First Version. Written from the Computer Science PO

### Subshifts as Models for MSO Logic

We study the Monadic Second Order (MSO) Hierarchy over colourings of the
discrete plane, and draw links between classes of formula and classes of
subshifts. We give a characterization of existential MSO in terms of
projections of tilings, and of universal sentences in terms of combinations of
"pattern counting" subshifts. Conversely, we characterise logic fragments
corresponding to various classes of subshifts (subshifts of finite type, sofic
subshifts, all subshifts). Finally, we show by a separation result how the
situation here is different from the case of tiling pictures studied earlier by
Giammarresi et al.Comment: arXiv admin note: substantial text overlap with arXiv:0904.245

### A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

We introduce the first complete and approximatively universal diagrammatic
language for quantum mechanics. We make the ZX-Calculus, a diagrammatic
language introduced by Coecke and Duncan, complete for the so-called Clifford+T
quantum mechanics by adding four new axioms to the language. The completeness
of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open
questions in categorical quantum mechanics. We prove the completeness of the
Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus,
a calculus dealing with integer matrices. We also prove that the Clifford+T
fragment of the ZX-Calculus represents exactly all the matrices over some
finite dimensional extension of the ring of dyadic rationals

### Structural aspects of tilings

In this paper, we study the structure of the set of tilings produced by any
given tile-set. For better understanding this structure, we address the set of
finite patterns that each tiling contains. This set of patterns can be analyzed
in two different contexts: the first one is combinatorial and the other
topological. These two approaches have independent merits and, once combined,
provide somehow surprising results. The particular case where the set of
produced tilings is countable is deeply investigated while we prove that the
uncountable case may have a completely different structure. We introduce a
pattern preorder and also make use of Cantor-Bendixson rank. Our first main
result is that a tile-set that produces only periodic tilings produces only a
finite number of them. Our second main result exhibits a tiling with exactly
one vector of periodicity in the countable case.Comment: 11 page

### Completeness of the ZX-Calculus

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum
mechanics and quantum information theory. It comes equipped with an equational
presentation. We focus here on a very important property of the language:
completeness, which roughly ensures the equational theory captures all of
quantum mechanics. We first improve on the known-to-be-complete presentation
for the so-called Clifford fragment of the language - a restriction that is not
universal - by adding some axioms. Thanks to a system of back-and-forth
translation between the ZX-Calculus and a third-party complete graphical
language, we prove that the provided axiomatisation is complete for the first
approximately universal fragment of the language, namely Clifford+T.
We then prove that the expressive power of this presentation, though aimed at
achieving completeness for the aforementioned restriction, extends beyond
Clifford+T, to a class of diagrams that we call linear with Clifford+T
constants. We use another version of the third-party language - and an adapted
system of back-and-forth translation - to complete the language for the
ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the
added axioms, and finally, we provide a complete axiomatisation for an altered
version of the language which involves an additional generator, making the
presentation simpler

### Tilings and model theory

ISBN 978-5-94057-377-7International audienceIn this paper we emphasize the links between model theory and tilings. More precisely, after giving the definitions of what tilings are, we give a natural way to have an interpretation of the tiling rules in first order logics. This opens the way to map some model theoretical properties onto some properties of sets of tilings, or tilings themselves

- …