44,356 research outputs found
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
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
Subshifts, MSO Logic, and Collapsing Hierarchies
We use monadic second-order logic to define two-dimensional subshifts, or
sets of colorings of the infinite plane. We present a natural family of
quantifier alternation hierarchies, and show that they all collapse to the
third level. In particular, this solves an open problem of [Jeandel & Theyssier
2013]. The results are in stark contrast with picture languages, where such
hierarchies are usually infinite.Comment: 12 pages, 5 figures. To appear in conference proceedings of TCS 2014,
published by Springe
Flexible RNA design under structure and sequence constraints using formal languages
The problem of RNA secondary structure design (also called inverse folding)
is the following: given a target secondary structure, one aims to create a
sequence that folds into, or is compatible with, a given structure. In several
practical applications in biology, additional constraints must be taken into
account, such as the presence/absence of regulatory motifs, either at a
specific location or anywhere in the sequence. In this study, we investigate
the design of RNA sequences from their targeted secondary structure, given
these additional sequence constraints. To this purpose, we develop a general
framework based on concepts of language theory, namely context-free grammars
and finite automata. We efficiently combine a comprehensive set of constraints
into a unifying context-free grammar of moderate size. From there, we use
generic generic algorithms to perform a (weighted) random generation, or an
exhaustive enumeration, of candidate sequences. The resulting method, whose
complexity scales linearly with the length of the RNA, was implemented as a
standalone program. The resulting software was embedded into a publicly
available dedicated web server. The applicability demonstrated of the method on
a concrete case study dedicated to Exon Splicing Enhancers, in which our
approach was successfully used in the design of \emph{in vitro} experiments.Comment: ACM BCB 2013 - ACM Conference on Bioinformatics, Computational
Biology and Biomedical Informatics (2013
- …