474 research outputs found
On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms
Uniquely closable skeletons of lambda terms are Motzkin-trees that
predetermine the unique closed lambda term that can be obtained by labeling
their leaves with de Bruijn indices. Likewise, uniquely typable skeletons of
closed lambda terms predetermine the unique simply-typed lambda term that can
be obtained by labeling their leaves with de Bruijn indices.
We derive, through a sequence of logic program transformations, efficient
code for their combinatorial generation and study their statistical properties.
As a result, we obtain context-free grammars describing closable and uniquely
closable skeletons of lambda terms, opening the door for their in-depth study
with tools from analytic combinatorics.
Our empirical study of the more difficult case of (uniquely) typable terms
reveals some interesting open problems about their density and asymptotic
behavior.
As a connection between the two classes of terms, we also show that uniquely
typable closed lambda term skeletons of size are in a bijection with
binary trees of size .Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
Controlled non uniform random generation of decomposable structures
Consider a class of decomposable combinatorial structures, using different
types of atoms \Atoms = \{\At_1,\ldots ,\At_{|{\Atoms}|}\}. We address the
random generation of such structures with respect to a size and a targeted
distribution in of its \emph{distinguished} atoms. We consider two
variations on this problem. In the first alternative, the targeted distribution
is given by real numbers \TargFreq_1, \ldots, \TargFreq_k such that 0 <
\TargFreq_i < 1 for all and \TargFreq_1+\cdots+\TargFreq_k \leq 1. We
aim to generate random structures among the whole set of structures of a given
size , in such a way that the {\em expected} frequency of any distinguished
atom \At_i equals \TargFreq_i. We address this problem by weighting the
atoms with a -tuple \Weights of real-valued weights, inducing a weighted
distribution over the set of structures of size . We first adapt the
classical recursive random generation scheme into an algorithm taking
\bigO{n^{1+o(1)}+mn\log{n}} arithmetic operations to draw structures from
the \Weights-weighted distribution. Secondly, we address the analytical
computation of weights such that the targeted frequencies are achieved
asymptotically, i. e. for large values of . We derive systems of functional
equations whose resolution gives an explicit relationship between \Weights
and \TargFreq_1, \ldots, \TargFreq_k. Lastly, we give an algorithm in
\bigO{k n^4} for the inverse problem, {\it i.e.} computing the frequencies
associated with a given -tuple \Weights of weights, and an optimized
version in \bigO{k n^2} in the case of context-free languages. This allows
for a heuristic resolution of the weights/frequencies relationship suitable for
complex specifications. In the second alternative, the targeted distribution is
given by a natural numbers such that
where is the number of undistinguished atoms.
The structures must be generated uniformly among the set of structures of size
that contain {\em exactly} atoms \At_i (). We give
a \bigO{r^2\prod_{i=1}^k n_i^2 +m n k \log n} algorithm for generating
structures, which simplifies into a \bigO{r\prod_{i=1}^k n_i +m n} for
regular specifications
Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata
Define a certain gambler's ruin process \mathbf{X}_{j}, \mbox{ \ }j\ge 0,
such that the increments
take values and satisfy ,
all , where if , and if .
Here denote persistence parameters and with
. The process starts at and terminates when
. Denote by , , and ,
respectively, the numbers of runs, long runs, and steps in the meander portion
of the gambler's ruin process. Define and let for some . We show exists in an explicit form. We obtain a
companion theorem for the last visit portion of the gambler's ruin.Comment: Presented at 8th International Conference on Lattice Path
Combinatorics, Cal Poly Pomona, Aug., 2015. The 2nd version has been
streamlined, with references added, including reference to a companion
document with details of calculations via Mathematica. The 3rd version has 2
new figures and improved presentatio
Target annihilation by diffusing particles in inhomogeneous geometries
The survival probability of immobile targets, annihilated by a population of
random walkers on inhomogeneous discrete structures, such as disordered solids,
glasses, fractals, polymer networks and gels, is analytically investigated. It
is shown that, while it cannot in general be related to the number of distinct
visited points, as in the case of homogeneous lattices, in the case of bounded
coordination numbers its asymptotic behaviour at large times can still be
expressed in terms of the spectral dimension , and its exact
analytical expression is given. The results show that the asymptotic survival
probability is site independent on recurrent structures (),
while on transient structures () it can strongly depend on the
target position, and such a dependence is explicitly calculated.Comment: To appear in Physical Review E - Rapid Communication
Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
The Airy distribution function describes the probability distribution of the
area under a Brownian excursion over a unit interval. Surprisingly, this
function has appeared in a number of seemingly unrelated problems, mostly in
computer science and graph theory. In this paper, we show that this
distribution also appears in a rather well studied physical system, namely the
fluctuating interfaces. We present an exact solution for the distribution
P(h_m,L) of the maximal height h_m (measured with respect to the average
spatial height) in the steady state of a fluctuating interface in a one
dimensional system of size L with both periodic and free boundary conditions.
For the periodic case, we show that P(h_m,L)=L^{-1/2}f(h_m L^{-1/2}) for all L
where the function f(x) is the Airy distribution function. This result is valid
for both the Edwards-Wilkinson and the Kardar-Parisi-Zhang interfaces. For the
free boundary case, the same scaling holds P(h_m,L)=L^{-1/2}F(h_m L^{-1/2}),
but the scaling function F(x) is different from that of the periodic case. We
compute this scaling function explicitly for the Edwards-Wilkinson interface
and call it the F-Airy distribution function. Numerical simulations are in
excellent agreement with our analytical results. Our results provide a rather
rare exactly solvable case for the distribution of extremum of a set of
strongly correlated random variables. Some of these results were announced in a
recent Letter [ S.N. Majumdar and A. Comtet, Phys. Rev. Lett., 92, 225501
(2004)].Comment: 27 pages, 10 .eps figures included. Two figures improved, new
discussion and references adde
Multiplicative anomaly and zeta factorization
Some aspects of the multiplicative anomaly of zeta determinants are
investigated. A rather simple approach is adopted and, in particular, the
question of zeta function factorization, together with its possible relation
with the multiplicative anomaly issue is discussed. We look primordially into
the zeta functions instead of the determinants themselves, as was done in
previous work. That provides a supplementary view, regarding the appearance of
the multiplicative anomaly. Finally, we briefly discuss determinants of zeta
functions that are not in the pseudodifferential operator framework.Comment: 20 pages, AIP styl
Counting, generating and sampling tree alignments
Pairwise ordered tree alignment are combinatorial objects that appear in RNA
secondary structure comparison. However, the usual representation of tree
alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce
identical sets of matches between identical pairs of trees. This ambiguity is
uninformative, and detrimental to any probabilistic analysis.In this work, we
consider tree alignments up to equivalence. Our first result is a precise
asymptotic enumeration of tree alignments, obtained from a context-free grammar
by mean of basic analytic combinatorics. Our second result focuses on
alignments between two given ordered trees and . By refining our grammar
to align specific trees, we obtain a decomposition scheme for the space of
alignments, and use it to design an efficient dynamic programming algorithm for
sampling alignments under the Gibbs-Boltzmann probability distribution. This
generalizes existing tree alignment algorithms, and opens the door for a
probabilistic analysis of the space of suboptimal RNA secondary structures
alignments.Comment: ALCOB - 3rd International Conference on Algorithms for Computational
Biology - 2016, Jun 2016, Trujillo, Spain. 201
Page usage in quadtree indexes
Résumé disponible dans le fichier PD
Uniform generation in trace monoids
We consider the problem of random uniform generation of traces (the elements
of a free partially commutative monoid) in light of the uniform measure on the
boundary at infinity of the associated monoid. We obtain a product
decomposition of the uniform measure at infinity if the trace monoid has
several irreducible components-a case where other notions such as Parry
measures, are not defined. Random generation algorithms are then examined.Comment: Full version of the paper in MFCS 2015 with the same titl
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