Approximating Petri Net Reachability Along Context-free Traces

We investigate the problem asking whether the intersection of a context-free language (CFL) and a Petri net language (PNL) is empty. Our contribution to solve this long-standing problem which relates, for instance, to the reachability analysis of recursive programs over unbounded data domain, is to identify a class of CFLs called the finite-index CFLs for which the problem is decidable. The k-index approximation of a CFL can be obtained by discarding all the words that cannot be derived within a budget k on the number of occurrences of non-terminals. A finite-index CFL is thus a CFL which coincides with its k-index approximation for some k. We decide whether the intersection of a finite-index CFL and a PNL is empty by reducing it to the reachability problem of Petri nets with weak inhibitor arcs, a class of systems with infinitely many states for which reachability is known to be decidable. Conversely, we show that the reachability problem for a Petri net with weak inhibitor arcs reduces to the emptiness problem of a finite-index CFL intersected with a PNL.Comment: 16 page

Gaussian distribution of short sums of trace functions over finite fields

We show that under certain general conditions, short sums of $\ell$-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalizing results of Erd\H{o}s-Davenport, Mak-Zaharescu and Lamzouri. In particular, this applies to exponential sums arising from Fourier transforms such as Kloosterman sums or Birch sums, as we can deduce from the works of Katz. By approximating the moments of traces of random matrices in monodromy groups, a quantitative version can be given as in Lamzouri's article, exhibiting a different phenomenon than the averaging from the central limit theorem.Comment: 42 page

Model for optical forward scattering by nonspherical raindrops

We describe a numerical model for the interaction of light with large raindrops using realistic nonspherical drop shapes. We apply geometrical optics and a Monte Carlo technique to perform ray traces through the drops. We solve the problem of diffraction independently by approximating the drops with areaequivalent ellipsoids. Scattering patterns are obtained for different polarizations of the incident light. They exhibit varying degrees of asymmetry and depolarization that can be linked to the distortion and thus the size of the drops. The model is extended to give a simplified long-path integration.Comment: 12 pages, 16 figure

Proving Safety with Trace Automata and Bounded Model Checking

Loop under-approximation is a technique that enriches C programs with additional branches that represent the effect of a (limited) range of loop iterations. While this technique can speed up the detection of bugs significantly, it introduces redundant execution traces which may complicate the verification of the program. This holds particularly true for verification tools based on Bounded Model Checking, which incorporate simplistic heuristics to determine whether all feasible iterations of a loop have been considered. We present a technique that uses \emph{trace automata} to eliminate redundant executions after performing loop acceleration. The method reduces the diameter of the program under analysis, which is in certain cases sufficient to allow a safety proof using Bounded Model Checking. Our transformation is precise---it does not introduce false positives, nor does it mask any errors. We have implemented the analysis as a source-to-source transformation, and present experimental results showing the applicability of the technique

pp-adic framed braids II

The Yokonuma-Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to Juyumaya and Lambropoulou (2007), we explore further the structures of the $p$-adic framed braids and the $p$-adic Yokonuma-Hecke algebras constructed in Juyumaya and Lambropoulou (2007), by means of dense sub-structures approximating $p$-adic elements. We also construct a $p$-adic Markov trace on the $p$-adic Yokonuma-Hecke algebras and we approximate the values of the $p$-adic trace on $p$-adic elements. Surprisingly, the Markov traces do not re-scale directly to yield isotopy invariants of framed links. This leads to imposing the `$E$-condition' on the trace parameters. For solutions of the `$E$-system' we then define 2-variable isotopy invariants of modular framed links. These lift to $p$-adic isotopy invariants of classical framed links. The Yokonuma-Hecke algebras have topological interpretations in the context of framed knots, of classical knots of singular knots and of transverse knots.Comment: 48 pages, 7 figures, added references, lighter notation, ameliorated presentation, corrected typos. To appear in Advances in Mathematic

Cellular Automata and Powers of p/qp/q

We consider one-dimensional cellular automata $F_{p,q}$ which multiply numbers by $p/q$ in base $pq$ for relatively prime integers $p$ and $q$. By studying the structure of traces with respect to $F_{p,q}$ we show that for $p\geq 2q-1$ (and then as a simple corollary for $p>q>1$) there are arbitrarily small finite unions of intervals which contain the fractional parts of the sequence $\xi(p/q)^n$, ($n=0,1,2,\dots$) for some $\xi>0$. To the other direction, by studying the measure theoretical properties of $F_{p,q}$, we show that for $p>q>1$ there are finite unions of intervals approximating the unit interval arbitrarily well which don't contain the fractional parts of the whole sequence $\xi(p/q)^n$ for any $\xi>0$.Comment: 15 pages, 8 figures. Accepted for publication in RAIRO-IT