Entropy of regular timed languages

For timed languages, we define size measures: volume for languages with a fixed finite number of events, and entropy (growth rate) as asymptotic measure for an unbounded number of events. These measures can be used for quantitative comparison of languages, and the entropy can be viewed as information contents of a timed language. For languages accepted by deterministic timed automata, we give exact formulas for volumes. We show that automata with non-vanishing entropy ("thick") have a normal (non-Zeno, discretizable etc.) behavior for typical runs. Next, we characterize the entropy, using methods of functional analysis, as the logarithm of the leading eigenvalue (spectral radius) of a positive integral operator. We devise a couple of methods to compute the entropy: a symbolical one for so-called "1 1 ⁄2-clock" automata, and a numerical one (with a guarantee of convergence)

Limit theorems for iterated random topical operators

Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively homogeneous) operators. Let $x(n,x_0)$ be defined by $x(0,x_0)=x_0$ and $x(n,x_0)=A(n)x(n-1,x_0)$. This can modelize a wide range of systems including, task graphs, train networks, Job-Shop, timed digital circuits or parallel processing systems. When A(n) has the memory loss property, we use the spectral gap method to prove limit theorems for $x(n,x_0)$. Roughly speaking, we show that $x(n,x_0)$ behaves like a sum of i.i.d. real variables. Precisely, we show that with suitable additional conditions, it satisfies a central limit theorem with rate, a local limit theorem, a renewal theorem and a large deviations principle, and we give an algebraic condition to ensure the positivity of the variance in the CLT. When A(n) are defined by matrices in the \mp semi-ring, we give more effective statements and show that the additional conditions and the positivity of the variance in the CLT are generic

Asymptotic behavior in a heap model with two pieces

International audienceIn a heap model, solid blocks, or pieces, pile up according to the Tetris game mechanism. An optimal schedule is an infinite sequence of pieces minimizing the asymptotic growth rate of the heap. In a heap model with two pieces, we prove that there always exists an optimal schedule which is balanced, either periodic or Sturmian. We also consider the model where the successive pieces are chosen at random, independently and with some given probabilities. We study the expected growth rate of the heap. For a model with two pieces, the rate is either computed explicitly or given as an infinite series. We show an application for a system of two processes sharing a resource, and we prove that a greedy schedule is not always optimal

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius

We present several results describing the interplay between the max algebraic joint spectral radius (JSR) for compact sets of matrices and suitably defined matrix norms. In particular, we extend a classical result for the conventional algebra, showing that the JSR can be described in terms of induced norms of the matrices in the set. We also show that for a set generating an irreducible semigroup (in a cone-theoretic sense), a monotone Barabanov norm always exists. This fact is then used to show that the max algebraic JSR is locally Lipschitz continuous on the space of compact irreducible sets of matrices with respect to the Hausdorff distance. We then prove that the JSR is Hoelder continuous on the space of compact sets of nonnegative matrices. Finally, we prove a strict monotonicity property for the max algebraic JSR that echoes a fact for the classical JSR

Land-Cover and Land-Use Study Using Genetic Algorithms, Petri Nets, and Cellular Automata

Recent research techniques, such as genetic algorithm (GA), Petri net (PN), and cellular automata (CA) have been applied in a number of studies. However, their capability and performance in land-cover land-use (LCLU) classification, change detection, and predictive modeling have not been well understood. This study seeks to address the following questions: 1) How do genetic parameters impact the accuracy of GA-based LCLU classification; 2) How do image parameters impact the accuracy of GA-based LCLU classification; 3) Is GA-based LCLU classification more accurate than the maximum likelihood classifier (MLC), iterative self-organizing data analysis technique (ISODATA), and the hybrid approach; 4) How do genetic parameters impact Petri Net-based LCLU change detection; and 5) How do cellular automata components impact the accuracy of LCLU predictive modeling. The study area, namely the Tickfaw River watershed (711mi²), is located in southeast Louisiana and southwest Mississippi. The major datasets include time-series Landsat TM / ETM images and Digital Orthophoto Quarter Quadrangles (DOQQ’s). LCLU classification was conducted by using the GA, MLC, ISODATA, and Hybrid approach. The LCLU change was modeled by using genetic PN-based process mining technique. The process models were interpreted and input to a CA for predicting future LCLU. The major findings include: 1) GA-based LCLU classification is more accurate than the traditional approaches; 2) When genetic parameters, image parameters, or CA components are configured improperly, the accuracy of LCLU classification, the coverage of LCLU change process model, and/or the accuracy of LCLU predictive modeling will be low; 3) For GA-based LCLU classification, the recommended configuration of genetic / image parameters is generation 2000-5000, population 1000, crossover rate 69%-99%, mutation rate 0.1%-0.5%, generation gap 25%-50%, data layers 16-20, training / testing data size 10000-20000 / 5000-10000, and spatial resolution 30m-60m; 4) For genetic Petri nets-based LCLU change detection, the recommended configuration of genetic parameters is generation 500, population 300, crossover rate 59%, mutation rate 5%, and elitism rate 4%; and 5) For CA-based LCLU predictive modeling, the recommended configuration of CA components is space 6025 * 12993, state 2, von Neumann neighborhood 3 * 3, time step 2-3 years, and optimized transition rules