4 research outputs found
On the Tightness of Bounds for Transients of Weak CSR Expansions and Periodicity Transients of Critical Rows and Columns of Tropical Matrix Powers
We study the transients of matrices in max-plus algebra. Our approach is
based on the weak CSR expansion. Using this expansion, the transient can be
expressed by , where is the weak CSR threshold and
is the time after which the purely pseudoperiodic CSR terms start to dominate
in the expansion. Various bounds have been derived for and ,
naturally leading to the question which matrices, if any, attain these bounds.
In the present paper we characterize the matrices attaining two particular
bounds on , which are generalizations of the bounds of Wielandt and
Dulmage-Mendelsohn on the indices of non-weighted digraphs. This also leads to
a characterization of tightness for the same bounds on the transients of
critical rows and columns. The characterizations themselves are generalizations
of those for the non-weighted case.Comment: 42 pages, 9 figure
Computation of the Transient in Max-Plus Linear Systems via SMT-Solving
This paper proposes a new approach, grounded in Satisfiability Modulo
Theories (SMT), to study the transient of a Max-Plus Linear (MPL) system, that
is the number of steps leading to its periodic regime. Differently from
state-of-the-art techniques, our approach allows the analysis of periodic
behaviors for subsets of initial states, as well as the characterization of
sets of initial states exhibiting the same specific periodic behavior and
transient. Our experiments show that the proposed technique dramatically
outperforms state-of-the-art methods based on max-plus algebra computations for
systems of large dimensions.Comment: The paper consists of 22 pages (including references and Appendix).
It is accepted in FORMATS 2020 First revisio
Formal Analysis and Verification of Max-Plus Linear Systems
Max-Plus Linear (MPL) systems are an algebraic formalism with practical
applications in transportation networks, manufacturing and biological systems.
In this paper, we investigate the problem of automatically analyzing the
properties of MPL, taking into account both structural properties such as
transient and cyclicity, and the open problem of user-defined temporal
properties. We propose Time-Difference LTL (TDLTL), a logic that encompasses
the delays between the discrete time events governed by an MPL system, and
characterize the problem of model checking TDLTL over MPL. We first consider a
framework based on the verification of infinite-state transition systems, and
propose an approach based on an encoding into model checking. Then, we leverage
the specific features of MPL systems to devise a highly optimized,
combinational approach based on Satisfiability Modulo Theory (SMT). We
experimentally evaluate the features of the proposed approaches on a large set
of benchmarks. The results show that the proposed approach substantially
outperforms the state of the art competitors in expressiveness and
effectiveness, and demonstrate the superiority of the combinational approach
over the reduction to model checking.Comment: 28 pages (including appendixes