Łukasz Stefanowicz

Application of Hypergraphs to SMCs Selection

The paper deals with selection of State Machine Components (SMCs) based on Hypergraphs theory. The entire selection process use Petri nets as benchmarks. As it is known, Petri nets are used for modeling of concurrency processes. The SMCs selection problem is classified as NP-Hard which means there does not exist polynomial algorithm which provides an exact solution. In the article we show three SMCs selection methods, advantages and disadvantages of each, results of comparison between traditional methods (exponential backtracking, polynomial greedy) and an exact transversal method based on hypergraphs theory, their efficiency and propriety. An exact transversal method allows to obtain exact solution in polynomial time if selection hypergraph belongs to xt-hypergraph class.

Theoretical Aspects of Petri Nets Decomposition Based on Invariants and Hypergraphs

Two methods of Petri nets decomposition into State Machine Components (SMCs) are shown in the paper. The first one bases on the well-known algorithm of place invariants (p-invariants) calculation. The second method applies hypergraph theory and computation of exact transversals. The aim of the paper is theoretical analysis of the effectiveness of presented methods. We show, that despite the high popularity, the achieved results generated by p-invariants are not always correct and spurious components ought to be eliminated. Furthermore, the effectiveness of the application of hypergraphs into Petri net decomposition is analysed.

Application of an Exact Transversal Hypergraph in Selection of SM-Components

The paper deals with the application of the hypergraph theory in selection of State Machine Components (SM-Components) of Petri nets [1,2].As it is known, Petri nets are widely used for modeling of concurrency processes. However, in order to implement the concurrent automaton, an initial Petri net ought to be decomposed into sequential automata (SM-Components), which can be easily designed as an Finite-State-Machine (FSM) or Microprogrammed Controller [3]. The last step of the decomposition process of the Petri nets is selection of SM-Components. This stage is especially important because it determines the final number of sequential automata. In the article we propose a new idea of SM-Components selection. The aim of the method is reduction of the computational complexity from exponential to polynomial. Such a reduction can be done if the selection hypergraph belongs to the exact transversal hypergraphs (xt-hypergraphs) class. Since the recognition and generation of the first transversal in the xt-hypergraphs are both polynomial, the complete selection process can be performed in polynomial time. The proposed ideas are an extension of the concept presented in [1].The proposed method has been verified experimentally. The conducted investigations have shown that for more than 85% of examined Petri nets the selection process can be done via xt-hypergraphs.

Exact cover of states in the discrete state-space system

Given the discrete state-space system, the set cover problem is defined as selection of the minimal number of global states to cover all the local states. Commonly known methods base on the matrix reduction, boolean function transformation or heuristics ideas. Most of them are inefficient because of computational/memory complexity or non-optimal results. We propose an application of xt-hypergraphs to compute the solution in case where the discrete system can be represented by an xt-hypergraph. Recognition, as well as computation of exact cover in case of xt-hypergraphs is bounded by a polynomial in the number of local states. Therefore, the whole cover process problem turns out to be polynomial.

Partial reconfiguration of concurrent logic controllers implemented in FPGA devices

Reconfigurable systems are recently used in many domains. Although the concept of multi-context logic controllers is relatively new, it may be noticed that the subject is receiving a lot of attention, especially in the industry. The work constitutes a stepping stone in design of reconfigurable logic controllers implemented in an FPGA device. An approach of designing of logic controllers oriented for further partial reconfiguration is proposed. A case study of a milling machine is used for an illustration.

Application of modified Martinez-Silva algorithm in determination of net cover

In the article we present the idea of modifications of Martinez-Silva algorithm, which allows for determination of place invariants (p-invariants) of Petri net. Their generation time is important in the parallel decomposition of discrete systems described by Petri nets. Decomposition process is essential from the point of view of discrete system design, as it allows for separation of smaller sequential parts. The proposed modifications of Martinez-Silva method concern the net cover by p-invariants and are focused on two important issues: cyclic reduction of invariant matrix and cyclic checking of net cover.