Bitcoin is under the threat of fork since it operates with a distributed ledger. Predicting the fork probability in advance is beneficial for taking early action to avoid malicious attacks. In this study, we compose a colored Petri net model of Bitcoin. Our model consists of a given number of nodes, and each node has five subpages representing the node structure: proof of work, broadcast blocks, verify blocks, and the process of adding blocks to blockchain, respectively. Simulation results of fork probability can be easily obtained and analyzed by observing the data in the measuring components of subpages. The results show that our model correctly simulates the fork probability: on recent Bitcoin data, compared with the results of the wide-known SimBlock simulator, a difference of some 4.3% has been obtained. Thus, taking into account vivid graphical representation, our model has certain advantages for the developing techniques of attack avoidance.

Solving linear Diophantine systems of equations is applied in discrete-event systems, model checking, formal languages and automata, logic programming, cryptography, networking, signal processing, and chemistry. For modeling discrete systems with Petri nets, a solution in non-negative integer numbers is required, which represents an intractable problem. For this reason, solving such kinds of tasks with significant speedup is highly appreciated. In this paper we design a new solver of linear Diophantine systems based on the parallel-sequential composition of the system clans. The solver is studied and implemented to run on parallel architectures using a two-level parallelization concept based on MPI and OpenMP. A decomposable system is usually represented by a sparse matrix; a minimal clan size of the decomposition restricts the granulation of the technique. MPI is applied for solving systems for clans using a parallel-sequential composition on distributed-memory computing nodes, while OpenMP is applied in solving a single indecomposable system on a single node using multiple cores. A dynamic task-dispatching subsystem is developed for distributing systems on nodes in the process of compositional solution. Computational speedups are obtained on a series of test examples, e.g., illustrating that the best value constitutes up to 45 times speedup obtained on 5 nodes with 20 cores each.

We show that Sleptsov place-transition nets (that allow transition firing in multiple instances at a step) run fast by implementing multiplication and division operations in polynomial time. In comparison, Petri nets (PNs) implement the mentioned operations in exponential time. Moreover, PNs are obtained as a special case of Sleptsov nets (SNs) using loops with places having unit marking attached to each transition. In addition, we develop basics of an SN programming technology including basic operations and program composition rules. We provide examples of programs written in SN language for encryption/decryption with the RSA algorithm, calculation of fuzzy logic functions, and parallel calculation of the solutions to Laplace equations. SN computers promise hyper-performance because of a concurrent programming style consisting of a concise graphical language and small granulation of parallel processes on the level of separate events.

A universal Petri net with 14 places, 42 transitions, and 218 arcs was built in the class of deterministic inhibitor Petri nets (DIPNs); it is based on the minimal Turing machine (TM) of Woods and Neary with 6 states, 4 symbols, and 23 instructions, directly simulated by a Petri net. Several techniques were developed, including bi-tag system (BTS) construction on a DIPN, special encoding of TM tape by two stacks, and concise subnets that implement arithmetic encoding operations. The simulation using the BTS has cubic time and linear space complexity, while the resulting universal net runs in exponential time and quadratic space with respect to the target net transitions' firing sequence length. The technique is applicable for simulating any TM by the Petri net.