Preface for special issue Petri/Sleptsov net based technology of programming for parallel, emergent and distributed systems

Schemata for concurrent processes sharing some resources first appeared for manufacture processes in works of family research duet Frank and Lillian Gilbreth in 1921. Flow chart standard, developed in 1947 by Herman Goldstine and John von Neumann, did not contain blocks for parallel processing of information. Program schemata, including concurrent programs, was developed in Novosibirsk, Russia, during 1950–1960 that includes program schemes of Yanov, Martiniuk, Ershov, and Lavrov for formal proof of (concurrent) program correctness andmaximising their efficiency. Place-transition nets, which were introduced in doctoral dissertation of Carl Petri in 1962 represent, in essence, a unification of the mentioned approaches having only two types of nodes in a directed bipartite graph; a crucial amendment of Carl Petri was the introduction of a dynamical process on the graphwith tokens situated inside places which were produced and consumed as a result of transitions firing. An attractionof Petri netswas apossibility of applying formal techniquesof analysis developed in 1970–1980by Hack, Molloy, TadaoMurata and others. Nets were widely applied for modelling concurrent programs, manufacturing and transportation systems etc. Concurrent programming languages appeared which were loading elements of Petri net graph by constructs of convectional programming languages. According to the Carl Petri firing rule, a single transition fires at a step chosen in nondeterminiatic way among the set of firable transitions. It was a brilliant abstraction to study behaviour of concurrent systems with regard to ALL permitted possibilities that was especially useful for verification of communication protocols. Concurrent (parallel) programming gains speed-up at the expense of doing a few actions simultaneously that becomes the basic motivation of introducing in 1990 the Anjey Salvitsky firing rule published in works of Hans-Dieter Burkhard. It was also called the maximal firing strategy and transformed the net into a Turing-complete system. In 1976, Tilak Agerwala proved Turing-completeness of inhibitor Petri nets which contain a special inhibitor arc to check whether a place marking is zero. After constructing a series of universal Petri nets in an explicit form by Dmitry Zaitsev in 2010–2020, including minimal ones containing about forty vertices, it became clear that a Petri net runs exponentially slower than a Turing machine because of incremental way of computing arithmetic functions (similar to recursive functions of Kleene and Minsky counter automata). Anatoly Sleptsov, an outstanding Ukrainian scientist in computer science, a PhD supervisor of Dmitry Zaitsev during 1988–1991, hinted to Dmitry Zaitsev an idea of firing a transition in a few instances at the same time. The corresponding theoryof timedPetri netswithmultichannel transitions, including state equation and algebraic equivalent transformations, was developed in Dmitry Zaitsev doctoral dissertation and applied for manufacture control andmanagement in software Opera-Topaz which class of nets was a forerunner of workflow nets. Later on, Dmitry Zaitsev introduced a class of place-transition nets with multiple firing of a transition and called it a Sleptsov net that was justified by the result that it runs exponentially faster than a place-transition net with the Carl Petri firing rule.