Petri Nets: Theory, Analysis, and the Case for Algebraic Structures -A Review from Foundations to 2026
Abstract
Petri nets, introduced by Carl Adam Petri in 1962, remain among the most enduring formalisms for modelling concurrent, distributed, and asynchronous systems. This paper reviews Petri net theory from its classical foundations to developments reported through 2026, organised around three strands: the behavioural and structural theory consolidated in the classical tutorial literature, most notably Murata's 1989 survey; the algebraic treatment of nets that began with incidence matrices and net languages and was deepened in dedicated studies such as Kunimochi's dissertation on the algebraic properties of Petri net languages and codes; and the more recent categorical, semigroup theoretic, and semiring based reformulations that support compositional modelling and performance analysis. Building on this synthesis, we argue that a systematic algebraic foundation for Petri nets, treating markings, firing sequences, and net morphisms as elements of explicitly defined semigroups, monoids, lattices, or categories, is a structural necessity rather than a cosmetic reformulation. Such structures unify decidability results, enable compositional construction of large nets from verified components, support invariant based verification, and connect Petri net theory to automata, formal language, and coding theory. We close with open problems whose resolution depends on a clearer algebraic axiomatisation of Petri net behaviour, and we recommend directions for future research.
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BibTeX
@misc{k2026petri,
title = {{Petri Nets: Theory, Analysis, and the Case for Algebraic Structures -A Review from Foundations to 2026}},
author = {Dr Girish K P and Saithalavi P},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05748056v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Petri Nets: Theory, Analysis, and the Case for Algebraic Structures -A Review from Foundations to 2026
AU - Dr Girish K P
AU - Saithalavi P
PY - 2026
DA - 2026/09/13
PB - HAL
UR - https://hal.science/hal-05748056v1
AB - Petri nets, introduced by Carl Adam Petri in 1962, remain among the most enduring formalisms for modelling concurrent, distributed, and asynchronous systems. This paper reviews Petri net theory from its classical foundations to developments reported through 2026, organised around three strands: the behavioural and structural theory consolidated in the classical tutorial literature, most notably Murata's 1989 survey; the algebraic treatment of nets that began with incidence matrices and net languages and was deepened in dedicated studies such as Kunimochi's dissertation on the algebraic properties of Petri net languages and codes; and the more recent categorical, semigroup theoretic, and semiring based reformulations that support compositional modelling and performance analysis. Building on this synthesis, we argue that a systematic algebraic foundation for Petri nets, treating markings, firing sequences, and net morphisms as elements of explicitly defined semigroups, monoids, lattices, or categories, is a structural necessity rather than a cosmetic reformulation. Such structures unify decidability results, enable compositional construction of large nets from verified components, support invariant based verification, and connect Petri net theory to automata, formal language, and coding theory. We close with open problems whose resolution depends on a clearer algebraic axiomatisation of Petri net behaviour, and we recommend directions for future research.
ER -