90 research outputs found
Process versus Unfolding Semantics for Place/Transition Petri Nets
In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical "token game," one can model the behaviour of Petri nets via non-sequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. In our formal development a relevant role is played by DecOcc, a category of occurrence nets appropriately decorated to take into account the history of tokens. The structure of decorated occurrence nets at the same time provides natural unfoldings for Place/Transition (PT) nets and suggests a new notion of processes, the decorated processes, which induce on Petri nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification
Between quantum logic and concurrency
We start from two closure operators defined on the elements of a special kind
of partially ordered sets, called causal nets. Causal nets are used to model
histories of concurrent processes, recording occurrences of local states and of
events. If every maximal chain (line) of such a partially ordered set meets
every maximal antichain (cut), then the two closure operators coincide, and
generate a complete orthomodular lattice. In this paper we recall that, for any
closed set in this lattice, every line meets either it or its orthocomplement
in the lattice, and show that to any line, a two-valued state on the lattice
can be associated. Starting from this result, we delineate a logical language
whose formulas are interpreted over closed sets of a causal net, where every
line induces an assignment of truth values to formulas. The resulting logic is
non-classical; we show that maximal antichains in a causal net are associated
to Boolean (hence "classical") substructures of the overall quantum logic.Comment: In Proceedings QPL 2012, arXiv:1407.842
On the Semantics of Petri Nets
Petri Place/Transition (PT) nets are one of the most widely used models of concurrency. However, they still lack, in our view, a satisfactory semantics: on the one hand the "token game"' is too intensional, even in its more abstract interpretations in term of nonsequential processes and monoidal categories; on the other hand, Winskel's basic unfolding construction, which provides a coreflection between nets and finitary prime algebraic domains, works only for safe nets. In this paper we extend Winskel's result to PT nets. We start with a rather general category {PTNets} of PT nets, we introduce a category {DecOcc} of decorated (nondeterministic) occurrence nets and we define adjunctions between {PTNets} and {DecOcc} and between {DecOcc} and {Occ}, the category of occurrence nets. The role of {DecOcc} is to provide natural unfoldings for PT nets, i.e. acyclic safe nets where a notion of family is used for relating multiple instances of the same place. The unfolding functor from {PTNets} to {Occ} reduces to Winskel's when restricted to safe nets, while the standard coreflection between {Occ} and {Dom}, the category of finitary prime algebraic domains, when composed with the unfolding functor above, determines a chain of adjunctions between {PTNets} and {Dom}
Petri Games: Synthesis of Distributed Systems with Causal Memory
We present a new multiplayer game model for the interaction and the flow of
information in a distributed system. The players are tokens on a Petri net. As
long as the players move in independent parts of the net, they do not know of
each other; when they synchronize at a joint transition, each player gets
informed of the causal history of the other player. We show that for Petri
games with a single environment player and an arbitrary bounded number of
system players, deciding the existence of a safety strategy for the system
players is EXPTIME-complete.Comment: In Proceedings GandALF 2014, arXiv:1408.556
Concurrent systems and inevitability
AbstractConcurrent systems viewed as partially ordered sets of states are considered. A property of system states is called inevitable, if the system will eventually reach a state with this property. This notion is discussed within the partial order framework
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