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Vector addition system

A vector addition system (VAS) is one of several mathematical modeling languages for the description of distributed systems. Vector addition systems were introduced by Richard M. Karp and Raymond E. Miller in 1969,[1] and generalized to vector addition systems with states (VASS) by John E. Hopcroft and Jean-Jacques Pansiot in 1979.[2] Both VAS and VASS are equivalent in many ways to Petri nets introduced earlier by Carl Adam Petri.

Example of a vector addition with states. In this VASS, e.g., q(1,2) can be reached from p(0,0), but q(0,0) cannot be reached from p(0,0).

Reachability in vector addition systems is Ackermann-complete (and hence nonelementary).[3][4]

  1. ^ Karp, Richard M.; Miller, Raymond E. (May 1969). "Parallel program schemata". Journal of Computer and System Sciences. 3 (2): 147–195. doi:10.1016/S0022-0000(69)80011-5.
  2. ^ Hopcroft, John E.; Pansiot, Jean-Jacques (1979). "On the reachability problem for 5-dimensional vector addition systems". Theoretical Computer Science. 8 (2): 135–159. doi:10.1016/0304-3975(79)90041-0. hdl:1813/6102.
  3. ^ Czerwiński, Wojciech; Orlikowski, Łukasz (2021). Reachability in Vector Addition Systems is Ackermann-complete. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). arXiv:2104.13866.
  4. ^ Leroux, Jérôme (2021). The Reachability Problem for Petri Nets is Not Primitive Recursive. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). arXiv:2104.12695.
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