A284276 Number of event structures with n labeled elements.

1, 4, 41, 916, 41099, 3528258, 561658287

Offset: 1

Author

Marco B. Caminati, Mar 24 2017

Keywords

• nonn
• hard
• more

Comments

Little is known about event structures enumeration. The entries were obtained by a dedicated algorithm recursively constructing all possible event structures. This algorithm has been formally verified to be correct by construction using the theorem prover Isabelle/HOL (see the Links section). The formal proof also formally certifies the correctness of other sequences already in the OEIS (quasi-orders, partial orders). Note that we count what are called "event structures" in the given References. Other sources, however, refer to the same objects as "prime event structures".

Examples

			An event structure is given by a poset and a conflict relation (denoted #) on it. The conflict relation is irreflexive and symmetric, and must propagate over the order: a<=b and a#c imply b#c.
For n=2, (i.e., two elements a and b), there are three possible posets: a<=b, b<=a, and neither of the two. For the first two cases, only the empty conflict is possible. For the third case, you can have either the empty conflict relation, or a#b. Hence the total number of event structures is 4.
		

Links

• Juliana Bowles and Marco B. Caminati, A Verified Algorithm Enumerating Event Structures, arXiv:1705.07228 [cs.LO], 2017.
• Marco B. Caminati, Isabelle/HOL code
• Marco B. Caminati and Juliana K. F. Bowles, Representation Theorems Obtained by Mining across Web Sources for Hints, Lancaster Univ. (UK, 2022).
• M. Nielsen, G. Plotkin, and G. Winskel, Petri nets, event structures and domains, part I, Theoretical Computer Science 13, no. 1 (1981): 85-108.
• G. Winskel and M. Nielsen, Models for concurrency, DAIMI Report Series 21, no. 429 (1992) (revised version).

Crossrefs

Cf. A001035 (generating all the event structures entails generating all the posets), A000798 (to generate all the posets we preemptively generated all the quasi-orders).

Extensions

a(7) from Marco B. Caminati, Aug 01 2017