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| Notations | Descriptions |
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| Relations of event, event sequence number, sender, and cycle number |
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| Seq(s, c) | The sequence number of the event sent for cycle c from sender s |
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| Seq(e) | The sequence number of event e |
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| e(s, j) | The event of sequence number j sent from s |
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| Deliver(e, c) | Return true if event e is deliverable at cycle c. Otherwise, return false |
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| DeliverCycle(e) | Return the cycle in which event e is delivered. If e has not been delivered, return ⊥. |
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| Receive(e, c) | Return true if e has been received at cycle c. Otherwise, return false. |
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| ReceiveCycle(e) | Return the cycle in which event e is received. If e has not been received, return ⊥. |
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| Event of cycle | The event sent by a sender for a cycle is called the event of the cycle. Given the first event sent from sender s at cycle c0. The event of cycle c satisfies Seq(e) = Seq(s, c) = (c - c0). |
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| Cycle open / close | If a replica ri has delivered the events for cycle c and moved to the next cycle c + 1, then c is closed and c + 1 is still open to ri. |
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| Other notations |
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| ri | Replica i |
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| rL | Group leader |
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| G | The set of group members |
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| S | The set of event senders |
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| U | Set of update recipients |
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| g | Replica group g |
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| s | Sender s∈S |
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| Index(s) | The index of s in S sorted by sender ID |
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| Sender(e) | The sender of event e |
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| c | Cycle number c |
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| ∆t | Cycle length |
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| tstart,s | The start time of the first cycle for s |
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| tsend,s(n) | The time to send the nth event from s |
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| trecv,s(n) | The time to receive the nth event sent from s |
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| tnow | Current time |
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| E / E(c) | The set of decided events / decided events of c |
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| Qd | Delivery queue, containing the sorted events to be delivered to the application in sequence |
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| (c, γ, e) | The event e delivered for cycle c with local index γ in Qd |
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| (λ, e) | The delivery sequence λ of event e in Qd, one-to-one mapped onto (c, γ, e) for the given e. |
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| Ω(s, c) | The set of expected events which can be delivered to Qd (called deliverable events) from s at cycle c |
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| Π(s, c) | The set of actual received events from s at cycle c |
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