Common use of Sequences of Messages Clause in Contracts

Sequences of Messages. We express the atomic broadcast algorithm using message sequences. In addition to the tradi- tional set operators, we use the concatenation operator ⊕ and the prefix operator ⊗ to handle sequences. def • Concatenation s1⊕s2 : The sequence s = s1⊕s2 is defined as s1 followed by s2\s1, that is, all the messages in s1 followed by all the messages in s2 that are not in s1 (in the same order as they appear in s2). For example, let s1 = (m0; m1; m2; m3; ), and s2 = (m0; m1; m4). We have s1 ⊕ s2 = (m0; m1; m2; m3; m4), and s2 ⊕ s1 = (m0; m1; m4; m2; m3). 5When reducing atomic broadcast to consensus, see [9], we get a solution in which the ordering oracle, used in the consensus algorithm, is decoupled from the atomic broadcast algorithm. def • Prefix s1 ⊗ s2 : The sequence s = s1 ⊗ s2 is defined as the longest common prefix of s1 and s2. The ⊗ operator is commutative and associative. For example, taking s1 and s2 as defined above, s1 ⊗ s2 = s2 ⊗ s1 = (m0; m1). We say that a sequence s is a prefix of another sequence sj, denoted s ≤ sj, iff s = s ⊗ sj. Notice that the empty sequence s is a prefix of every sequence.

Sequences of Messages. We express the atomic broadcast algorithm using message sequences. In addition to the tradi- tional traditional set operators, we use the concatenation operator ⊕ and the prefix prefix operator ⊗ to handle sequences. 5 When reducing atomic broadcast to consensus, see [6], we get a solution in which the ordering oracle, used in the consensus algorithm, is decoupled from the atomic broadcast algorithm. def • – Concatenation s1⊕s2 s1 ⊕ s2 : The sequence s = s1⊕s2 s ⊕ s2 is defined as s1 followed \ by s2\s1s2 s1, that is, all the messages in s1 followed by all the messages in s2 that are not in s1 (in the same order as they appear in s2). For example, let s1 = (m0; m1; m2; m3; ), and s2 = (m0; m1; m4). We have s1 ⊕ s2 = (m0; m1; m2; m3; m4), and s2 ⊕ s1 = (m0; m1; m4; m2; m3). 5When reducing atomic broadcast to consensus, see [9], we get a solution in which the ordering oracle, used in the consensus algorithm, is decoupled from the atomic broadcast algorithm. def • – Prefix s1 ⊗ s2 : The sequence s = s1 s ⊗ s2 is defined as the longest common ⊗ ≤ ⊗ ⊗ ( ) ⊗ prefix of s1 and s2. The ⊗ operator is commutative and associative. For example, taking s1 and s2 as defined above, s1 ⊗ s2 = s2 ⊗ s1 = (m0; m1)m1 . We say that a sequence s is a prefix of another sequence sj, denoted s ≤ sj, iff if and only if s = s ⊗ sj. Notice that the empty sequence s is a prefix of every sequence.

Sequences of Messages. We express the atomic broadcast algorithm using message sequences. In addition to the tradi- tional traditional set operators, we use the concatenation operator ⊕ and the prefix prefix operator ⊗ to handle sequences. 5 When reducing atomic broadcast to consensus, see [6], we get a solution in which the ordering oracle, used in the consensus algorithm, is decoupled from the atomic broadcast algorithm. def • – Concatenation s1⊕s2 s1 ⊕ s2 : The sequence s = s1⊕s2 s ⊕ s2 is defined defined as s1 followed \ by s2\s1s2 s1, that is, all the messages in s1 followed by all the messages in s2 that are not in s1 (in the same order as they appear in s2). For example, let s1 = (m0⟨m0; m1; m2; m3; )⟩, and s2 = (m0⟨m0; m1; m4). m4⟩. We have s1 ⊕ s2 = (m0⟨m0; m1; m2; m3; m4)m4⟩, and s2 ⊕ s1 = (m0⟨m0; m1; m4; m2; m3). 5When reducing atomic broadcast to consensus, see [9], we get a solution in which the ordering oracle, used in the consensus algorithm, is decoupled from the atomic broadcast algorithm. def • m3⟩. – Prefix s1 ⊗ s2 : The sequence s = s1 s ⊗ s2 is defined defined as the longest common prefix ⊗ ≤ ⊗ ⊗ ⟨ ⟩ ⊗ prefix of s1 and s2. The ⊗ operator is commutative and associative. For example, taking s1 and s2 as defined defined above, s1 ⊗ s2 = s2 ⊗ s1 = (m0; m1)m1 . We say that a sequence s is a prefix prefix of another sequence sjs′, denoted s ≤ sjs′, iff if and onlyif s = s ⊗ sj. s′. Notice that the empty sequence s emptysequence ϵ is a prefix prefix of every sequence.