Mathematical equivalence treats two system states or outcomes as interchangeable under an application-defined criterion, even when their representations are not equal. The practical question is not “Are they identical?” but “Do their differences matter here?” section 4.4 section 8.2
For message processing, this can be written P(m) ≡ P(m) • … P(m): processing a message repeatedly produces an equivalent system state. Duplicate log entries may differ literally while remaining inconsequential to the application. section 4.4
When are two books one item?
A library may count editions of the same work as one logical item, while another perspective counts only a particular language or edition. The physical inventory is unchanged; the equivalence criterion changes the answer. section 8.2
Table 8.1 Book editions
| Name |
Edition |
Year |
| SICP |
First |
1984 |
| SICP |
Second |
1996 |
| SICP |
JavaScript |
2022 |
Compare the listed editions to see how different representations can be grouped under a shared logical identity. section 8.2
Equivalence also connects higher-level abstractions with lower-level executions: reasoning at either level should produce the same result. Transactions use this relationship to hide concurrency and failure behind simpler definitions. section 5.1
A useful guarantee has explicit limits:
- Define equivalence for the application; different observers may judge the same difference important or irrelevant. section 4.4
- Treat exactly-once processing as an end-to-end property requiring at-least-once delivery and processing by both sender and receiver. section 4.4
