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A concurrent algorithm coordinates activities whose execution steps can overlap in time or be interleaved. Correctness must cover the allowed thread interleavings, including executions in which one activity pauses while another continues. Merely testing one convenient schedule does not establish the required property.

Suppose two activities increment a shared counter initially zero by separately reading, adding one and writing. Both can read zero before either writes, so the final value is one rather than two. An atomic read-modify-write can make each increment indivisible; mutual exclusion can instead protect a larger critical section. The appropriate choice depends on which shared-state invariant must survive each permitted execution step.

Leslie Lamport's Lamport's bakery algorithm illustrates the distinction between a coordination rule and assumptions about hardware. Its mutual exclusion argument orders ticket/process-number pairs, and the original proof explicitly treats reads overlapping writes rather than silently assuming every shared-memory access behaves atomically.^[ref] Concurrent correctness can include safety, such as never entering two critical sections together, and progress, such as a waiting activity eventually proceeding under stated scheduling assumptions.

Lamport's bakery algorithm is a mutual exclusion concurrent algorithm in which each participating process chooses a numbered ticket before entering a critical section. Process $i$ announces that it is choosing, sets its ticket to one more than the largest observed ticket, and finishes choosing. It then waits for each other process to finish choosing and for every smaller nonzero pair $(\text{ticket},\text{process identifier})$ to leave. Comparing identifiers breaks ties between equal tickets; setting one's ticket to zero announces departure.^[ref]

For example, simultaneous choices can give processes 2 and 5 the same ticket 7; pair $(7,2)$ precedes $(7,5)$. The choosing flag prevents treating an unfinished choice as an absent contender. Leslie Lamport's 1974 proof even permits an overlapping read to return an arbitrary value under the specified shared-register model. The mathematical model uses unbounded tickets; a bounded machine implementation must also address overflow and its memory model.^[ref]