Sometimes 'Some' is as Good as 'All'

The representation of partial order semantics as an equivalence relation on interleaving sequences was shown to extend the expressive power of interleaving semantics. A specification formalism called existential specification is introduced in which a formula is interpreted over equivalence classes of sequences by asserting that some (at least one but not necessarily all) sequences from each equivalence class satisfy a given property. It differs from the more common universal specification, which is interpreted over all sequences in all classes. Its advantage over other formalisms that deal with partial order executions lies in its simplicity: any syntax that is defined over interleaving sequences, e.g., linear temporal logic, can be adopted. It is shown how under an appropriate semantical construction, an exact existential specification of a program (i.e., each property of the program expressed using the same formalism is a consequence of this specification) can be given. Moreover, under such a construction, no information about the program is lost by choosing exact existential specification rather than exact universal specification; it is possible to generalise, by means of a proof system, an exact existential specification into an exact universal specification. Applications of these results to achieving compositional proof rules are shown.