Defining conditional independence using collapses

Trace semantics is extended to allow conditional commutativity among operations. Conditional commutativity is obtained by identifying the context (the set of global states) in which operations are commutative using special predicates. These predicates allow collapsing execution histories into equivalence classes of conditional traces. Using this approach, it is possible that the execution of two operations will be dependent in one context and independent in another. The predicates allow defining a family of possible semantic definitions for each language, where each is an extension of previous standard definitions. Examples are shown when such a semantics is desired. As an example of an application, a proof method for total correctness is introduced.