equivalentsterminato

Equivalentsterminato is a term used in formal semantics and program analysis to describe a relation between expressions that share identical termination behavior under a given evaluation strategy. It is a theoretical construct applied to reason about when two program fragments can be considered interchangeable with respect to whether they terminate and what results they produce upon termination. Although not universally standardized, it is often discussed alongside contextual and observational notions of program equivalence that focus specifically on termination properties.

Definition and scope

Two closed terms t1 and t2 in a language L are said to be equivalentsterminato with respect to a semantics S if, for every closing context C in L, the composed term C[t1] terminates if and only if C[t2] terminates. If they terminate, the observable outcomes of C[t1] and C[t2] are required to be observationally equivalent under the chosen observation criteria. Some variants strengthen the definition by requiring termination times to match exactly or to lie within a specified bound, while others allow differences in non-terminating contexts to be ignored.

Relation to other concepts

Equivalentsterminato is closely related to contextual equivalence but concentrates specifically on termination behavior. It is a stronger form than may-termination equivalence and is often studied using logical relations, bisimulation techniques, or operational semantics. It can serve as a basis for validating optimizations and program transformations that preserve not only results but also the property of terminating.

Applications and examples

In compiler correctness and program transformation, proving equivalentsterminato for original and transformed code ensures that optimizations do not alter termination behavior. For example, two expressions that compute the same final value through different evaluation orders but always terminate are often considered equivalentsterminato under standard call-by-value or call-by-name semantics.

See also

contextual equivalence, termination, observational equivalence, bisimulation, operational semantics.