Abstraction
It is often considered important to connect denotational semantics with operational semantics. This is especially important when the denotational semantics is rather mathematical and abstract, and the operational semantics is more concrete or closer to the computational intuitions. The following properties of a denotational semantics are often of interest.
- Syntax independence: The denotations of programs should not involve the syntax of the source language.
- Soundness: All observably distinct programs have distinct denotations;
- Full abstraction: Two programs have the same denotations precisely when they are observationally equivalent. For semantics in the traditional style, full abstraction may be understood roughly as the requirement that "operational equivalence coincides with denotational equality". For denotational semantics in more intensional models, such as the Actor model and process calculi, there are different notions of equivalence within each model, and so the concept of full abstraction is a matter of debate, and harder to pin down. Also the mathematical structure of operational semantics and denotational semantics can become very close.
Additional desirable properties we may wish to hold between operational and denotational semantics are:
- Constructivism: Constructivism is concerned with whether domain elements can be shown to exist by constructive methods.
- Independence of denotational and operational semantics: The denotational semantics should be formalized using mathematical structures that are independent of the operational semantics of a programming language; However, the underlying concepts can be closely related. See the section on Compositionality below.
- Full completeness or definability: Every morphism of the semantic model should be the denotation of a program.
Read more about this topic: Denotational Semantics
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