Editing Denotational semantics (section)

==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.
#'''Adequacy (or soundness)''': All [[observational equivalence|observably distinct]] programs have distinct denotations;
#'''Full abstraction''': All observationally equivalent programs have equal denotations.

For semantics in the traditional style, adequacy and 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 concepts of adequacy and of full abstraction are 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 (mathematics)|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 [[Denotational semantics#Compositionality|Compositionality]] below.
#'''Full completeness''' or '''definability''': Every morphism of the semantic model should be the denotation of a program.<ref>{{cite journal
  | last = Curien	
  | first = Pierre-Louis
  | title = Definability and Full Abstraction
  | journal = Electronic Notes in Theoretical Computer Science
  | volume = 172
  | pages = 301–310
  | doi = 10.1016/j.entcs.2007.02.011
  | year = 2007 | doi-access = free
  }}</ref>