Theory Semantic_Domains
section ‹Interpretation of syntax in semantic domains›
theory Semantic_Domains imports Iteration
begin
text ‹In this section, we employ our iteration principle
to obtain interpretation of syntax in semantic domains via valuations.
A bonus from our Horn-theoretic approach is the built-in
commutation of the interpretation with substitution versus valuation update,
a property known in the literature as the ``substitution lemma".›
subsection ‹Semantic domains and valuations›
text‹
Semantic domains are for binding signatures
what algebras are for standard algebraic signatures. They fix carrier sets for each sort,
and interpret each operation symbol as an operation on these sets
%
\footnote{
To match the Isabelle type system, we model (as usual) the family of carrier sets as a
``well-sortedness" predicate taking sorts and semantic items from a given
(initially unsorted) universe into booleans,
and require the operations, considered on the unsorted universe, to preserve well-sortedness.
}
%
of corresponding arity, where:
%
\\- non-binding arguments
are treated as usual (first-order) arguments;
%
\\- binding arguments are treated as second-order (functional) arguments.
%
\footnote{
In other words, syntactic bindings are captured semantically as functional bindings.}
%
In particular, for the untyped and simply-typed $\lambda$-calculi,
the semantic domains become the well-known (set-theoretic) Henkin models.
We use terminology and notation according to our general methodology employed so far:
the inhabitants of semantic domains are referred to as ``semantic items";
we prefix the reference to semantic items with an ``s": sX, sA, etc.
This convention also applies to the operations on semantic domains: ``sAbs", ``sOp", etc.
We eventually show that the function spaces consisting of maps
from valuations to semantic items form models;
in other words,
these maps can be viewed as ``generalized items"; we use for them
term-like notations ``X", ``A", etc.
(as we did in the theory that dealt with iteration).
›
subsubsection ‹Definitions:›