General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

all possible substitutions to it. Thus rules represent ground constructor subterm replacement
actions in a computations, where we are allowed to replace all ground instances of the left hand
side of the rule by the corresponding ground instance of the right hand side.
A Second Abstract Interpreter
Unfortunately, constructor terms are still not enough to write down rules, as rules also contain
the symbols from the abstract procedures.
Are Constructor Terms Really Enough for Rules?
Example 155 ρ(cons(n, l)) @(ρ(l), cons(n, nil)). (ρ is not a constructor)
Idea: need to include defined procedures.
Definition 156 Let A := 〈S 0 , D〉 be an abstract data type with A ∈ S, f ∈ D be a symbol,
then we call a pair [f : A] a procedure declaration for f over S.
We call a finite set Σ of procedure declarations a signature over A, if Σ is a partial function.
(unique sorts)
add the following rules to the definition of constructor terms
T ∈ S 0 and [p: T] ∈ Σ, or
t is of the form f(a), where a is a term of sort A and there is a procedure declaration
[f : A → T] ∈ Σ.
we call the the resulting structures simply “terms” over A, Σ, and V (the set of variables we
use). We denote the set of terms of sort A with TA(A, Σ; V).
c○: Michael Kohlhase 96
Again, we combine all of the rules for the inductive construction of the set of terms in one slide
for convenience.
Terms: The Complete Definition
Idea: treat procedures (from Σ) and constructors (from D) at the same time.
Definition 157 Let 〈S 0 , D〉 be an abstract data type, and Σ a signature over A, then we
call a representation t a term of sort T (over A and Σ), iff
T ∈ S 0 and [t: T] ∈ D or [t: T] ∈ Σ, or
t ∈ VT and T ∈ S 0 , or
T = A × B and t is of the form 〈a, b〉, where a and b are terms of sorts A and B, or
t is of the form c(a), where a is a term of sort A and there is a constructor declaration
[c: A → T] ∈ D or a procedure declaration [c: A → T] ∈ Σ.
Subterms
c○: Michael Kohlhase 97
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