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

Axiom 134 equality on ground constructor terms is trivial
Axiom 135 only ground constructor terms of sort T are in T (induction axioms)
c○: Michael Kohlhase 88
Example 136 (An Abstract Data Type of Truth Values) We want to build an abstract
data type for the set {T, F } of truth values and various operations on it: We have looked at the abbreviations
∧, ∨, ¬, ⇒ for “and”, “or”, “not”, and “implies”. These can be interpreted as functions
on truth values: e.g. ¬(T ) = F , . . . . We choose the abstract data type 〈{B}, {[T : B], [F : B]}〉,
and have the abstract procedures
∧ : 〈∧::B × B → B ; {∧(T, T ) T , ∧(T, F ) F , ∧(F, T ) F , ∧(F, F ) F }〉.
∨ : 〈∨::B × B → B ; {∨(T, T ) T , ∨(T, F ) T , ∨(F, T ) T , ∨(F, F ) F }〉.
¬ : 〈¬::B → B ; {¬(T ) F , ¬(F ) T }〉,
Now that we have established how to represent data, we will develop a theory of programs, which
will consist of directed equations in this case. We will do this as theories often are developed;
we start off with a very first theory will not meet the expectations, but the test will reveal how
we have to extend the theory. We will iterate this procedure of theorizing, testing, and theory
adapting as often as is needed to arrive at a successful theory.
A First Abstract Interpreter
Let us now come up with a first formulation of an abstract interpreter, which we will refine later
when we understand the issues involved. Since we do not yet, the notions will be a bit vague for
the moment, but we will see how they work on the examples.
But how do we compute?
Problem: We can define functions, but how do we compute them?
Intuition: We direct the equations (l2r) and use them as rules.
Definition 137 Let A be an abstract data type and s, t ∈ T g
T (A) ground constructor terms
over A, then we call a pair s t a rule for f, if head(s) = f.
Example 138 turn λ(nil) = o and λ(cons(n, l)) = s(λ(l))
to λ(nil) o and λ(cons(n, l)) s(λ(l))
Definition 139 Let A := 〈S 0 , D〉, then call a quadruple 〈f::A → R ; R〉 an abstract procedure,
iff R is a set of rules for f. A is called the argument sort and R is called the result
sort of 〈f::A → R ; R〉.
Definition 140 A computation of an abstract procedure p is a sequence of ground constructor
terms t1 t2 . . . according to the rules of p. (whatever that means)
Definition 141 An abstract computation is a computation that we can perform in our
heads. (no real world constraints like memory size, time limits)
Definition 142 An abstract interpreter is an imagined machine that performs (abstract)
computations, given abstract procedures.
c○: Michael Kohlhase 89
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