Specification of Reactive Hardware/Software Systems - Electronic ...

238 Introduction to the Semantics of POOSL
A rule has zero or more premises and one conclusion. It tells us how we can construct a
complex mathematical object (the conclusions) from simpler ones (the premises). A rule
may have a condition which has to be fulfilled whenever the rule is to be applied. If this
condition is vacuously true, it is omitted. Rules with premises that are vacuously true
are called axioms. Axioms define what are considered primitive mathematical objects.
Together, the axioms and rules define a collection of mathematical objects. This collection
contains those and only those objects that are derivable from these axioms and rules.
Inductive Proofs
Axioms and rules are used to define mathematical objects. The same axioms and rules,
however, can also be used to prove some property about these objects. First the property
is proved for all primitive objects defined by the axioms. Then, for each rule, the property
is proved for the complex objects defined by the conclusion of that rule. Such a proof
uses the assumption that the property already holds for the objects used in the premises
of the rule. This assumption is called the induction hypothesis. The proofs themselves are
called proofs by rule induction.
Mathematical Induction
Mathematical induction is probably the most well-known form of induction. It is based
on the collection of natural numbers. is inductively defined by one axiom and one
rule:
¡£
¡£ ¡£
¡ To a property, say P(n), one can use rule induction. One starts by
0
n
n 1
prove for each n
proving P(0). Then P(n 1) is proved under the assumption that P(n) already holds.
This form of induction is commonly called mathematical induction (on n).
Inductive Function Definitions
We will repeatedly define functions in an inductive way. As a simple example, consider
the well-known faculty function fac : £ . It is defined by one axiom and one rule:
fac(n) k
if ¡£ n
fac(0) 1 fac(n 1) k(n 1)
It is however quite unusual to define functions this way. We will therefore follow the
convention and use the more usual notation:
fac(0) 1 and fac(n 1) (n 1)fac(n) for n ¡£
Using rule induction one is able to prove that each object fac(n) satisfies some property
P. It is also possible, however, to use mathematical induction on n. Then the axiom and
rule in the definition of are used as a basis for induction. The axiom and rule of the
definition of fac are used as well, but they serve a different purpose. Both of these forms
of induction are applied in this thesis.
Backus Naur Form (BNF)
As we explained in Subsection 7.4.1, we will occasionally have to define collections of
syntactic elements. Consider, for instance, the collection Bin of binary representations
of natural numbers. Bin is defined as