Download Algebraic Systems: Denning, Dennis, Qualitz

COMPONENTS
• Set of objects, S
• Collection of functions relating objects in
S
• Set of axioms
•that specify membership in S
•specify properties of functions
EXAMPLE
The system of natural numbers is an
algebraic system (N,S) where
• N is a set
• S is a function that satisfies three
particular axioms
AXIOMS
• There is a zero element in N (0 є N)
• There is a one-to-one successor function
𝐒: 𝐍 → 𝑵 − 𝟎 , w𝒉𝒆𝒓𝒆 ∀ 𝒏 ∈ 𝑵, 𝒔 𝒏 → 𝒏′
•
If M ⊆ 𝑵 𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕 0 ∈ 𝑴 𝒂𝒏𝒅
𝒏′ ∈ 𝑴 𝒘𝒉𝒆𝒏𝒆𝒗𝒆𝒓 𝒏 ∈ 𝑴
then M = N
IN ENGLISH
Axioms 1 and 2 assert that the o bjects
0,0’,0’’,0’’’,… are contained in N. These are
the natural numbers, for which we use the
standard notation, 0,1,2,3
Axiom 3 requires that N contain only the natural
numbers. That is, there must be no smaller
set M that satisfies axioms 1 and 2
IMPLICATIONS
• All operations on integers may be defined in
terms of (N,S)
• Can be shown that the principle of
mathematical induction follows from (N,S)
• if p(0) is valid and p(k) -> p(k’), then p(n) is
valid for each natural number, n.
SOURCE
Denning, P., Dennis, J., Qualitz, J. (1978) Machines Languages and Computation.
Englewood Cliffs, NJ: Prentice-Hall.