F = {(a
1
,b
1
), … (a
n
,b
n
)}. Each pair is called a rule and may be denoted as a fraction b
i
/a
i
or a grammar style rule a
i
x
→
b
i
x or as a pair as shown in the definition.
A configuration (ID), x, of a FRS is a positive integer.
A rule b
i
/a
i
is enabled by some ID x, if x is divisible by a
i
.
Computation is defined by multiplying x by the fraction b
i
/a
i
.provided the rule is enabled
by x.
We define derivation in F by x
⇒
F
y iff y = x
×
b
i
/a
i
.where x is divisible by a
i
. .
The concept of derivation in zero or more steps,
⇒
F
* is then the reflexive, transitive
closure of
⇒
F
. As usual, we omit the F if it is understood by context.
The derivation or word problem for F is then the problem to determine of two arbitrary
positive integers x and y, whether or not x
⇒
F
* y.
Ordered Factor Replacement Systems are merely FRS’s where the rules are totally
ordered and
x
⇒
F
y iff y = x
×
b
i
/a
i
.where x is divisible by a
i
and x is not divisible by any a
j,
j<i.
The halting problem for an ordered FRS F is the problem to determine of an arbitrary
positive integer x whether or not there is some y such that x
⇒
F
* y and y is terminal,
meaning that there is no z such that y
⇒
F
z. This means that y enables no rules. An
alternative version says there is no z, z
≠
y, such that y
⇒
F
z. This allows termination by
reaching a fixed point, rather than having no applicable rule.
FRS’s are not computationally complete, whereas ordered FRS’s are.
The notion of computation with ordered FRS’s is typically to start with inputs as the
exponents of successive primes, starting with 3, e.g., if we call the 0-th prime 2, the first
3, etc, then 3
x1
5
x2
… p
n
xn
represents a start with input x1, x2, …, xn. When termination
occurs, we have the answer as the exponent of p
0
= 2. That is, for function f, FRS F
computes f, if, for all x1, x2, …, xn,
3
x1
5
x2
… p
n
xn
⇒
F
* p
0
f(x1, x2, …, xn)
Z. where Z contains no factors of 2.
Moreover, the last ID above is terminal (or a fixed point).