Next we use each φ associated with ψ to compute a

Next we use each φ associated with ψ to compute a semimeasure μ by
approximation from below. In the algorithm, at each stage of the com-
putation the local variable μ contains the current approximation to the
function μ . This is doable because the nonzero part of the approximation
is always finite.
We describe a sequence of lower semicomputable semimeasures ψ
k
( x )
computed from φ ( x,k ) such that if φ ( x,k ) represents a lower semicom-
putable semimeasure ψ ( x ), then lim
k →∞
ψ
k
( x ) = ψ ( x ) .
Step 1. Initialize by setting μ ( x ) := ψ
k
( x ) := 0, for all x in B
∗
and
k ∈ N ; and set k := 0.

296 4. Algorithmic Probability
Step 2. Set k := k + 1. Compute φ ( x,k ) and set ψ
k
( x ) := φ ( x,k ) for
all x ∈ B k . { If the computation does not terminate, then μ will not
change any more and is trivially a semimeasure }
Step 3. For i := k − 1 ,k − 2 ,..., 0 do
for each x of length i do
search for the least K>k such that φ ( x,K ) >
∑
b ∈B
ψ
k
( xb );
set ψ
k
( x ) := φ ( x,K );
if ψ
k
( ǫ ) ≤ 1 then μ := ψ
k
else terminate.
{ Step 3 tests whether the new values in Step 2 satisfy the semimea-
sure requirements; note that if ψ is a lower semicomputable semimea-
sure, then the K ’s always exist, since for each x we have
∑
b ∈B
ψ
k
( xb )
<
∑
b ∈B
ψ ( xb ) ≤ ψ ( x ) and lim
k →∞
φ ( x,k ) = ψ ( x ) }
Step 4. Go to Step 2.
Since φ represents ψ ( x ), by monotonicity of φ we have ψ
k
( x ) ≥ φ ( x,k )
for all x of length at most k , which implies lim
k →∞
ψ
k
( x ) = ψ ( x ). If ψ
is already a semimeasure, then μ := ψ and the algorithm never finishes
but continues to approximate μ from below. If for some k and x ∈ B
k
the value of φ ( x,k ) is undefined, then the values of μ do not change any
more even though the computation of μ goes on forever. If the condition
in Step 3 is violated, then the algorithm terminates, and the constructed
μ is a semimeasure—even a computable one. Clearly, in all cases, μ is a
lower semicomputable semimeasure.
The current construction was suggested by J. Tyszkiewicz [personal commu-
nication of April 1996] and assumes a finite set B of basic elements. It can
be made to handle B = N if in the construction of ψ
k
one considers and
gives possibly nonzero measures to only sequences of length at most k and
consisting of natural numbers ≤ k.
Executing the above procedure on all functions in the list φ
1
,φ
2
,...
yields an effective enumeration μ
1
,μ
2
,... of all lower semicomputable
semimeasures.
Stage 2 Let α : N → R be any lower semicomputable function satisfy-
ing α ( j ) > 0 for all j and
∑
j
α ( j ) ≤ 1. Define the function μ
0
from B
∗
into [0 , 1) as
μ
0
( x ) =
summationdisplay
j
α ( j ) μ
j
( x ) .
We show that μ
0
is a universal lower semicomputable semimeasure. The
first condition in Definition 4.2.1 of being a semimeasure is satisfied,

4.5. Continuous Sample Space 297
since
μ
0
( ǫ ) =
summationdisplay
j
α ( j ) μ
j
( ǫ ) ≤
summationdisplay
j
α ( j ) ≤ 1 .
The second condition in Definition 4.2.1 of being a semimeasure is sat-
isfied, since, for all x in B
∗
,
μ
0
( x ) =
summationdisplay
j
α ( j ) μ
j
( x ) ≥
summationdisplay
j
α ( j )
summationdisplay
b ∈B
μ
j
( xb ) =
summationdisplay
b ∈B
μ
0
( xb ) .
The function μ
0
is lower semicomputable, since the μ
j
( x )’s are lower
semicomputable in j and x . (Use the universal partial recursive function
φ
0
and the construction above.)
Finally, μ
0
multiplicatively dominates each μ
j
, since μ
0
( x ) ≥ α ( j ) μ
j
( x ).