Next we use eachφassociated withψto compute a semimeasureμbyapproximation from below. In the algorithm, at each stage of the com-putation the local variableμcontains the current approximation to thefunctionμ. This is doable because the nonzero part of the approximationis always finite.We describe a sequence of lower semicomputable semimeasuresψk(x)computed fromφ(x,k) such that ifφ(x,k) represents a lower semicom-putable semimeasureψ(x), then limk→∞ψk(x) =ψ(x).Step 1.Initialize by settingμ(x) :=ψk(x) := 0, for allxinB∗andk∈ N; and setk:= 0.

2964.Algorithmic ProbabilityStep 2.Setk:=k+ 1. Computeφ(x,k) and setψk(x) :=φ(x,k) forallx∈ Bk.{If the computation does not terminate, thenμwill notchange any more and is trivially a semimeasure}Step 3. Fori:=k−1,k−2,...,0doforeachxof lengthidosearch for the leastK>ksuch thatφ(x,K)>∑b∈Bψk(xb);setψk(x) :=φ(x,K);ifψk(ǫ)≤1thenμ:=ψkelseterminate.{Step 3 tests whether the new values in Step 2 satisfy the semimea-sure requirements; note that ifψis a lower semicomputable semimea-sure, then theK’s always exist, since for eachxwe have∑b∈Bψk(xb)<∑b∈Bψ(xb)≤ψ(x) and limk→∞φ(x,k) =ψ(x)}Step 4. Go toStep 2.Sinceφrepresentsψ(x), by monotonicity ofφwe haveψk(x)≥φ(x,k)for allxof length at mostk, which implies limk→∞ψk(x) =ψ(x). Ifψis already a semimeasure, thenμ:=ψand the algorithm never finishesbut continues to approximateμfrom below. If for somekandx∈ Bkthe value ofφ(x,k) is undefined, then the values ofμdo not change anymore even though the computation ofμgoes on forever. If the conditionin Step 3 is violated, then the algorithm terminates, and the constructedμis a semimeasure—even a computable one. Clearly, in all cases,μis alower semicomputable semimeasure.The current construction was suggested by J. Tyszkiewicz [personal commu-nication of April 1996] and assumes a finite setBof basic elements. It canbe made to handleB=Nif in the construction ofψkone considers andgives possibly nonzero measures to only sequences of length at mostkandconsisting of natural numbers≤k.Executing the above procedure on all functions in the listφ1,φ2,...yields an effective enumerationμ1,μ2,...of all lower semicomputablesemimeasures.Stage 2Letα:N → Rbe any lower semicomputable function satisfy-ingα(j)>0 for alljand∑jα(j)≤1. Define the functionμ0fromB∗into [0,1) asμ0(x) =summationdisplayjα(j)μj(x).We show thatμ0is a universal lower semicomputable semimeasure. Thefirst condition in Definition 4.2.1 of being a semimeasure is satisfied,

4.5.Continuous Sample Space297sinceμ0(ǫ) =summationdisplayjα(j)μj(ǫ)≤summationdisplayjα(j)≤1.The second condition in Definition 4.2.1 of being a semimeasure is sat-isfied, since, for allxinB∗,μ0(x) =summationdisplayjα(j)μj(x)≥summationdisplayjα(j)summationdisplayb∈Bμj(xb) =summationdisplayb∈Bμ0(xb).The functionμ0is lower semicomputable, since theμj(x)’s are lowersemicomputable injandx. (Use the universal partial recursive functionφ0and the construction above.)Finally,μ0multiplicatively dominates eachμj, sinceμ0(x)≥α(j)μj(x).

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Computational complexity theory, Algorithmic information theory