 # Next we use each φ associated with ψ to compute a

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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 allxinBandk∈ 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:=k1,k2,...,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 haveb∈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 numbersk.Executing the above procedure on all functions in the listφ12,...yields an effective enumerationμ12,...of all lower semicomputablesemimeasures.Stage 2Letα:N → Rbe any lower semicomputable function satisfy-ingα(j)>0 for alljandjα(j)1. Define the functionμ0fromBinto [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
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