This post is part of the Asymptotic Logical Uncertainty series. In this post, I will give an algorithm, BenfordLearner, that passes the Benford test. This algorithm further satisfies ${lim}_{N\to \infty ,N\in S}BenfordLearner\left(N\right)=p$ whenever $S$ is an irreducible pattern with probability $p$.

In the following algorithm, we define $\log q$ to be 1 whenever $q\le 2$.

BenfordLearner(N)
 1: P=0
 2: M=N
 3: for j=0 to N
 4:   M_Y=0
 5:   for Y a TM expressible in K_Y <log N bits
 6:     M_X=N
 7:     for Y a TM expressible in K_X <log N bits
 8:       if UTM(X,N) and UTM(Y,N) both accept in time T(N)
 9:         A=0
10:         R=0
11:         i=1
12:         while i<=N
13:           if UTM(X,i) and UTM(Y,i) both accept in time T(i)
14:             if UTM(L,i) accepts in time T(N)
15:               A=A+1
16:             else if UTM(L,i) rejects in time T(N)
17:               R=R+1
18:             else
19:               i=N
20:           i=i+1
21:         F=A/(A+R)
22:         Q=A+R
23:         if max(K_X,|F-j/N|*sqrt(Q)/K_Y/sqrt(log log Q))<M_X
24:           M_X=max(K_X,|F-j/N|*sqrt(Q)/K_Y/sqrt(log log Q))
25:     if M_X>M_Y
26:       M_Y=M_X
27:   if M_Y<M
28:     M=M_Y
29:     P=j/N
30: return P

Let $TM\left(N\right)$ be the set of all Turing machines $X$ expressible in at most $\log N$ bits such that $UTM(X,N)$ accepts in time at most $T\left(N\right)$. The encoding of Turing machines must be prefix-free, which in particular means that no Turing machine is encoded in 0 bits. Let $J_N$ denote the set of rational numbers of the form $\frac{j}{N}$ with $j=0,\dots ,N$.

For $X$ and $Y$ Turing machines, let $K\left(X\right)$ be the number of bits necessary to encode $X$. Let $S^{\prime }(X,Y)$ be the subset of natural numbers $i$ which are accepted by both $UTM(X,i)$ and $UTM(Y,i)$ in time at most $T\left(i\right)$. Let $Q_N(X,Y)$ be the greatest number less than or equal to $N$ such that for every $s$ in the first $Q_N(X,Y)$ elements of $S^{\prime }$, $UTM(L,s)$ halts in time $T\left(N\right)$. Let $F_N(X,Y)$ be the proportion of the first $Q_N(X,Y)$ elements of $S^{\prime }$ which $L$ accepts. Let $$B_N(X,Y,P)=max(K\left(X\right),\frac{|F_N(X,Y)-P|\sqrt{Q_N(X,Y)}}{K\left(Y\right)\sqrt{\log \log Q_N(X,Y)}}).$$

Lemma: The output of $BenfordLearner\left(N\right)$ is in $$\begin{array}{c}argmin\\ P\in J_N\end{array}\begin{array}{c}max\\ Y\in TM\left(N\right)\end{array}\begin{array}{c}min\\ X\in TM\left(N\right)\end{array}{B}_N(X,Y,P).$$

Proof: The algorithm has three for loops, the outer ranging over $j=0,\dots N$ and the inner two ranging over $Y$ and $X$ respectively, both restricted to Turing machines expressible in $\log N$ bits. The condition on line 8 means that $X$ and $Y$ effectively range over all Turing machines in $TM\left(N\right)$, and $P=\frac{j}{N}$ ranges over $J_N$.

The inner while loop will increment the variables $A$ or $R$ a total of exactly $Q_N(X,Y)$ times. Thus, $Q$ is set to $Q_N(X,Y)$ in line 22. Similarly, $F$ is sent to $F_N(X,Y)$ in line 21. Clearly $K_X$ and $K_Y$ are $K\left(X\right)$ and $K\left(Y\right)$ respectively. Therefore, the expression on lines 23 and 24 is $B_N(X,Y,P).$

Considering the for loops from inner to outer, we minimize this quantity in $X$, maximize it in $Y$, and find $P$ of the form $j/N$ minimizing the whole quantity. The $P$ returned is therefore a minimizer of $$\underset{Y\in TM\left(N\right)}{max}\underset{X\in TM\left(N\right)}{min}{B}_N(X,Y,P).$$ $\square$

Our goal is to quickly assign logical probabilities. The code is not optimized to run efficiently. The following proposition is just to ensure that the run time is not far off from $T\left(N\right)$.

Proposition: The run time of $BenfordLearner\left(N\right)$ is in $O\left(T\right(N\left)N^4\log T\right(N\left)\right))$.

Simulating $UTM$ on any input for $T$ time steps can be done in time $T$ in time $cT\log T$ for some fixed constant $c$ by simulating $U$ for $T$ steps \cite{Hennie1966}. The bulk of the run time comes from simulating Turing machines on lines 8, 13, 14, and 16. Each of these lines takes at most $cT\left(N\right)\log T\left(N\right)$ time, and we enter each of these lines at most $N^4$ times. Therefore, the program runs in time $O\left(T\right(N\left)N^4\log T\right(N\left)\right)$.

$\square$

In the next post, we will use the above Lemma and the notion of irreducible patterns to show that this algorithm passes the Benford Test.