Normal number

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A different topic is treated in the article titled normal number (computing).

In mathematics, a normal number is, roughly speaking, a real number whose digits show a random distribution, with all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. "Digits" refers to the finitely many digits before the decimal point (the integer part of the real number) and the infinite sequence of digits after the point (the fractional part of the real number).

While a general proof can be given that "almost all" numbers are normal, this proof is not constructive and only very few concrete numbers have been shown to be normal. It is for instance widely believed that the numbers √2, π and e are normal, but a proof remains elusive.

Definitions

Let [\Sigma] be a finite alphabet of b digits. Let [S\in\Sigma^\infty] be an infinite sequence drawn from the alphabet [\Sigma]. Let [w\in\Sigma^*] be a finite string drawn from the alphabet [\Sigma]. Let n be a positive integer. Define N_S(w,n) to be the number of times the string w appears as a substring in the first n digits of the sequence S. (For instance, if S = 01010101..., then N_S(010, 8) = 3.) S is normal if, for all finite strings [w\in\Sigma^*],

[\lim_ \frac = \frac]

(where [|w|] denotes the length of the string w; see also limit.) In other words, S is normal if all strings of equal length occur with equal asymptotic frequency. For example, in a normal binary sequence (a sequence over the alphabet ), 0 and 1 each occur with frequency 1/2; 00, 01, 10, and 11 each occur with frequency 1/4; 000, 001, 010, 011, 100, 101, 110, and 111 each occur with frequency 1/8, etc. Roughly speaking, the probability of finding the string w in any given position in S is precisely that expected if the sequence had been produced at random.

Suppose now that b is an integer greater than 1 and x is a real number. Consider the infinite digit sequence expansion S_x,b of x in the base b positional number system (we ignore the decimal point). We say x is normal in base b if the sequence S_x,b is normal. The number x is called a normal number (or sometimes an absolutely normal number) if it is normal in base b for every positive integer b.

A given infinite sequence is either normal or not normal, whereas a real number, having a different base-b expansion for each integer [b \geq 2], may be normal in one base but not in another (Cassels 1959 and Schmidt 1960).

A weaker property than normality is simple normality. A number is simply normal in base b if each individual digit appears with frequency 1/b.

Properties and examples

The concept of a normal number was introduced by Émile Borel in 1909. Using the Borel-Cantelli lemma, he proved the normal number theorem: almost all real numbers are normal, in the sense that the set of non-normal numbers has Lebesgue measure zero. (Borel 1909) This theorem established the existence of normal numbers, but Waclaw Sierpinski in 1917 was the first to give an example of one.

The set of non-normal numbers, even though "small" in the sense of being a null set, is "large" in the sense of being uncountable. Indeed, there are uncountably many numbers whose decimal expansion does not contain the digit 5, and none of these is normal.

Champernowne's number

0.1234567891011121314151617...,

obtained by concatenating the decimal representations of the natural numbers in order, is normal in base 10, but it is not normal in some other bases. The Copeland-Erdős constant

0.235711131719232931374143...,

obtained by concatenating the prime numbers in base 10, is also known to be normal in base 10.

No rational number is normal to any base, since the digit sequences of rational numbers are eventually periodic.

An example of a normal number is given by Chaitin's constant [\ \Omega]. Indeed, every algorithmically random sequence over the alphabet [\Sigma = \] is the base-b expansion of a normal number. A computable absolutely normal number was constructed in (Becher 2002).

It is extremely hard to prove the normality of numbers which were not explicitly constructed for the purpose. It is for instance unknown whether √2, π, ln(2) or e is normal (but all of them are strongly conjectured to be normal, because of some empirical evidence). Proofs are out of reach: it is not even known which digits occur infinitely often in the decimal expansions of those constants. David H. Bailey and Richard E. Crandall conjectured in 2001 that every irrational algebraic number is normal; while no counterexamples are known, there also exists no number that has been proven to be normal in some base and algebraic.

A disjunctive sequence is a sequence in which every finite string appears. A normal sequence is disjunctive, but a disjunctive sequence need not be normal.

Additional properties of normal numbers include:

• Every positive number is the product of two normal numbers. This follows from the general fact that every number is the product of two numbers from a set [X\subseteq\R^+] if the complement of X has measure 0.
• If x is normal in base b and q is a rational number, then [x \cdot q] is normal in base b. (Wall 1949)

• If [A\subseteq\N] is dense (for every [\alpha<1] and for all sufficiently large n, [|A \cap \| \geq n^\alpha]) and [a_1,a_2,a_3,\ldots] are the base-b expansions of the elements of A, then the number [0.a_1a_2a_3\ldots], formed by concatenating the elements of A, is normal in base b (Copeland and Erdős 1946). From this it follows that Champernowne's number is normal (since the set of all positive integers is obviously dense) and that the Copeland-Erdős constant is normal (since the prime number theorem implies that the set of primes is dense).

• A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the first length-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work of Ziv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodchandran (2005).

• A number is normal in base b if and only if it is simply normal in base b^k for every integer [k \geq 1]. This follows from the previous block characterization of normality: Since the n^th block of length k in its base b expansion corresponds to the n^th digit in its base b^k expansion, a number is simply normal in base b^k if and only if blocks of length k appear in its base b expansion with equal frequency.

• A number is normal if and only if it is simply normal in every base. This follows from the previous characterization of base b normality.

• The set of normal sequences is closed under finite variations: adding, removing, or changing a finite number of digits in any normal sequence leaves it normal.

Connection to finite-state machines

Agafonov showed an early connection between finite-state machines and normal sequences: every subsequence selected from a normal sequence by a regular language is also normal. In other words, if one runs a finite-state machine on a normal sequence, where the finite-state machine outputs the digit it just read whenever it enters an accepting state, then the sequence it outputs will be normal. (Agafonov 1968)

A deeper connection exists with finite-state gamblers (FSG's) and information lossless finite-state compressors (ILFSC's).

 A finite-state gambler (a.k.a. finite-state martingale) is a finite-state machine over a finite alphabet [\Sigma], each of whose states is labelled with percentages of money to bet on each digit in [\Sigma]. For instance, for a FSG over the binary alphabet [\Sigma = \], the current state q bets some percentage [q_0 \in [0,1]] of the gambler's money on the bit 0, and the remaining [q_1 = 1-q_0] fraction of the gambler's money on the bit 1. The money bet on the digit that comes next in the input (total money times percent bet) is multiplied by [|\Sigma|], and the rest of the money is lost. After the bit is read, the FSG transitions to the next state according to the input it received. A FSG d succeeds on an infinite sequence S if, starting from $1, it makes unbounded money betting on the sequence; i.e., if

[\limsup_ d(S \upharpoonright n) = \infty,]

where [d(S \upharpoonright n)] is the amount of money the gambler d has after reading the first n digits of S (see limit superior).

 A finite-state compressor is a finite-state machine with output strings labelling its state transitions, including possibly the empty string. (Since one digit is read from the input sequence for each state transition, it is necessary to be able to output the empty string in order to achieve any compression at all). An information lossless finite-state compressor is a finite-state compressor whose input can be uniquely recovered from its output and final state. In other words, for an finite-state compressor C with state set Q, C is information lossless if the function [f: \Sigma^* \to \Sigma^* \times Q], mapping the input string of C to the output string and final state of C, is 1-1. Compression techniques such as Huffman coding or Shannon-Fano coding can be implemented with ILFSC's. An ILFSC C compresses an infinite sequence S if

[\liminf_ \frac> < 1,]

where [|C(S \upharpoonright n)|] is the number of digits output by C after reading the first n digits of S. Note that the compression ratio (the limit inferior above) can always be made to equal 1 by the 1-state ILFSC that simply copies its input to the output.

Schnorr and Stimm showed that no FSG can succeed on any normal sequence, and Bourke, Hitchcock and Vinodchandran showed the converse. Therefore:

A sequence is normal if and only if there is no finite-state gambler that succeeds on it.

Ziv and Lempel showed:

A sequence is normal if and only if it is incompressible by any information lossless finite-state compressor

(they actually showed that the sequence's optimal compression ratio over all ILFSC's is exactly its entropy rate, a quantitative measure of its deviation from normality, which is 1 exactly when the sequence is normal). Since the LZ compression algorithm compresses asymptotically as well as any ILFSC, this means that the LZ compression algorithm can compress any non-normal sequence. (Ziv Lempel 1978)

These characterizations of normal sequences can be interpreted to mean that "normal" = "finite-state random"; i.e., the normal sequences are precisely those that appear random to any finite-state machine. Compare this with the algorithmically random sequences, which are those infinite sequences that appear random to any algorithm (and in fact have similar gambling and compression characterizations with Turing machines replacing finite-state machines).

Connection to equidistributed sequences

A number x is normal in base b if and only if the sequence [_^\infty] is equidistributed modulo 1, or equivalently, using Weyl's criterion, if and only if

[\lim_\frac\sum_^e^=0 \quad\mbox m\geq 1.]

References

• Agafonov, V. N. "Normal sequences and finite automata." Soviet Mathematics Doklady, 9:324-325, 1968.
• Bailey, D. H. and Crandall, R. E. "On the Random Character of Fundamental Constant Expansions." Experimental Mathematics 10, 175-190, 2001. [online version]
• Becher, V. and Figueira, S. "An example of a computable absolutely normal number", Theoretical Computer Science, 270, pp. 947-958, 2002. [online]
• Borel, E. "Les probabilités dénombrables et leurs applications arithmétiques." Rend. Circ. Mat. Palermo 27, 247-271, 1909.
• Bourke, C. , Hitchcock, J. M., and Vinodchandran, N. V. "Entropy rates and finite-state dimension." Theoretical Computer Science, 2005.
• Cassels, J. W. S. "On a problem of Steinhaus about normal numbers." Colloquium Mathematicum, 7:95-101, 1959.
• Champernowne, D. G. "The Construction of Decimals Normal in the Scale of Ten." Journal of the London Mathematical Society 8, 254-260, 1933.
• Copeland A. H. and Erdős P. "Note on normal numbers." Bull. Amer. Math. Soc., (52):857-860, 1946.
• Schmidt, W. "On normal numbers." Pacific Journal of Mathematics, 10:661-672, 1960.
• Schnorr, C. P. and Stimm, H. "Endliche Automaten und Zufallsfolgen." Acta Informatica, 1:345-359, 1972.
• Sierpinski, W. "Démonstration élémentaire d'un théorème de M. Borel sur les nombres absolutment normaux et détermination effective d'un tel nombre." Bull. Soc. Math. France 45, 125-144, 1917.
• Wall. D. D. "Normal Numbers." Ph.D. thesis, University of California, Berkeley, California, USA, 1949.
• Ziv, J. and Lempel, A. "Compression of individual sequences via variable-rate coding." IEEE Transaction on Information Theory, 24:530-536, 1978.

 

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