There Are Fewer Facts Than Words: Communication With A Growing Complexity
Abstract
We present an impossibility result, called a theorem about facts and words, which pertains to a general communication system.
The theorem states that the number of distinct words used in a ﬁnite text is roughly greater than the number of independent elementary persistent facts described in the same text.
In particular, this theorem can be related to Zipf’s law, power-law scaling of mutual information, and powerlaw-tailed learning curves.
The assumptions of the theorem are: a ﬁnite alphabet, linear sequence of symbols, complexity that does not decrease in time, entropy rate that can be estimated, and ﬁniteness of the inverse complexity rate
Introduction
In several recent large-scale computational experiments in statistical language modeling, there were observed power-law tails of learning curves [Takahira et al., 2016, Hestness et al., 2017, Hahn and
Futrell, 2019, Braverman et al., 2020, Kaplan et al., 2020, Henighan et al., 2020, Hernandez et al., 2021, Tanaka-Ishii, 2021].
Namely, the difference between the cross entropy rate of the statistical language model and the entropy rate of natural language decays as a power law with respect to the amount of training data.
Equivalently, this is tantamount to a power-law growth of mutual information between increasing blocks of text—the ﬁrst observation thereof attributed to Hilberg [1990], see also [Crutchﬁeld and Feldman, 2003].
This power-law growth occurs for languages typologically as diverse as English, French, Russian, Chinese, Korean, and Japanese.
Moreover, we observe a universal language-independent value of the power-law exponent: the mutual information between two blocks of length n is proportional to n 0 .
[Takahira et al., 2016, Tanaka-Ishii, 2021].
In this paper, we would like to advertise some mathematical theory of this phenomenon, covering its potential causes and effects.
We have been developing this theory for several years.
Our results were resumed in the recently published book [D˛ebowski, 2021a] and the subsequent article [D˛ebowski, 2021b].
This paper supplies an updated brief overview for a venue of machine learning.
The novel thing is a simple generalization to non-stationary processes with a growing complexity.
The basic theory of power-law-tailed learning curves can be simply stated as furnishing the proof of a general statement of form:
The number of distinct words used in a ﬁnite text is roughly greater than the number of independent elementary persistent facts described in this text.
We call this sort of a statement a theorem about facts and words.
In fact, the theorems about facts and words come into a few distinct ﬂavors and can be proved relatively easily provided a certain attention is paid to the formal understanding of the concepts of a fact and of a word.
The theorems about facts and words can be regarded as an impossibility result that pertains to a general communication system.
Simply speaking, one cannot communicate about a certain amount of independent facts in a repetitive fashion without effectively using at least as many distinct words.
This result seems paradoxical since we might think that combining words we may express many more independent facts.
Moreover, the above result links information theory and discrete stochastic processes with linguistics, semantics, and cognition.
In principle, it applies to any kind of a communication system consisting of a ﬁnite number of discrete signs stringed into longer messages.
Besides natural language, some obvious examples are computer programs, DNA, and music.
In particular, the statements and the proofs of the theorems about facts and words combine:
• Zipf’s and Herdan-Heaps’ laws for word frequency distributions [Zipf, 1935, Mandelbrot, 1954, Guiraud, 1954, Herdan, 1964, Heaps, 1978], • universal coding based on grammars [de Marcken, 1996, Kieffer and Yang, 2000, Charikar et al., 2005] and on normalized maximum likelihood [Shtarkov, 1987, Ryabko, 1988, 2008], • consistent (hidden)
Markov order estimators [Merhav et al., 1989, Ziv and Merhav, 1992], • the concept of inﬁnite excess entropy [Hilberg, 1990, Ebeling and Nicolis, 1991, Ebeling and Pöschel, 1994, Bialek et al., 2001, Crutchﬁeld and Feldman, 2003], • the ergodic theorem and the ergodic decomposition [Birkhoff, 1932, Rokhlin, 1962, Gray and Davisson, 1974], and • Kolmogorov complexity and algorithmic randomness [Kolmogorov, 1965, Martin-Löf, 1966, Li and Vitányi, 2008].
As we can see, there are many interacting mathematical concepts.
There are also many open problems in the surrounding theory.
In the following, we present some particular version of a theorem about facts and words, which pertains to algorithmic randomness [D˛ebowski, 2021a] and consistent Markov order estimation [D˛ebowski, 2021b]. An informal discussion is relegated to Appendix A.
Preliminaries
Consider a string x kj :
=
( x j , x j +1 , ..., x k ) over a countable alphabet.
Its preﬁx-free Kolmogorov complexity is denoted K ( x kj )
[Li and Vitányi, 2008].
The algorithmic mutual information between strings u and v is J ( u, v ) :
= K ( u ) + K ( v ) − K ( u, v ) .
The expectation of random variable X is denoted E X .
The Shannon entropy of X is H ( X ) :
= E [ − log P ( X )]
[Cover and Thomas, 2006].
As in [D˛ebowski, 2021a, Deﬁnition 8.1], we will use the Hilberg exponent of a sequence, deﬁned as
[Equation]
so that hilb n →∞ n β = β for β ≥ 0 .
We recall that if limit s = lim n →∞ S ( n ) /n exists then [Equation] For a discrete one-sided stochastic process ( X i ) i ∈ N , we consider conditions:
(A) The complexity rate h
= lim n →∞ E K ( X n 1 ) /n exists and hilb n →∞ [ hn − H ( X n 1 )]
= 0 .
(B) The complexity does not decrease in time: E K ( X n 1 ) ≤ E K ( X 2 nn +1 ) . (C)
The inverse complexity rate is ﬁnite, H :=
lim sup n →∞ E n K ( X n 1 ) <
∞ (thus h > 0 for (A)).
(D)
The alphabet is ﬁnite: X i : Ω → { a 1 , a 2 , ..., a D } , where D ∈ N .
In particular, conditions (A) and (B) are satisﬁed by any stationary process that satisﬁes (D). For conditions (A) and (B), we also obtain that the power-law rate of redundancy is dominated by the power-law rate of mutual information,
[Equation]
Condition hilb n →∞ E J
( X n 1 ; X nn +1 ) > 0 is called the Hilberg condition, after Hilberg [1990].
Facts
The concept of independent elementary persistent facts can be most easily understood on the example of a certain stationary ergodic process over a countably inﬁnite alphabet called a Santa Fe process [D˛ebowski, 2011].
Let ( K i ) i ∈ N be an IID process in natural numbers with the Zipﬁan marginal distribution
[Equation]
where k ∈ N , α > 1 is a ﬁxed parameter, and ζ ( α ) :
= (cid:80) ∞
k
=1 k − α is the Riemann zeta function.
Distribution (4) is a formal model of Zipf’s law from quantitative linguistics [Zipf, 1935, Mandelbrot, 1954].
Moreover, let ( z k ) k ∈
N be an algorithmically random sequence, i.e., a sequence of particular (= ﬁxed) bits (= coin ﬂips) such that the Kolmogorov complexity of any string z k 1 is the highest possible,
K ( z k 1 ) ≥ k − c for a certain constant c < ∞ and all lengths k ∈ N
[Li and Vitányi, 2008].
Then the
Santa Fe process ( X i ) i ∈ N is a sequence of pairs
[Equation]
In the following, bits z k will be called facts.
The Santa Fe process can be understood as a model of an inﬁnite text that consists of random statements of form „the k -th fact equals z k ”.
Importantly, these statements are non-contradictory, namely, if statements X i and X j describe the same fact ( K i = K j ) then they assert the same value of this fact ( z K i =
z K j ).
Moreover, we observe that facts z k are in some sense independent (the
Kolmogorov complexity of their concatenation is the highest possible), elementary (they assume only two distinct values), and persistent (described faithfully at any time instant i ).
We will say that a ﬁnite text x n 1 describes exactly ﬁrst l facts of a ﬁxed sequence ( z k ) k ∈ N by means of a computable function g
if l = U g ( x n 1 ; z ∞ 1 ) − 1 , where
[Equation]
For a Santa Fe process we can easily construct a function g such that U g ( X n 1 ; z ∞ 1 )
=
min { k ∈ N : k (cid:54)∈ K n 1 } by reading off the values of z k for all k ∈ K n 1 .
Hence it can be proved that the expected number of initial facts described by a random text X n 1 grows as a power law
[Equation]
Power laws (4) and (7) are related to each other as Zipf’s law is related to Herdan-Heaps’ law [Guiraud, 1954, Herdan, 1964, Heaps, 1978].
Processes ( X i ) i ∈ N such that hilb n →∞ E
U g
( X n 1 ; z ∞ 1 ) >for a certain algorithmically random sequence ( z k ) k ∈ N and a certain computable function g are called perigraphic.
All perigraphic processes have incomputable probability distributions.
We notice that
Santa Fe processes are perigraphic.
These results can be linked to the redundancy rate.
As shown in [D˛ebowski, 2021a, Eqs. (8.117)– (8.119)], for any discrete stochastic process ( X i ) i ∈ N , any algorithmically random sequence ( z k ) k ∈ N , and any computable function g , we obtain inequality
[Equation]
where constant c g < ∞ depends on function g .
Hence, the power-law rate of the number of facts is dominated by the power-law rate of redundancy,
[Equation]
if condition (A) holds. The above inequality can be chained with inequality (3).
Words
In the remaining move, we will relate the algorithmic mutual information to the number of words used in a given text.
This can be done in several ways.
One way, pursued by D˛ebowski [2011], is to apply the minimal grammar-based coding [Kieffer and Yang, 2000, Charikar et al., 2005], which applies a mathematical concept of a word that resembles words in a linguistic sense [de Marcken, 1996] but this method is a bit lengthy to describe formally.
Therefore, here, we will apply a different approach which is based on consistent Markov order estimation [Merhav et al., 1989, Ziv and Merhav, 1992]—that approach was pursued by D˛ebowski [2021b].
There is a function called subword complexity that counts how many distinct substrings of a given length there are in a given string, namely,
[Equation]
Function V ( k | x n 1 ) will be a proxy for the number of distinct words in text x n 1 .
The only problem is to choose a motivated length k of the substrings.
In principle, this k may depend on string x n 1 .
Quite a natural choice of k is the estimator of the Markov order of the process deﬁned as
[Equation]
where K ( x n 1 ) is the Kolmogorov complexity and L k ( x n 1 ) is the maximum likelihood of order k ,
[Equation]
Function M ( x n 1 ) is a strongly consistent and asymptotically unbiased estimator of the Markov order.
Namely, for any stationary ergodic process ( X i ) i ∈ N over a ﬁnite alphabet we have
[Equation]
surely
[Equation]
where we denote the Markov order of the process as
[Equation]
Let us write succinctly the number of substrings of the supposedly optimal length as
[Equation]
Then, using the universal code by Ryabko [1988], we can prove inequality
[Equation]
where D is the cardinality of the alphabet and c i < ∞ are certain small constants [D˛ebowski, 2020, Theorems 11–12].
Applying Hilberg exponents and expectations, we obtain that the power-law rate of mutual information is dominated by the power-law rate of the number of words,
[Equation]
[Equation]
if conditions (C) and (D) are satisﬁed.
The above inequality can be chained with inequalities (3) and (9).
The asymptotic power law hilb n →∞ E V ( X n 1 ) > 0 resembles Herdan-Heaps’ law for words in the linguistic sense [Guiraud, 1954, Herdan, 1964, Heaps, 1978].
Conclusion
Chaining inequalities (3), (9), and (17) under conditions (A)–(D), we obtain the sandwich bound
[Equation]
[Equation]
which yields a formal statement of a certain theorem about facts and words.
In particular, we can infer from inequalities (18) that no perigraphic process can be a Markov process, i.e., a process with a ﬁnite Markov order, M < ∞ .
These two classes of processes are disjoint.
Of course, the above inequalities rest on a repeated use of Kolmogorov complexity.
Therefore they are ineffective in some sense.
We note that there are other statements of theorems about facts and words that apply effective notions.
However, they are more complicated to formulate. See also
Appendix A in this paper for some informal discussion of our formal assumptions. The theorems about facts and words, as an impossibility result, raise questions about their applicability to empirical data as well as questions about further examples of perigraphic processes that are more complex or more realistic than Santa Fe processes.
Some of these questions were addressed or stated as open problems in book [D˛ebowski, 2021a] and article [D˛ebowski, 2021b]. A An informal discussion of the adopted mathematical model A certain problem about the presented theorems about facts and words is that they appear quite abstract.
This abstraction is an advantage from a mathematical point of view since it allows to apply the same result to very different systems—for example to discuss the internal complexity of communication in the ﬁeld of mathematics.
However, any abstraction also begs for some concrete examples of applications that would explain the intuitions standing behind the adopted formal models.
As a mathematician, we prefered to present the hard results in the main matter of this paper whereas we relegate the intuitions to the present appendix—written upon the request of the reviewers.
First of all one should be aware that theorems about facts and words are an asymptotic result—for the aesthetic virtue of applying the power law rate exponents, called succinctly Hilberg exponents.
The discussed inequalities for Hilberg exponents stem, however, from two non-asymptotic inequalities (8) and (16).
Thus, if we could somehow make an informed (and necessarily fallible) guess about the
Kolmogorov complexity of a particular text in natural language or another communication system then we might expect a sort of a theorem about facts and words also for ﬁnite amounts of empirical data.
Assumptions such as perfect stationarity or operations like the expectation were applied to get rid of some relatively small deviations from aesthetically appealing general trends.
Another important remark, theorems about facts and words are just an impossibility result:
In a certain precise sense, there are always roughly fewer facts described than words used.
For Santa Fe processes, these two quantities are of a similar order.
For other processes or communication systems, the number of facts can be signiﬁcantly smaller than the number of words.
If it were so for natural language, it would be extremely interesting.
In such a case, language communication would be much less complex than suggested by the mere power-law growth of the lexicon.
This hypothesis is so counterintuitive that it asks for a further consideration.
From this point of view, it is advisable to bridge the adopted formal notions of a fact and a word with a more usual understanding of these concepts.
Let us begin with the concept of a fact.
As noted by one reviewer, the representational content of a fact is irrelevant for our theorem.
This makes the theorem both powerful from a mathematical point of view and quite difﬁcult to digest by empirical researchers.
However, asking what the facts are in their essence is not the most fortunate question to answer.
It is more appropriate to ask what the distinctive features of facts are in our model, namely, how they behave.
Taking Santa Fe processes as a working mathematical model of facts, there are three basic properties of our facts.
First, they are assumed to be binary variables, like bits, coin ﬂips, or spins.
Second, they are independent, either probabilistically or algorithmically, namely, their Kolmogorov complexity is the highest possible.
Third, they are persistent, recurrent, or eternal, in the sense that the same fact is repeatedly described inﬁnitely many times by the information source that generates the text.
The question whether such eternal and algorithmically random facts can be actually described at a power law rate by the totality of human culture is intellectually challenging but important.
It concerns not only the approximate or real randomness of cultural conventions but also their origins in the physical sources of noise.
The question cannot be honestly answered unless we know exactly how much randomness there is in the physical world.
Fortunately, in the spirit of algorithmic information theory, we may ultimately falsify some class of wrong answers—given enough computation time.
Moreover, it may be helpful to imagine our sequence of facts as a sort of an unknown real parameter of an information source.
When we observe or learn this source, we learn particular binary digits or facts of this parameter at a rate that is speciﬁc for the information source.
For Santa Fe processes, this rate is given by the power law (7).
For Bernoulli processes or ﬁnite-state sources, the same number of described facts grows only logarithmically, so the respective Hilberg exponent is zero.
By contrast, for natural language, we cannot be sure how large it is.
However—by the theorems about facts and words—we already know that the number of independent facts is upper bounded by the estimates of mutual information or by the number of distinct words, which grow roughly as n 0 .
8 , where n is the length of the text [Takahira et al., 2016, Tanaka-Ishii, 2021].
In this way, we proceed smoothly to discuss the concept of a word.
From the point of view of formal linguistics, the only rigorous deﬁnition of a word that makes sense is by enumeration of the lexicon.
This can be successful only if the lexicon is ﬁnite or given by a ﬁnite formal grammar.
Nevertheless, this deﬁnition cannot fully accommodate creativity and openness of the lexicon and of other language or culture conventions.
Thus in order to count words, we need a certain operational deﬁnition of what we want to count in general.
One such way is to count strings of letters that appear empirically and are delimited by separators such as spaces or pauses.
However, when we count such strings in a stream of probabilistically independent letters then we obtain a spurious power law [Mandelbrot, 1954, Miller, 1957].
We have a general intuition that a memoryless mechanism cannot be responsible for the power-law distribution of words in natural language.
Thus we had better made words operational in a different way.
For example, we may use phrases deﬁned by universal coding.
Since there are many different universal codes of varying encoding rates for ﬁnite data, it is not clear which one we should use.
Quite an interesting idea seems to apply grammar-based codes.
These codes represent a text as a speciﬁc context-free grammar that produces the text as the only production.
If we heuristically minimize the length of such a grammar then we obtain non-terminals that usually span across whole words [de Marcken, 1996].
That is, such minimal non-terminals can be considered a proxy for orthographic words.
There is also a theorem about facts and words that applies to the number of minimal nonterminals [D˛ebowski, 2011].
The problem is that this theorem applies the global minimization of the grammar length, which is likely intractable [Charikar et al., 2005].
For this reason, we searched for yet another approach.
In the main matter of this paper, we presented an operational proxy for words as overlapping strings of the constant length equal to the Markov order estimate computed for the considered text.
This approach is also quite intuitive.
Its advantage is that it does not lead to a spurious discovery of an unbounded vocabulary in Markovian sources.
However, the Markov order estimate is usually smaller than the average length of a word for moderately sized texts.
Thus the number of Markovian substrings is only an imperfect proxy for the number of words.
Thus there remains an open problem of ﬁnding an operational deﬁnition of a word that could be applied to any sort of symbolic data and would possess the following properties:
The number of distinct operational words should be theoretically lower bounded by the block mutual information—and hence by the number of facts.
(Thus, a rich semantics implies a rich vocabulary.)
The number of distinct operational words should be theoretically upper bounded for a ﬁnite Markov order of the process.
(Thus, a meager semantics implies a meager vocabulary.)
Parsing of the input text into operational words should be efﬁciently computable in a time close to linear in the length of the text.
The operational words should be similar in shape and in number to the orthographic words for natural language data.
We hope that such a satisfactory operational deﬁnition of a word exists.
Concluding this appendix, there seem to be a few somewhat different formal statements that fall under the umbrella of theorems about facts and words.
In our work, we tried to identify a few of them but the topic has not been exhausted, in our opinion.