Incompressibility method

The incompressibility method is a proof method such as the probabilistic method, the counting method, or the pigeonhole principle. The method proceeds as follows: In order to prove that an object in a certain class on average satisfies a certain property, select an object of that class that is incompressible. Subsequently it is shown that if it does not satisfy the property then it can be compressed by clever computable coding. Since in general it can be proved that almost all objects of a given class are incompressible, the argument shows that almost all objects in the class have the property involved (and not just the average). To select an incompressible object is not effective: it cannot be done by a computer program. But a simple counting argument usually shows that almost all objects of a given class can be compressed by but a few bits (are incompressible).

History

The application of the incompressibity method depended on an objective and fixed notion of incompressibility. Such a notion was provided by the Kolmogorov complexity theory named for Andrey Kolmogorov^[1]

The Kolmogorov complexity of an object (represented by a finite binary string) is the length of a shortest binary program on a fixed optimal universal Turing machine (optimality here has a technical meaning). Since the machine is fixed and the program concerned is shortest, the Kolmogorov complexity is a definite positive integer. The Kolmogorov complexity of an object is thus the length of a shortest binary program from which it can be computed. Therefore, it is a lower bound on the length of a computably compressed version (in bits) of that object by any existing or future compression program.

One of the first uses of the incompressibility method with Kolmogorov complexity the theory of computation was in ^[2] proving that the running time of a 1-tape Turing machine is quadratic for accepting a palindromic language and that sorting algorithms require at least $n\log n$ time for sorting $n$ items. The initially most influential paper using the incompressibility method was in 1980.^[3] Hereafter the method was applied in many fields and codified and acquiered its name in the textbook.^[4]

Applications

Number theory

An elegant proof of Euclid shows that there are infinitely many prime numbers. Eventually Riemann showed that the number of primes less than a given number is connected with the 0s of the Riemann zeta function. Hadamard and de la Vallee Poussin proved in the 19th century (1896) that this number of primes is asymptotic to ${\displaystyle n/\ln n}$. (Use $\ln$ for the natural logarithm an $\log$ for the binary logarithm.) Using the incompressibility method G.J. Chaitin (attributed) argued as follows. For each $n$ we can describe it by giving its prime factorization ${\displaystyle n=p_{1}^{n_{1}}\cdots p_{k}^{n_{k}}}$ (which is unique) where ${\displaystyle p_{1},\ldots ,p_{k}}$ are the first $k$ primes which are at most $n$ and the exponents are possibly 0. Each exponent is at most $\log n$ and can be described by ${\displaystyle \log \log n}$ bits. Therefore, the description of $n$ can be given in ${\displaystyle k\log \log n}$ bita provided we know the value of $\log \log n$ enabling one to parse the consecutive blocks of exponents. To describe $\log \log n$requires only ${\displaystyle \log \log \log n}$ bits. Using the incompressibility of most positive integers: for each $k>0$ there is a positive integer $n$ of binary length ${\displaystyle l\approx \log n}$ that cannot be described in less than $l$ bits. Together this shows that the number of primes $\pi (n)$ less than $n$ satisfies

${\displaystyle \pi (n)\geq {\frac {\log n}{\log \log n}}-o(1).}$

A more sophisticated approach attributed to Piotr Berman (present proof partially by John Tromp) describes every incompressible $n$ by $k$ and $n/p_k$, where $p_{k}$ is the largest prime number dividing $n$. Since $n$ is incompressible the length of this description must exceed $\log n$. To be able to parse the first block of the description $k$ must be given in prefix form ${\displaystyle P(k)=\log k+\log \log k+\log \varepsilon (k)}$ where ${\displaystyle \varepsilon (k)}$ is an arbitrary small positive function. Therefore, ${\displaystyle \log p_{k}\leq P(k)}$. Hence ${\displaystyle p_{k}\leq n_{k}}$ with ${\displaystyle n_{k}=\varepsilon (k)k\log k}$ for a special sequence of values ${\displaystyle n_{1},n_{2},\ldots }$.This shows that the expression below holds for this special sequence and a simple extension shows that it holds for every ${\displaystyle n>0}$:

${\displaystyle \pi (n)\geq {\frac {n}{\varepsilon (n)\log n}}.}$

Both proofs are given in more detail in.^[4]

Graph theory

A labeled graph $G=(V,E)$ with $n$ nodes can be represented by a string $E(G)$ of ${n \choose 2}$ bits where each bit indicates the presence or absence of an edge between the pair of nodes in that position. Let ${\displaystyle K(G)\geq {n \choose 2}}$.. Then the vertex degree $d$ of each vertex satisfies

${\displaystyle |d-n/2|=O\left({\sqrt {n\log n}}\right).}$

To prove this by the incompressibity method.one shows that if the deviation is larger than we can compress the description of $G$ below ${\displaystyle K(G)}$. This gives the required contradiction. This theorem is required in a more complicated proof where the incompressibility argument is used many times to show that th number of unlabeled graphs is

${\displaystyle \sim {\frac {2^{n \choose 2}}{n!}},}$

see.^[5]

Combinatorics

A transitive tournament is a complete directed graph, $G=(V,E)$, such that if ${\displaystyle (i,j),(j,k)\in E}$ then also ${\displaystyle (i,k)\in E}$. Consider the set of all transitive tournaments on $n$ nodes. Since a tournament is a labeled directed complete graph it can be encoded by a string $E(G)$ of ${n \choose 2}$ bits where each bit indicates the direction of the edge between the pair of nodes in that position. Using this encoding it is shown that every transitive tournament contains a transitive subtournament on at least $v(n)$ vertices with

${\displaystyle v(n)\leq 1+\lfloor 2\log n\rfloor .}$

This was shown (as the first problem) in.^[6] It is easily solved by the incompressibility method in,^[7] as are the coin-weighing problemn, the number of covering families, and expected properties (for example at least a fraction of $1-1/n$ of all transitive tournaments on $n$ vertices have transitive subtournaments on not more than ${\displaystyle 1+2\lceil 2\log n\rceil }$ vertices for $n$ is large enough).

Lovász local lemma. If a number of events are independent (in the sense of Probability theory) of one another then the probability that none of the events occur can be nonzero and is easily calculated. If the events are dependent then the problem gets hard. Lovasz local lemma.^[8] is a celebrated principle which tells us that if the events are mostly independent of one another and have individually small probability then there is positive probability that none of them occur. A constructive proof appeared in.^[9] It was shown to be a proof by the incompressibility method in.^[10]

Expanders. Using the incompressibility method, various versions of expanders and superconcentrator graphs were shown to exist using the incompressibility method.^[11] It was shown that the best known bounds on the size of expanders and superconcentrators can be attained based on this method

Topological combinatorics

Heilbronn triangle problem. Throw $n$ points in the unit square and determine the maximum of the minimal area of a triangle formed by three of those points over all possible arrangements. This problem was solved for small arrangements and a lot of work was done on the asymptotic expression as a function of $n$. The original conjecture of Heilbronn is ${\displaystyle O(1/n^{2})}$ in the early 1950s. Paul Erdos proved that this bound is correct for $n$ is a prime number. The general problem remains unsolved apart from the best known lower bound ${\displaystyle \Omega ((\log n)/n^{2})}$ (achievable and hence Heilbronn's conjecture is not correct for general $n$) and upper bound ${\displaystyle \exp(c{\sqrt {\log n}})/n^{8/7}}$ proven by Komlos, Pintsz & Szemeredi in 1982 and 1981, respectively. Using the incompressibility method the average case was studied. In^[12] it was proved that if the area is too small it can be compressed to below the Kolmogorov complexity of a uniformly random arrangement (high Kolmogorov complexity) and if it is too large as well. This proves that for the overwhelming majority of the arrangements (and the expectation) the area of the smallest triangle formed by three of $n$ points thrown uniformly at random in the unit square is ${\displaystyle \Theta (1/n^{3})}$. In this case the incompressibility method proves both lower bound and upper bound of the property involved.

Probability

The Law of the iterated logarithm, the Strong Law of Large Numbers and the recurrence property, were shown to hold using the incompressibility method,^[13] Kolmogorov zero-one law in,^[14] and Normal numbers expressed as binary strings in the sense of E. Borel; more in general the distribution of 0s and 1s in binary strings of high Kolmogorov complexity in.^[15]

Turing machines time complexity

The basic Turing machine as conceived by Alan Turing in 1936 consists of a memory that is a tape of cells in which a symbol can be written, potentially infinite, and a finite control with a read-write head attached which scans a cell on the tape. At each step the read-write head can change the symbol in the cell under scan, move one cell left, right, or not at all, according to instruction from the finite control. For convenience consider Turing machines with two tape symbols (but this is not essential).

In 1968 F.C. Hennie showed that such a Turing machine requires order $n^{2}$ to recognize the language of binary palindromes in the worst-case. In 1977 W. J. Paul^[2] gave an incompreessibility proof which showed that order $n^{2}$ time is required in the average-case. Namely, for every integer $n$ consider all words of that length. For convenience we consider only words with the middle third of the word consisting of 0's. Moreover, the accepting Turing machine ends with an accept state on the left (the beginning of the tape). A computation of a Turing Machine on a given word gives for each location (boundary between adjacent cells) a sequence of crossings from left-to-right and from right-to-left, each crossing in a particular state of the finite control. Consider positions in the middle third of a candidate word. Either they all have a crossing sequence of length $O(n)$ in which case the total computation time is $O(n^{2})$, or some position has a crossing sequence of $o(n)$. In the last case the word, if it is a palindrome, can be identified by that crossing sequence. Namely if other palindromes (ending in an accepting state on the left) have the same crossing sequence then the word considering of a prefix (up to the position of the involved crossing sequence) of the original palindome concatenated with a suffix (of the remaining length) of the other palindrome would be accepted as well. Taking the palindrome of $\Omega (n)$ Kolmogorov complexity we have just described it by $o(n)$ bits: contradiction. Since the overwhelming majority of binary palindromes have this high Kolmogorov complexity this gives a lower bound on the average-case running time. The result in^[3] is much more difficult and shows that Turing machines with $k+1$ work tapes are more powerful than those with $k$ work tapes in real-time (here: one symbol per step).

In 1984 W. Maass^[16] and M. Li and P.M.B. Vitanyi ^[17] showed that the simulation of two work tapes by one work tape (of a Turing machine) takes $\Theta (n^{2})$ time deterministically (this is optimal and solved a 30-year open problem) and ${\displaystyle \Omega (n^{2}/(\log n\log \log n))}$ time nondeterministically ^[17] (in ^[16] this is ${\displaystyle \Omega (n^{2}/(\log ^{2}n\log \log n))}$. In ^[17] there are more results concerning tapes, stacks, and queues both deterministically and nondeterministically.

Many more results in this area were proven using the incompressibiity method.^[4]

Theory of computation

Heapsort is a sorting method invented by J.W. J. Williams and improved by R. W. Floyd. This method always runs in $O(n\log n)$ time. The question to decide is whether Floyd's method is better than Williams method on average (it is better in the worst-case). Using the incompressibility method it was shown^[4] that Williams method runs on average in ${\displaystyle 2n\log n+O(n)}$ time and Floyd's method runs on average in ${\displaystyle n\log n+O(n)}$ time. The proof was suggested by Ian Munro.

Shellsort, discovered by Donald Shell in 1959, is a comparison sort which splits the list to be sorted in scattered sublists and sorts these separately. Subsequently the sorted sublists are merged to reconstitute a partially sorted list. This process repeats a number of times: the number of passes. The difficulty of analyzing the complexity of the sorting process is that it depends on the number $n$ of keys to be sorted, on the number $p$ of passes, but also on the increments governing the scattering in each pass. That is, the sublist is the list of keys that are the increment parameter apart. Although this sorting method gave rise to a large number of papers only the worst-case was established. For the average-case running time only the best case for 2-pass Shellsort was established,^[18] and an upper bound on the best case for 3-pass Shellsort.^[19] A general lower bound on the average-case of $p$-pass Shelllsort was established in ^[20] making a first advance on this problem in four decades, The idea is as follows. In every pass the comparison sort moves a key to another place a certain distance: a path length. Code all these path lengths logarithmically in their length in the correct order (of passes and keys). This allows to reconstruct the unsorted list from the sorted list. Now let the unsorted list be incompressible (or almost so). Since the sorted list has almost zero Kolmogorov complexity, and the path lengths together give a certain code length, the sum must be at least as large as the Kolmogorov complexity of the original list. The sum of the path lengths correspond to the running time. It turns out that the running time is lower bounded by this argument by ${\displaystyle \Omega (pn^{1+1/p})}$.

Assume ${\displaystyle n,r,s}$ are natural numbers and ${\displaystyle 2\log n\leq r,s\leq n/4}$. It was shown that for every $n$ there is a boolean $n\times n$ matrix such that every ${\displaystyle s\times (n-r)}$ submatrix has rank at least $n/2$ by the incompressibility method.

Logic

In Gödel's incompleteness theorems the first one states that in every formal system with computably enumerable theorems/proofs that is strong enough to contain Peano Arithmetic there are true but unprovable statements (theorems). This is proved as follows by the incompressibility method. Every formal system $F$ as above can be described finitely say in $f$ bits. In such a formal system we can express ${\displaystyle K(x)\geq |x|}$ since it contains arithmetic. Given $F$ we can search exhaustively for a proof that some string $y$ of length ${\displaystyle n\gg f}$ satisfies ${\displaystyle K(y)\geq n}$. In this way we obtain the first such string effectively. Therefore, ${\displaystyle K(y)\leq \log n+f}$: contradiction.^[21] (We have ignored some lower order logarithmic terms which do not matter anyway.).

Comparison with other methods

While the probabilistic method generally shows the existence of an object with a certain property in a class, the incompressibility method tends to show that the overwhelming majority of objects in the class (and hence the average or the expectation) has that property. It is sometimes easy to turn a probabilistic proof in an incompressibility proof or vice versa. In some cases it seems hard or impossible to turn a proof by incompresibility in a probabilistic or counting proof. In virtually all cases of Turing machine time complexity cited above, the incompressibility method solved problems which were open for decades. No other proofs are known. Sometimes a proof by incompressibility can be turned into a proof by counting, as happened in the case of the general lower bound on the running time of Shellsort.^[20] Since this problem was open for almost half a century, and well-known, this shows that thinking about coding as in the incompressibility method can be easier than thinking about probability or counting.

References

1. A. N. Kolmogorov, Three approaches to the definition of the concept "quantity of information", Probl. Peredachi Inf., 1:1 (1965), 3–11
2. W.J. Paul, Kolmogorov's complexity and lower bounds, pp 325-333 in: L. Budach Ed., Proc. 2nd Int. Conf. Fund. Comput. Theory, 1979.
3. W.J. Paul, J.I. Seiferas, J. Simon, An information-theoretic approach to time bounds for on-line computation (preliminary version),Proc. 12th ACM Symp. Theory Comput (STOC), 357–367, 1980.
4. M. Li, P.M.B. Vitanyi, An Introduction to Kolmogorov Ccomplexity and Its Applications, Springer, 1993, 1997, 2008, Chapter 6
5. H.M. Buhrman, M. Li, J.T. Tromp, P.M.B. Vitanyi, Kolmogorov random graphs and the incompressibility method, SIAM J. Comput., 29:2(1999), 590–599
6. P. Erdos, J. Spencer, Probabilistic methods in combinatorics, Academic Press, 1974
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8. P. Erdos, L. Lovasz, Problems and results on 3-chromatic hypergraphs and some related questions,in A. Hajnal, R. Rado, and V. T. Sós, eds. Infinite and Finite Sets (to Paul Erdős on his 60th birthday). North-Holland. pp. 609–627
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10. L. Fortnow, A Kolmogorov Complexity Proof of the Lovász Local Lemma, Computational Complexity Weblog, 2 June 2009
11. U. Schoning, Construction of expanders and superconcentrators using Kolmogorov complexity, Random Structures & Algorithms, 17:1(2000), 64–77
12. T. Jiang, M. Li, P.M.B. Vitanyi, The average‐case area of Heilbronn‐type triangles, Random Structures & Algorithms, 20:2(2002), 206–219
13. V.G. Vovk, The law of the iterated logarithm for random Kolmogorov, or chaotic, sequences, Theory Probab. Appl. 3:32(1988), 413–426.
14. M. Zimand, A high-low Kolmogorov complexity law equivalent to the 0–1 law, Inform. Process. Letters, 57:2(1996), 59–84
15. M. Li, P.M.B. Vitanyi, Statistical properties of finite sequences with high Kolmogorov complexity, Mathematical Systems Theory, 27(1994), 365–376
16. W. Maass, Combinatorial lower bound arguments for deterministic and nondeterministic Turing machines, Trans. Amer. Math. Soc. 292 (1985), 675–693
17. M.Li, P.M.B. Vitanyi, Tape versus queue and stacks: The lower bounds, Information and Computation, 78:1(1988), 56–85
18. D.E. Knuth, Sorting and Searching (Vol. 3 The Art of Computer Programming), 2nd Ed. Addison-Wesley, 1998, pp 83–95
19. S. Janson, D.E. Knuth, Shellsort with three increments, Random Structures Algorithms 10:1–2(1997), 125–142
20. T. Jiang, M. Li, P.M.B. Vitanyi, A lower bound on the average-case complexity of Shellsort, Journal of the ACM (JACM), 47:5(2000) 905–911
21. G.J. Chaitin, Algorithmic Information Theory, Cambridge University Press, 1977