Kolmogorov Complexity Need for Approximate . . . I-Complexity Good Properties of I- . . . I-Complexity and Discrete Towards Precise . . . Derivative of Logarithms: Our Result Proof A Group-Theoretic Acknowledgments References Explanation Home Page Title Page Vladik Kreinovich and Jaime Nava ◭◭ ◮◮ Department of Computer Science ◭ ◮ University of Texas at El Paso 500 W. University Page 1 of 12 El Paso, TX 79968, USA Emails: vladik@utep.edu, jenava@miners.utep.edu Go Back Full Screen Close Quit
Kolmogorov Complexity Need for Approximate . . . 1. Kolmogorov Complexity I-Complexity • The best way to describe the complexity of a given Good Properties of I- . . . string s is to find its Kolmogorov complexity K ( s ). Towards Precise . . . Our Result • K ( s ) is the shortest length of a program that com- Proof putes s . Acknowledgments • For example, a sequence is random if and only if its References Kolmogorov complexity is close to its length. Home Page • We can check how close are two DNA sequences s and Title Page s ′ by comparing K ( ss ′ ) with K ( s ) + K ( s ′ ): ◭◭ ◮◮ – if they are unrelated , the only way to generate ss ′ ◭ ◮ is to generate s and then generate s ′ , so Page 2 of 12 K ( ss ′ ) ≈ K ( s ) + K ( s ′ ); Go Back – if they are related , we have K ( ss ′ ) ≪ K ( s )+ K ( s ′ ). Full Screen Close Quit
Kolmogorov Complexity Need for Approximate . . . 2. Need for Approximate Complexity I-Complexity • The big problem is that the Kolmogorov complexity is, Good Properties of I- . . . in general, not algorithmically computable . Towards Precise . . . Our Result • Thus, it is desirable to come up with computable ap- Proof proximations. Acknowledgments • At present, most algorithms for approximating K ( s ): References – use some loss-less compression technique to com- Home Page press s , and Title Page – take the length � K ( s ) of the compression as the de- ◭◭ ◮◮ sired approximation. ◭ ◮ • However, this approximation has limitations: for ex- ample, Page 3 of 12 – in contrast to K ( s ), where a change (one-bit) change Go Back in x cannot change K ( s ) much, Full Screen – a small change in s can lead to a drastic change Close in � K ( s ). Quit
Kolmogorov Complexity Need for Approximate . . . 3. I-Complexity I-Complexity • Limitation of � K ( s ): a small change in s = ( s 1 s 2 . . . s n ) Good Properties of I- . . . can lead to a drastic change in � K ( s ). Towards Precise . . . Our Result • To overcome this limitation, V. Becher and P. A. Heiber Proof proposed the following new notion of I-complexity . Acknowledgments • For each position i , we find the length B s [ i ] of the References largest repeated substring within s 1 . . . s i . Home Page • For example, for aaaab , the corresponding values of Title Page B s ( i ) are 01233. ◭◭ ◮◮ � n def • We then define I ( s ) = f ( B s [ i ]), for an appropriate ◭ ◮ i =1 decreasing function f ( x ). Page 4 of 12 • Specifically, it turned out that the discrete derivative Go Back of the logarithm works well: f ( x ) = dlog( x + 1), where Full Screen def dlog( x ) = log( x + 1) − log( x ) . Close Quit
Kolmogorov Complexity Need for Approximate . . . 4. Good Properties of I-Complexity I-Complexity � n Good Properties of I- . . . • Reminder: I ( s ) = f ( B s [ i ]), where: i =1 Towards Precise . . . • B s [ i ] is the length of the largest repeated substring Our Result within s 1 . . . s i , and Proof Acknowledgments • f ( x ) = log( x + 1) − log( x ) . References • Similarly to K ( s ): Home Page • If s starts s ′ , then I ( s ) ≤ I ( s ′ ). Title Page • We have I (0 s ) ≈ I ( s ) and I (1 s ) ≈ I ( s ). ◭◭ ◮◮ • We have I ( ss ′ ) ≤ I ( s ) + I ( s ′ ). ◭ ◮ • Most strings have high I-complexity. Page 5 of 12 • In contrast to K ( s ) : I-complexity can be computed in Go Back linear time. Full Screen • A natural question : why this function f ( x )? Close Quit
Kolmogorov Complexity Need for Approximate . . . 5. Towards Precise Formulation of the Problem I-Complexity • We view the desired function f ( x ) as a discrete ana- Good Properties of I- . . . logue of an appropriate continuous function F ( x ): Towards Precise . . . � x +1 Our Result f ( x ) = g ( y ) dy = F ( x + 1) − F ( x ) . Proof x Acknowledgments • Which function F ( x ) should we choose? References Home Page • In the continuous case, the numerical value of each quantity depends: Title Page ◭◭ ◮◮ – on the choice of the measuring unit and ◭ ◮ – on the choice of the starting point. • By changing them, we get a new value x ′ = a · x + b . Page 6 of 12 Go Back • For length x , the starting point 0 is fixed. Full Screen • So, we only have re-scaling x → x ′ = a · x . Close Quit
Kolmogorov Complexity Need for Approximate . . . 6. Our Result I-Complexity • By changing a measuring unit, we get x ′ = a · x . Good Properties of I- . . . Towards Precise . . . • When we thus re-scale x , the value y = F ( x ) changes, to y ′ = F ( a · x ). Our Result Proof • It is reasonable to require that the value y ′ represent Acknowledgments the same quantity. References • So, we require that y ′ differs from y by a similar re- Home Page scaling: Title Page y ′ = F ( a · x ) = A ( a ) · F ( x )+ B ( a ) for some A ( a ) and B ( a ) . ◭◭ ◮◮ • It turns out that all monotonic solutions of this equa- ◭ ◮ tion are linearly equivalent to log( x ) or to x α , i.e.: Page 7 of 12 a · x α + � a · ln( x ) + � F ( x ) = � b or F ( x ) = � b. Go Back • So, symmetries do explain the selection of the function Full Screen F ( x ) for I-complexity. Close Quit
Kolmogorov Complexity Need for Approximate . . . 7. Proof I-Complexity • Reminder: for some monotonic function F ( x ), for ev- Good Properties of I- . . . ery a , there exist values A ( a ) and B ( a ) for which Towards Precise . . . Our Result F ( a · x ) = A ( a ) · F ( x ) + B ( a ) . Proof • Known fact: every monotonic function is almost every- Acknowledgments where differentiable. References Home Page • Let x 0 > 0 be a point where the function F ( x ) is dif- Title Page ferentiable. ◭◭ ◮◮ • Then, for every x , by taking a = x/x 0 , we conclude that F ( x ) is differentiable at this point x as well. ◭ ◮ Page 8 of 12 • For any x 1 � = x 2 , we have F ( a · x 1 ) = A ( a ) · F ( x 1 )+ B ( a ) and F ( a · x 2 ) = A ( a ) · F ( x 2 ) + B ( a ). Go Back • We get a system of two linear equations with two un- Full Screen knowns A ( a ) and B ( a ). Close Quit
Kolmogorov Complexity Need for Approximate . . . 8. Proof (cont-d) I-Complexity • We get a system of two linear equations with two un- Good Properties of I- . . . knowns A ( a ) and B ( a ): Towards Precise . . . Our Result F ( a · x 1 ) = A ( a ) · F ( x 1 ) + B ( a ) . Proof F ( a · x 2 ) = A ( a ) · F ( x 2 ) + B ( a ) . Acknowledgments • Thus, both A ( a ) and B ( a ) are linear combinations of References differentiable functions F ( a · x 1 ) and F ( a · x 2 ). Home Page • Hence, both functions A ( a ) and B ( a ) are differentiable. Title Page • So, F ( a · x ) = A ( a ) · F ( x ) + B ( a ) for differentiable ◭◭ ◮◮ functions F ( x ), A ( a ), and B ( a ). ◭ ◮ • Differentiating both sides by a , we get Page 9 of 12 x · F ′ ( a · x ) = A ′ ( a ) · F ( x ) + B ′ ( a ) . Go Back • In particular, for a = 1, we get x · dF Full Screen dx = A · F + B , Close def def = A ′ (1) and B = B ′ (1). where A Quit
Kolmogorov Complexity Need for Approximate . . . 9. Proof (final part) I-Complexity • Reminder: x · dF Good Properties of I- . . . dx = A · F + B . Towards Precise . . . A · F + b = dx dF Our Result • So, x ; now, we can integrate both sides. Proof • When A = 0 : we get F ( x ) Acknowledgments = ln( x ) + C , so b References F ( x ) = b · ln( x ) + b · C. Home Page Title Page d � = F + b F = dx def • When A � = 0 : for � F A , we get x , so A · � ◭◭ ◮◮ F 1 A · ln( � F ( x )) = ln( x )+ C , and ln( � F ( x )) = A · ln( x )+ A · C . ◭ ◮ Page 10 of 12 def • Thus, � F ( x ) = C 1 · x A , where C 1 = exp( A · C ). Go Back F ( x ) − b A = C 1 · x A − b • Hence, F ( x ) = � A . Full Screen • The theorem is proven. Close Quit
Kolmogorov Complexity Need for Approximate . . . 10. Acknowledgments I-Complexity This work was supported in part: Good Properties of I- . . . Towards Precise . . . • by the National Science Foundation grants HRD-0734825 Our Result and DUE-0926721, and Proof • by Grant 1 T36 GM078000-01 from the National Insti- Acknowledgments tutes of Health. References Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 12 Go Back Full Screen Close Quit
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