### Wit and Wisdom from Kolmogorov

I just learned about a result of Kolmogorov from the '50s that ought to interest TCS fans. Consider circuits operating on real-variable inputs, built from the following gates:

-sum gates, of unbounded fanin;

-arbitrary continuous functions of a single variable.

How expressive are such circuits? Kolmogorov informs us that they can compute any continuous function on n variables. Wow! In fact, one can achieve this with poly(n)-sized, constant-depth formulas alternating between sums and univariate continuous functions (two layers of each type, with the final output being a sum).

Note that sums can also be computed in a tree of fanin-two sums, so this theorem (whose proof I've not yet seen) tells us that fanin-two continuous operations capture continuous functions in the same way that fanin-two Boolean operations capture Boolean computation (actually, even more strongly in light of the poly-size aspect).

This result (strengthening earlier forms found by Kolmogorov and by his student Arnol'd) may, or may not, negatively answer one of Hilbert's problems, the thirteenth--it's not entirely clear even to the experts what Hilbert had intended to ask. But a fun source to learn about this material is a survey/memoir written by A. G. Vitushkin, a former student of Kolmogorov. [a gated document, unfortunately...]

"Kolmogorov was fluent in French and German and read English well, but his spoken English was not very good. Shannon, with some sympathy, expressed his regret that they could not understand each other well. Kolmogorov replied that there were five international languages, he could speak three of them, and, if his interlocutor were also able to speak three languages, then they would have no problems."

-sum gates, of unbounded fanin;

-arbitrary continuous functions of a single variable.

How expressive are such circuits? Kolmogorov informs us that they can compute any continuous function on n variables. Wow! In fact, one can achieve this with poly(n)-sized, constant-depth formulas alternating between sums and univariate continuous functions (two layers of each type, with the final output being a sum).

Note that sums can also be computed in a tree of fanin-two sums, so this theorem (whose proof I've not yet seen) tells us that fanin-two continuous operations capture continuous functions in the same way that fanin-two Boolean operations capture Boolean computation (actually, even more strongly in light of the poly-size aspect).

This result (strengthening earlier forms found by Kolmogorov and by his student Arnol'd) may, or may not, negatively answer one of Hilbert's problems, the thirteenth--it's not entirely clear even to the experts what Hilbert had intended to ask. But a fun source to learn about this material is a survey/memoir written by A. G. Vitushkin, a former student of Kolmogorov. [a gated document, unfortunately...]

The freewheeling article starts off with Hilbert's problem, but also contains interesting anecdotes about Kolmogorov and the academic scene he presided over in Moscow. Towards the end we get some amusing partisan invective against Claude Shannon, who, scandalously, "entirely stopped his research at an early stage and still kept his position of professor at the Massachusetts Institute of Technology". Vitushkin (who passed away in 2004) apparently bore a grudge over the insufficient recognition of Soviet contributions to information theory. My favorite story relates a not-so-successful visit by Shannon to meet Kolmogorov and features a deft mathematical put-down:

"Kolmogorov was fluent in French and German and read English well, but his spoken English was not very good. Shannon, with some sympathy, expressed his regret that they could not understand each other well. Kolmogorov replied that there were five international languages, he could speak three of them, and, if his interlocutor were also able to speak three languages, then they would have no problems."

Labels: general math