Time-Bounded Kolmogorov Complexity May Help in Search for Extra Terrestrial Intelligence (SETI)

One of the main strategies in Search for Extra Terrestrial Intelligence SETI is trying to overhear communications between advanced civilizations However there is a seeming problem with this ap proach advanced civilizations most probably save communication expenses by maximally compressing their messages and the notion of a maximally compressed message is naturally formalized as a message x for which Kolmogorov complexity C x is close to its length l x i e as a random message In other words a maximally compressed message is indistinguishable from the truly random noise and thus trying to detect such a message does not seem to be a working SETI strategy We show that this argument does not take into consideration the time necessary for compression and decompression of message If we take this time into consideration and therefore consider time bounded versions of Kolmogorov complexity then the above problem disappears We also show which version of time bounded Kolmogorov complexity is most appropriate for formalizing SETI strategies According to modern physics extra terrestrial intelligence is very probable According to modern physics our Universe is largely homogeneous hence in many locations it is possible to have the same physical conditions as the ones that led to the emergence of intelligent life on Earth It is therefore reasonable to conclude that in many of these places intelligent life did emerge A contact with an extra terrestrial intelligence could be highly bene cial for our civilization hence SETI The processes which lead to the emergence and development of intelligent life are far from being deterministic there are many random factors involved As a result the rates with which di erent civilizations appear and progress are expected to be drastically di erent Hence we can expect both civi lizations which are largely behind us in technological and scienti c progress as well as civilizations which are way ahead of ours A contact with such an advanced civilization can bring new knowledge help us solve our problems and give a big boost to our civilization In view of this potential bene t a lot of e ort goes into Search for Extra Terrestrial Intelligence SETI see e g and references therein A seeming problem with the current SETI strategy If there are advanced civilizations out there they most probably know about each other and therefore intensely communicate with each other In view of this communication one of the main current SETI strategies consists of watching the radio signals in the sky and trying to determine which of them correspond to communications between advanced civilizations There is however a seeming problem in this approach Of course it is possible to distinguish between a meaningful signal and a random noise which typically comes from di erent celestial objects like galaxies quasars etc However it would be a waste of energy to send a pure signal and saving energy is extremely important for long distance inter civilization communications which require a lot of energy The signal will be most probably sent compressed The more advanced the civilization the more it knows how to compress its signals and the more perfect this compression will be As a result if we try to overhear the conversation between di erent civilizations what we will see is a signal x which cannot be further compressed This un compressibility admits a natural formalization Since most modern measuring devices have binary output we can assume that the signal x is a binary sequence i e a word in a binary alphabet f g By a compression intuitively we mean a pair consisting of a compressed signal c and an instruction i for some universal computer which enables us to reconstruct the original signal x from its compression c In other words a compression is a program c i for a universal computer which generates x Therefore the idea that the word x cannot be further compressed can be formalized as follows for some universal computer U every program p which generates x has the same length l p as x i e l p l x or the length which is larger than the length of x i e l p l x The shortest length of a program which generates x is called its Kolmogorov complexity C x In terms of Kolmogorov complexity the idea that the word x cannot be further compressed can be thus formalized as C x l x This approximate equality however is exactly the idea of formalizing of the intuitive notion of a random sequence x Thus we come to a conclusion that signals exchanged by advanced civilizations must be random i e such signals are indistinguishable from random noise This indistinguishability seems to invalidate the main current SETI strategy In this paper we show how this problem can be overcome How to resolve this problem enter time bounded Kolmogorov complexity The above negative conclusion is correct if we deal with chit chat However civilizations communicate not only to simply exchange news one of the main reasons for their communication is that they want to help each other solve problems by relaying known solutions This is true for an exchange between a less advanced and a more advanced civilization this is exactly what we hope for when we spend e orts on SETI This is also true for an exchange between equally advanced civilizations these civilizations will encounter many similar problems so all of them will bene t if instead of duplicating their e orts in each civilization solving all these problems they will divide these problems between themselves solve their share of the problems and exchange the solutions This only makes sense for problems for which the solution is di cult to get but easy to check and for which the solution has a reasonable length In other words we are talking about the problems of the following type given a word w nd a new word s whose length l s is bounded by some reasonable function of the length l w of the input w and for which some easily checkable property P w s holds Most problems solved by engineering and science are of this type e g in mathematics given a formulation w of a statement we want to nd a proof s of either this statement w or of its negation w checking whether a given step by step proof is correct is easy computers could do it even in the s in engineering we formulate the requirements w for e g a bridge and we want to nd a design s which satis es all these requirements in physics we have data w and we want to nd a simple law s which describes all this data etc If we follow the standard formalizations of reasonable function of length as polynomial function of length and easily checkable as checkable in polynomial time we conclude that a general problem is a problem from the class NP or to be more precise search problem in the sense of L A Levin For such problems maximal compression for which the length of a compressed message is equal to its Kolmogorov complexity does not make much sense Indeed we want to send a solution s to the problem w However we can always compress this solution into the formulation w because for problems from the class NP there is a simple algorithm for nding a solution exhaustive search So if we follow the idea of sending the maximally compressed message then we could as well send the original problem w instead of its solution and thus not send the solution at all The main idea of sending solutions is to save time on solving the corresponding problem and there is no sense in wasting all the saved time on de compressing the message When we formalized the incompressibil ity in terms of Kolmogorov complexity we neglected computation time with this time saving application in mind we must take this time into consideration e g consider a time bounded version of Kolmogorov complexity instead of the the original one In other words instead of considering C x min l p where min is taken over all programs generating x we must consider C x min f t p l p where t p is the running time of a program p and f t l is a function of two variables The less time and or memory compression decompression requires the better Therefore we should require that f t l be a non decreasing function of t and l The choice of an appropriate time bounded version of Kolmogorov complexity Which function f t l should we choose We talk about transmitting solutions s to problems w from the class NP The decompression time t p takes the smallest possible value if we simply send the uncompressed solution s Realistically the only processing time t p for the received signal is the time which is necessary to check that s is indeed a solution i e that the desired property P w s is true Alternatively we can send all the bits of s but one then to reconstruct the desired solution we must check both possible completions of the sent signal completion by and completion by Then the length of the sent signal decreased by one bit l p l p while the processing time increases twice t p t p Similarly we can skip bit from any message at the cost of increasing the processing time twice This skipping a single bit should not change the fact that what we are considering is an intelligent message therefore it is reasonable to require that the function f t l used in our distinction between a message and a random noise be invariant under such transformation In other words we require that the desired function f t l satisfy for every two integers t and l the equation f t l f t l This equation has also been obtained in where it appeared in a slightly di erent situation in these papers it is shown that a function f t l satis es this equation if and only if f t l F t l for some function F z of one variable Since we required that the function f t l is monotonically non decreasing in both variables we can conclude that the function F z is non decreasing too So looking for the best compression means looking for a program p for which the product t p l p takes the smallest possible value The corresponding modi ed Kolmogorov complexity C x min t p l p where min is taken over al