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The Structure of Logarithmic Advice Complexity Classes

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The Structure of Logarithmic Advice Complexity Classes The Structure of Logarithmic Advice Complexity Classes Jose L Balcazar Department of Software LSI Universitat Politecnicade Catalunya Pau Gargallo E Barcelona Spain balquilsiupces Montserrat Hermo Department of Software LSI Universidad del Pas Vasco PO Box E San Sebastian Spain jiphehumsiehues May Abstract A nonuniform class called here FullPlog due to Ko is studied It corresp onds to p olynomial time with logarithmically long advice Its imp ortance lies in the struc tural prop erties it enjoys more interesting than those of the alternative class Plog sp ecically its introduction was motivated by the need of a logarithmic advice class closed under p olynomialtime deterministic reductions Several characterizations of FullPlog are shown formulated in terms of various sorts of tally sets with very small information content A study of its inner structure is presented by considering the most usual reducibilitie s and lo oking for the relationships among the corresp onding reduction and equivalence classes dened from these sp ecial tally sets Partially supp orted by the EU through the ESPRIT Long Term Research Pro ject ALCOM IT and through the HCM Network CHRXCT COLORET by the Spanish DGICYT through pro ject PB KOALA and by Acciones Integradas HispanoAlemanas HAB and ALB Introduction Nonuniform complexity classes were essentially introduced in where the main rela tionships to uniform classes were already shown In order to capture characteristics of nonuniform mo dels of computation in which xed input lengths are compulsory in the denition of nonuniform classes a b ounded amount of extra information the advice dep endent of the length of the input is provided The two most natural families of b ound functions for the advice information are p olynomials as in the class Pp oly and loga rithms as in the class Plog In structural terms the class Pp oly has a high interest due to its many characterizations and has b een studied in considerable depth However Plog not b eing closed under the most usual reducibilities its structural study has less interest We consider in this pap er a variant prop osed by KerI Ko in and also treated marginally in and of the complexity class Plog in which closure under p olynomial time reductions is obtained via a more restrictive condition in the denition Our aim is to argue here that this relative of Plog that we call FullPlog not only is closed under reducibilities but also has an internal structure worth study comparable to that of Pp oly with many similarities and a few interesting dierences This internal structure has also consequences for the learnability of certain circuit ex pression classes of logarithmic Kolmogorov complexity The reason is that FullPlog char acterizes the concepts that can b e describ ed by these representation classes similarly to the characterization of Pp oly by p olynomial size circuits This asp ect of FullPlog is treated in depth elsewhere We must mention in passing another motivation to study this class In a mo del of neural networks is describ ed and the language recognition p ower of these networks is characterized in terms of the types of numbers employed as weights Also a precise corresp ondence is establish b etween the choice of integer rational or real weights and the resp ective classes of languages When the computation time of the networks is constrained to b e p olynomial in the input size the classes recognized by the resp ective nets are regular P and Pp oly It may b e argued that any net with real weights that is computationally feasible to implement must admit a short description of its realvalued weights Thus setting a logarithmic b ound on the resourceb ounded Kolmogorov complexity of the real weights it is proved in that the languages recognized corresp ond to the class FullPlog This pap er is structured in two dierent parts The rst one shows several characteriza tions of FullPlog as a reduction class The pro ofs are quite interesting in that they require to dene another technical variant of the class based on prexclosed advice words and to introduce a new pro of technique by which information is selected at a doubly exp onential rate skipping all information corresp onding to intermediate advice words This kind of argument has b een heavily used in As applications of this technique we obtain sev eral characterizations of FullPlog as the reduction class of sp ecial sets tally sets whose words follow a given regular pattern and tally sets that are regular in a resourceb ounded Kolmogorov complexity sense The second part fo cuses on the inner structure of FullPlog trying to understand the p ower of this sort of sp ecial tally sets when they are used as oracles To do that we explain some of the relationships which exist among reductions classes dened from them In addition we describ e how equivalence classes dened in terms of the same tally sets are related The results of sections to app eared in preliminary form in The results of section app eared in preliminary form in Almost all the results describ ed here are from Preliminaries An alphab et is any nonempty nite set We use here the alphab ets f g and fg Given any alphab et a nite string or word over is a nite sequence of symbols from We denote words by lower case Latin letters such as x y Any set nite of innite of these strings is called a language The language of all p ossible nite strings over is denoted by There is a sp ecial string in that is the unique word consisting of zero symbols We call it Given a set A we indicate the cardinality of A with the expression jAj In the same way when we refer to the number of symbols in some string x we use jxj Regarding words x means the substring formed by x from the ith symbol on up ij to and including the j th symbol As our strings are usually sequences over f g sometimes we sp eak ab out the j th bit of x instead of the j th symbol A natural op erator the concatenation is dened among the strings of Given two words x y the concatenation of x with y pro duces the new word xy A prex of a word y is any word z such that for some word w y z w The notation z v y denotes the fact that z is a prex of y Let A b e a language over some alphab et The set of all strings x in A whose length is n less than or equal to n jxj n is denoted by A When we refer to the strings of A that n have exactly length n we use the notation A We express that the set A is included in the set B by writing A B but when the inclusion is strict we say that A B The complement of a language A is denoted by A when is known we omit it leaving just A Given any with at least two symbols say and the join or marked union of two languages A and B over is A B fx j x Ag fx j x B g Denition A language is tal ly if and only if it is included in fg The class of all tally sets is denoted by Tally The upp er case initial is used to express the complete class while the lower case initial means the prop erty of b elonging to Tally Next let us dene the concept of characteristic sequence of a particular set Denition Given a language A A Its characteristic sequence is an innite string of f g such that for all n A the nth bit of is if the nth nite string of ordered by lengths and in lexicographical order within each length b elongs to A Otherwise this bit is n A Its characteristic sequence up to a particular length n is a nite string fullling n n A n A A that j j and v A When the set A is a tally set is the characteristic sequence relative to fg that n means we only take into account the words for any n In the same way slightly abusing n the notation when we use sets L such that L f j n INg the characteristic sequence L will b e relative to the language formed by a p ower of many s More generally arbitrary innite strings are denoted by Greek letters as Throughout the pap er all logarithms are to base Dierent Types of Reductions Our uniform computational mo del is the multitape Turing machine with a readonly input tap e An oracle Turing machine has an additional writeonly oracle tap e The machines can b e deterministic DTM or nondeterministic NDTM We work with classes of languages recognized by oracle Turing machines that work in p olynomial time Since the access to the oracle can b e restricted in dierent ways the following reducibilities can b e obtained Denition A language A is r reducible to B in p olynomial time A P B if and r only if there exists an r restricted deterministic p olynomialtime oracle Turing machine M such that A is the language recognized by M querying oracle B Now we explain the meaning of r for a xed reduction When r is the Turing reduction r T there are no restrictions on the oracle machine M When r is the truthtable reduction r tt the machine M is able to write down on a separate tap e all the queries to b e made during the computation b efore the rst word is queried A particular case of the truthtable reduction is the k truthtable reduction written r k tt when k Now there exist p olynomialtime computable functions f and h such that for all x f x is a list of k strings f x hx x x i hx is a truth k table with k variables and x A if and only if the truthtable hx evaluates to true on the b o olean k tuple hx B x B x B i k Set A is b ounded truthtable reducible r btt to set B if there is a p ositive k such that A is k tt reducible to B When r is the manyone reduction r m it is only allowed to make a unique query to the oracle in the last step
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