On reductions to sets that avoid EXPSPACE y z x V Arvind J Kobler M Mundhenk Abstract Aset B is called EXPSPACEavoidingifevery subset of B in EXPSPACE is sparse Sparse sets and sets of high information density called HI G H sets in are shown to b e EXPSPACEavoiding Investigating the complexityof sets A in EXPSPACE that honestly reduce to EXPSPACEavoiding sets we show that if the reducibili ty used has a prop ertycalled rangeconstructibility then A must also reduce to a sparse set under the same reducibility Keywords Computational Complexity Reducibilities Sparse Sets Intro duction The study of reductions to low information content sets has received much attention in structural complexity theory research in recentyears There is a series of results showing that complexity classes containing intractable problems cannot b e reduced to sets of low information content unless there is an unlikely collapse of complexity classes The class of sparse sets is an example of a wellstudied class of low information content sets A researc h trend is to identify dierent classes of low information content sets and to study the consequences of the existence of hard sets of low information contentforintractable complexity classes under dierent kinds of reducibilities Recently Bo ok and Lutz intro duced and studied sets whose characteristic sequences are of very high spaceb ounded Kolmogorov complexity they call the class of such sets HI G H The existence of HI G H sets follows from the fact that RAN D the class of algorithmically random languages is of measure and is a sub class of HI G H In it is shown that every set in ESPACE that is p oly nomial time b ounded truthtable reducible to a set in HI G H is actually b ounded A preliminary version was presented at MFCS y Department of Computer Science Institute of Mathematical Sciences CIT Campus Madras India Work done at Universitat Ulm Supp orted in part by an Alexander von Humb oldt research fellowship z Abt Theoretische Informatik Universitat Ulm D Ulm Germany x FB IV Informatik Universitat Trier D Trier Germany Work supp orted in part by the DAAD through Acciones Integradas AIeeszk truthtable reducible to some sparse set it is shown in that this even holds for O log ntruthtable reducibilities The reason for considering the class ESPACE is that most intractable complexity classes of interest like NP PSPACE etc are contained in ESPACE and HI G H itself is dened using exp onential spaceb ounded Kolmogorov complexity Consequently if an NPcomplete set saySAT b ounded truthtable reduces to a set in HI G H then SAT b ounded truthtable reduces to a sparse set and by it follows that PNP This and similar consequences for other complexity classes eg PSPACE and PP are derived in In this pap er we address the following question are there further p olynomial time reducibilities such that every set in ESPACE that is reducible to a set in HI G H is actually reducible to some sparse set The answer weprovide to the ab ove question is based on the following obser vation which is easy to prove see Theorem for any A HIGHit holds that every subset of A in EXPSPACE is sparse This prop erty is captured bya class of sets larger than HI G H whichwe call EXPSPACEavoidingAsetA is voiding abbreviated as EAifevery subset of A in EXPSPACE is EXPSPACEa sparse Theorem rephrased states that every set in HI G H is in EA Next we identify a simple prop erty for reducibilities called rangeconstructibility Informally sp eaking whenever A honestly reduces to B via a rangeconstructible reducibi lity then Areduces to a subset C of B such that C can b e constructed byan ESPACE machine that uses A as oracle Weshow that several natural reducibili ties are rangeconstructible for example the manyone conjunctive and Hausdor reducibilities In Theorem weshow that every set in EXPSPACE which honestly reduces to a set in EA via a rangeconstructible reducibility in fact reduces to a sparse set From known collapse results for reductions to sparse sets under the considered rangeconstructible reducibilities it follows that the existence of hard sets in EA under honest rangeconstructible reductions for complexity classes like UP NP PP or PSPACE implies unlikely collapses of complexity classes It turns out that these collapse consequences hold for rangeconstructible reductions to sets in HI G H without the honesty assumption Preliminaries Letf g b e the standard alphab et and let A b e a set The length n n of a string x is denoted by jxj A A denotes the set of all strings in A of length n up to length n resp ectively denotes the characteristic function of A A A denotes the characteristic sequence of A for all strings up to length n n A n A ie j j and the ith bit of equals s where s is the ith A i i n n string in in lexicographic order The cardinalityof A is denoted by jAjThe n n census function of a set A is census jA jAsetS is called sparse if its A veby a p olynomial A set T is called a tally set if census function is b ounded ab o T We use T ALLY and SPARSE to denote the classes of tally and sparse O n sets resp ectivelyESPACE denotes the complexity class DSPACE whereas O n EXPSPACE DSPACE The deterministic and nondeterministic reducibilities discussed in this pap er are the standard p olynomialtime reducibilities dened in and the Hausdor reducibilityintro duced in p Denition A is Hausdor reducible to B in symbols A B hd if there exists a polynomialtime computable function f mapping every string x toasequence of queries such that for al l x iff xhy y i k then y B implies y B for al l i k and i i A maxfj j j k y B gfg is odd x j In this context the ith query y i k xcomputedbyf x is also i p denotedbyf x i We cal l f abounded Hausdor reduction A B if bhd the number k x of queries producedbyf on x is bounded byaconstant for al l x conp B A set A is conp manyone reducible to a set B denoted A m if there exists a polynomialtime nondeterministic Turing transducer M such that for every x x A if and only if al l outputs of M on input x are members of B corp cf A set A is corp manyone reducible to a set B denoted A B m if thereexistapolynomialtime computable function f and a polynomial q such that for al l x f x w B and x A Prob q jxj w x A Prob f x w B q jxj q jxj w q jxj Here the string w is chosen uniformly at random from the set corp corp p B B andA B implies A Note that for all sets A and B A m m c conp implies A B A reduction f of some typ e fromasetA to a set B is m called honest if there is a p olynomial p such that for every x and for every query jy j For any reducibilitytyp e y generated by f x it holds that jxjp Note that for a reduction f of any reducibilitytyp e discussed in this pap er it is meaningful to talk of queries generated by f on input x EXPSPACEavoiding sets Bo ok and Lutz measured the information contentofasetby the Kolmogorov complexity of its characteristic sequences Let M be a Turing machine z b e a string and let d s b e natural numbers Wesay that z KS d s if M on some input M of length at most d outputs z using space at most s In other words KS d s M is the set of strings whose sspacebounded Kolmogorov complexity relativetoM is b ounded by dWell known simulationtechniques see show that there is a Universal Turing machine U such that for every machine M there is a constant c c cs c Henceforth wex U as such such that for all d s KS d s KS d M U a Universal Turing machine and omit the subscript Note that there is a constant A n c such that for every set A and for all n the characteristic sequence of A is n n cn in KS c A set A is said to have maximum information content if for every c and almost every n A n cn KS n n As in HI G H is used to denote the class of sets of maximum information con tent Intuitively there is a large gap b etween the information contentofsetsin EXPSPACE and sets in HI G H We capture this prop erty in the following deni tion CEavoiding in short B is in EA if every Denition A set B is EXPSPA subset of B in EXPSPACE is sparse This can b e seen as a weaker form of EXPSPACEimmunitycf since only nonsparse subsets in EXPSPACE are forbidden in EXPSPACEavoiding sets We show next that only sparse subsets of a HI G H set can b e in EXPSPACE and therefore HI G H E A Since sparse sets are in EA and not in HI G H the next theorem implies that HI G H is a prop er subset of EA Theorem Every set A in HI G H is in EAMoreover even the padded version n pad Afx j x A n g of A is in EA Proof Assume that A is in HI G H but pad A is not in EA Then there exists a nonsparse subset B of pad A which is in EXPSPACE Since pad A is paddable it follows that there exists a nonsparse subset C of pad A whichisinESPACE k n For every nlet A b e the subset fx jk n jxjx C g of A and n n A let v b e the length jA j substring of obtained by deleting all s of n n n A which corresp ond to some x A Since C is nonsparse it follows that for all n n qn Then it is clear p olynomials q there exist innitely many n such that jA j n A O n that the following algorithm outputs and can b e implemented in space n denotes the empty string This and subsequent results concerning EXPSPACE in the pap er can b e easily extended to EXPSPACEpoly n input v n i n for x to in lexicographic order do if x A then output n else i i output the ith bit of v n n n A Since jv j jA j n it follows that there is a constant c such that n n n n cn KS q n for every p olynomial q and innitely