Separating Complexity Classes Using Autoreducibility

Separating Complexity Classes using Autoreducibility z y Dieter van Melkeb eek Lance Fortnow Harry Buhrman The University of Chicago The University of Chicago CWI x Leen Torenvliet University of Amsterdam Abstract A set is autoreducible if it can b e reduced to itself by a Turing machine that do es not ask its own input to the oracle We use autoreducibility to separate the p olynomialtime hi erarchy from p olynomial space by showing that all Turingcomplete sets for certain levels of the exp onentialtime hierarchy are autoreducible but there exists some Turingcomplete set for doubly exp onential space that is not Although we already knew how to separate these classes using diagonalization our pro ofs separate classes solely by showing they have dierent structural prop erties thus applying Posts Program to complexity theory We feel such techniques may prove unknown separations in the future In particular if we could settle the question as to whether all Turingcomplete sets for doubly exp onential time are autoreducible we would separate either p olynomial time from p olynomial space and nondeterministic logarithmic space from nondeterministic p olynomial time or else the p olynomialtime hierarchy from exp onential time We also lo ok at the autoreducibility of complete sets under nonadaptive b ounded query probabilistic and nonuniform reductions We show how settling some of these autoreducibility questions will also lead to new complexity class separations Key words complexity classes completeness autoreducibility coherence AMS classication Q Q D Part of this research was done while visiting the Univ Politecnica de Catalunya in Barcelona Partially supp orted by the Dutch foundation for scientic research NWO through NFI Pro ject ALADDIN under contract NF and a TALENT stip end Email buhrmancwinl URL httpwwwcwinlcwipeopleHarryBuhrmanhtml y Supp orted in part by NSF grant CCR Research partly done while visiting CWI Email fortnowcsuchicagoedu URL httpwwwcsuchicagoedufortnow z Partially supp orted by the Europ ean Union through Marie Curie Research Training Grant ERBGT by the US National Science Foundation through grant CCR and by the Fields Institute Re search partly done while visiting CWI and the University of Amsterdam Email dietercsuchicagoedu URL httpwwwcsuchicagoedudieter x Partially supp orted by HCM grant ERBPL Email leenwinsuvanl URL httpturingwinsuvanlleen Intro duction While complexity theorists have made great strides in understanding the structure of complexity classes they have not yet found the prop er to ols to do nontrivial separation of complexity classes such as P and NP They have develop ed sophisticated diagonalization combinatorial and algebraic techniques but none of these ideas have yet proven very useful in the separation task Back in the early days of computability theory Post wanted to show that the set of noncom putable computably enumerable sets strictly contains the Turingcomplete computably enumerable sets In what we now call Posts Program see Post tried to show these classes dier by nding a prop erty that holds for all sets in the rst class but not for some set in the second We would like to resurrect Posts Program for separating classes in complexity theory In particular we will show how some classes dier by showing that their complete sets have dierent structure While we do not separate any classes not already separable by known diagonalization techniques we feel that renements to our techniques may yield some new separation results In this pap er we will concentrate on the prop erty known as autoreducibility A set A is autoreducible if we can decide whether an input x b elongs to A in p olynomialtime by making queries ab out memb ership of strings dierent from x to A Trakhtenbrot rst lo oked at autoreducibility in b oth the computability theory and space b ounded mo dels Ladner showed that there exist Turingcomplete computably enumerable sets that are not autoreducible Amb osSpies rst transferred the notion of autoreducibility to the p olynomialtime setting More recently Yao and Beigel and Feigenbaum have studied a probabilistic variant of autoreducibility known as coherence In this pap er we ask for what complexity classes do all the complete sets have the autore ducibility prop erty In particular we show EXP EXP All Turingcomplete sets for are autoreducible for any constant k where denotes k k P the sets that are exp onentialtime Turingreducible to k There exists a Turingcomplete set for doubly exp onential space that is not autoreducible EXP Since the union of all sets coincides with exp onentialtime hierarchy we obtain a separation k of the exp onentialtime hierarchy from doubly exp onential space and thus of the p olynomialtime hierarchy from exp onential space Although these results also follow from the space hierarchy theorems which we have known for a long time our pro of do es not directly use diagonalization rather separates the classes by showing that they have dierent structural prop erties Issues of relativization do not apply to this work b ecause of oracle access see A p olynomial time autoreduction can not view as much of the oracle as an exp onential or doubly exp onential computation To illustrate this p oint we show that there exists an oracle relative to which some complete set for exp onential time is not autoreducible Note that if we can settle whether the Turingcomplete sets for doubly exp onential time are all autoreducible one way or the other we will have a ma jor separation result If there ex ists a Turingcomplete set for doubly exp onential time that is not autoreducible then we get that the exp onentialtime hierarchy is strictly contained in doubly exp onential time thus that the p olynomialtime hierarchy is strictly contained in exp onential time If all of the Turingcomplete sets for doubly exp onential time are autoreducible we get that doubly exp onential time is strictly contained in doubly exp onential space and thus p olynomial time strictly in p olynomial space We will also show that this assumption implies a separation of nondeterministic logarithmic space from nondeterministic p olynomial time Similar implications hold for space b ounded classes see Section Autoreducibility questions ab out doubly exp onential time and exp onential space thus remain an exciting line of research We also study the nonadaptive variant of the problem Our main results scale down one exp onential as follows P P All truthtablecomplete sets are truthtableautoreducible for any constant k where k k P denotes that sets p olynomialtime Turingreducible to k There exists a truthtablecomplete set for exp onential space that is not truthtableautoreducible Again nding out whether all truthtablecomplete sets for intermediate classes namely p olynomial space and exp onential time are truthtableautoreducible would have ma jor implications In contrast to the ab ove results we exhibit the limitations of our approach For the restricted reducibility where we are only allowed to ask two nonadaptive queries all complete sets for EXP EXPSPACE EEXP EEXPSPACE etc are autoreducible We also argue that uniformity is crucial for our technique of separating complexity classes b e cause our nonautoreducibility results fail in the nonuniform setting Razb orov and Rudich show that if strong pseudorandom generators exist natural pro ofs cannot separate certain nonuniform complexity classes Since this pap er relies on uniformity in an essential way their result do es not apply Regarding the probabilistic variant of autoreducibility mentioned ab ove we can strengthen our results and construct a Turingcomplete set for doubly exp onential space that is not even probabilistically autoreducible We leave the analogue of this theorem in the nonadaptive setting op en Do es there exist a truthtable complete set for exp onential space that is not probabilistically truthtable autoreducible We do show that every truthtable complete set for exp onential time is probabilisticall y truthtable autoreducible So a p ositive answer to the op en question would establish that exp onential time is strictly contained in exp onential space A negative answer on the other hand would imply a separation of nondeterministic logarithmic space from nondeterministic p olynomial time Here is the outline of the pap er First we intro duce our notation and state some preliminaries in Section Next in Section we establish our negative autoreducibility results for the adaptive as well as the nonadaptive case Then we prove the p ositive results in Section where we also briey lo ok at the randomized and nonuniform settings Section discusses the separations that follow from our results and would follow from improvements on them Finally we conclude in Section and mention some p ossible directions for further research Errata in conference version An previous version of this pap er erroneously claimed pro ofs showing all Turing complete sets for EXPSPACE are autoreducible and all truthtable complete sets for PSPACE are nonadaptively autoreducible Combined with the additional results in this version we would have a separation of NL and NP see Section However the pro ofs in the earlier version failed to account for the growth of the running time when recursively computing previous players moves We use the pro of technique in Section though unfortunately we get considerably weaker theorems The original results claimed in the previous version remain imp ortant op en questions as resolving them either way will

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