Boolean Hierarchies

Bo olean Hierarchies On Collapse Prop erties and Query Order Dissertation zur Erlangung des akademischen Grades do ctor rerum naturalium Dr rer nat vorgelegt dem Rat der Fakultat fur Mathematik und Informatik der FriedrichSchillerUniversitat Jena von DiplomMathematiker Harald Hemp el geb oren am August in Jena Gutachter Prof Dr Gerd Wechsung Prof Edith Hemaspaandra Prof Dr Klaus Wagner Tag des Rigorosums Tag der oentlichen Verteidigung To my family Acknowledgements Words can not express my deep gratitude to my advisor Professor Gerd Wechsung Gen erously he oered supp ort guidance and encouragement throughout the past four years Learning from him and working with him was and still is a pleasure and privilege I much ap preciate Through all the ups and downs of my research his optimism and humane warmth have made the downs less frustrating and the ups more encouraging I want to express my deep gratitude to Professor Lane Hemaspaandra and Professor Edith Hemaspaandra Allowing me to become part of so many joint pro jects has been a wonderful learning exp erience and I much b eneted from their scien tic exp ertise Their generous help and advice help ed me to gain insights into how research is done and made this thesis p ossible For serving as referees for this thesis I am grateful to Professor Edith Hemaspaandra and Professor Klaus Wagner Iwant to thank all my colleagues at Jena esp ecially HaikoMuller Dieter Kratsch Jorg Rothe Johannes Waldmann and Maren Hinrichs for generously oering help and supp ort A regarding the many little things that scientic work seems to require these days L T X E unix email etc Working in the theory group of Professor Gerd Wechsung has made the past years a wonderful time Iwant to thank myparents who brought ab out myinterest into mathematics For their encouragement and supp ort I am grateful to my familymy motherinlaw and most of all myloving wife Ines The research underlying this thesis has b een supp orted in part bygrant NSFINT DAADPROfoab and was in part done while visiting Le Moyne College Chapter and Section are based on jointwork with Edith Hemaspaandra and Lane Hemaspaandra HHHd and part of it app eared in journal as HHHe The research underlying Section is joint work with Edith Hemaspaandra and Lane Hemaspaandra HHHa The results of Section were discovered jointly with Lane Hemaspaandra and Gerd Wechsung and will app ear in journal as HHW Section is based on jointwork with Edith Hemaspaandra and Lane Hemaspaandra and has partially app eared in journal as HHHa Contents Intro duction The Bo olean and Polynomial Hierarchies Connection Downward Collapse Query Order Preliminaries Strings Languages and Op erators Turing Machines and Reductions Central Complexity Classes and Hierarchies The Polynomial Hierarchy The Bo olean Hierarchy The BoundedTruthTable and the BoundedQuery Hierarchies p A Renement of the Bo olean Hierarchyover k k Query Order Miscellaneous The Bo olean and Polynomial Hierarchies Connection Intro duction A Review Previous Results The Development of the EasyHard Technique A Close Lo ok at the Pro ofs A New Result A Deep er Induced Collapse Concluding Remarks Downward Collapse Intro duction A Review Previous Results The Development of the Pro of Metho d A Detailed Analysis of the Pro ofs viii CONTENTS A New Result The EasyHard Technique in Double Use Applications and Concluding Remarks Query Order Intro duction Query Order in the Bo olean Hierarchy The Base Case The General Case Remarks Query Order in the Polynomial Hierarchy Previous ResultsAn Overview The Missing Case Applications of Query Order and Related Results Base Classes Other than P Results Related to Query Order Bibliography List of Symb ols Sub ject Index List of Figures Example Graphs The Roman Semicircular Arch The Polynomial Hierarchy The Bo olean BoundedQuery and BoundedTruthTable Hierarchies p The Rened Bo olean Hierarchyover k k Inclusion Structure and Results Overview Inclusion Structure and Results Overview Query Tree Rened Query Tree Example Query Tree Chapter Intro duction Complexity theory studies the inherent computational complexity diculty of problems By problem we usually mean decision problems the question of whether a given ob ject b elongs to a certain set in other words has a certain prop erty or not But how do we measure diculty Quite naturally the time needed to come up with a solution is a reasonable and often used measure More precisely the time measure of a problem A is a function t satisfying that for each input x the answer to the question x A can be found in at most tjxj time units where jxj is the length of the representation for x In complexity theory time is measured in terms of steps needed by a Turing machine to solve the problem Turing machines rst dened by Turing Tur and Post Pos are a computational mo del that is simple yet powerful enough to capture the very notion of computability The time measure allows to classify problems Those b eing computable by a deterministic Turing machine with a time function that is b ounded by a p olynomial form the class P Edm The class P is a theoretical concept intended to capture the spirit of feasible computation The word feasible should b e viewed in a rather theoretical context Clearly problems having a time function that is a p olynomial of high degree say n are certainly not considered to be eciently solvable However p olynomials do not grow to o fast and p ossess a numb er of nice prop erties for instance they are closed under comp osition Furthermore the class P is a very robust notion not dep ending on the denitional variations of the underlying computational mo del In contrast problems having a time function that cn is b ounded by where c is some constant and n the length of the input are until one shows a b etter time b ound not considered to be feasibly computable The class E is the collection of all those problems It is known that there are problems in E that are not in P HS Examples of problems b ers The EULER TOUR in P are addition of natural numbers and sorting natural num problem emerging from Eulers famous Konigsb erger Bruc kenProblem Eul that given a graph asks whether it is p ossible to walk through the graph along its edges in sucha way that every edge is touched exactly once is another prominentmemb er of the class P A seemingly slight but crucial as we will see in a moment variation of this problem leads to the socalled HAMILTON CIRCUIT problem HC Given a graph one is asked Intro duction Figure The left graph contains both an Euler tour and a Hamilton Circuit whereas the right graph contains neither an Euler tour nor a Hamilton circuit to nd a way of moving through the graph along its edges such that starting from some vertex one returns to it while visiting every other vertex exactly once The b est known algorithms for HC need exp onential time HC E and consist essentially in testing for all permutations of the vertices of the input graph whether the vertices can be visited in the order given by the p ermutation It is not known whether HC P and p eople have tried to understand why this ques tion has resisted all solution eorts It turned out that a variant of deterministic Turing machines socalled nondeterministic Turing machines can solve HC in p olynomial time the amount of nondeterminism needed is growing exp onentially with the size of the input graph This gives rise to a new class of problems b etween P and E namely NP the class of problems that can b e solved by some nondeterministic p olynomialtime Turing machine It is known that HC is among the hardest problems in NP in the sense that HC Pwould immediately imply P NP In fact the question P NP is the most famous op en question in complexity theory having reformulations in many of its areas While b eing concerned with resolving the P NP question researchers quickly found out that there is a much richer structure of complexityclassesbetween P and E Variations to the acceptance mech anism of nondeterministic Turing machines and the notion of oracle Turing machines are a few examples for other computing paradigms that allow to exactly pinp ointmany naturally arising computation

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