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Contributions to the Study of Resource-Bounded Measure

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Contributions to the Study of Resource-Bounded Measure Contributions to the Study of ResourceBounded Measure Elvira Mayordomo Camara Barcelona abril de Contributions to the Study of ResourceBounded Measure Tesis do ctoral presentada en el Departament de Llenguatges i Sistemes Informatics de la Universitat Politecnica de Catalunya para optar al grado de Do ctora en Ciencias Matematicas por Elvira Mayordomo Camara Dirigida p or el do ctor Jose Luis Balcazar Navarro Barcelona abril de This dissertation was presented on June st in the Departament de Llenguatges i Sistemes Informatics Universitat Politecnica de Catalunya The committee was formed by the following members Prof Ronald V Bo ok University of California at Santa Barbara Prof Ricard Gavalda Universitat Politecnica de Catalunya Prof Mario Ro drguez Universidad Complutense de Madrid Prof Marta Sanz Universitat Central de Barcelona Prof Jacob o Toran Universitat Politecnica de Catalunya Acknowledgements Iwant to thank sp ecially two p eople Jose Luis Balcazar and Jack Lutz Without their help and encouragement this dissertation would never have b een written Jose Luis Balcazar was my advisor and taughtmemostofwhatIknow ab out Structural Complexity showing me with his example how to do research I visited Jack Lutz in Iowa State University in several o ccasions from His patience and enthusiasm in explaining to me the intuition behind his theory and his numerous ideas were the starting p ointofmostoftheresults in this dissertation Several coauthors have collab orated in the publications included in the dierent chapters Jack Lutz Steven Fenner Ronald Bo ok Jose Luis Balcazar Monsterrat Hermo and Harry Buhrman In addition a number of other p eople have contributed with dierent ideas among others Peter Miltersen Eric Allender and Martin Strauss Thanks to all the p eople that help ed me during my visits to Iowa State University the University of Ulm the University of California at Santa Barbara and the University of Aarhus I sp ecially appreciate the hospitalityof Robyn Lutz and Celia Wrathall The Spanish Ministerio de Educacion y Ciencia supp orted this work through grant FPI PN I received partial supp ort from Accion Integrada HA by the ESPRIT EC pro ject ALCOM and by National Science Foundation Grant CCR The p eople in the Department de Llenguatges i Sistemes Informatics have given me their friendship and help through these years In the section of Theoretical Computer Science I found lots of interesting discussions and p eople that listened to my often still incoherent ideas FinallyI thank my family for b eing there And Juanjo for his love and supp ort Contents Intro duction and preliminaries Intro duction Main contributions Preliminaries Resourceb ounded measure Some technical lemmas measurability and the Kolmogorov law Measuring in PSPACE Intro duction Measure in PSPACE additivity in PSPACE Measure versus category the Pbiimmune sets Intro duction Pbiimmunity and resourceb ounded measure Pbiimmunity and resourceb ounded category Measure of nonuniform complexity classes Intro duction Advice complexity classes Weaksto chasticity c Measure of P DENSE n tt Pp oly inside the Exp onential Hierarchy If NP is not small Intro duction If NP do es not have pmeasure Separating completeness notions in NP Separating reducibilities in NP Further results and op en problems Cones Intro duction Weaklyuseful languages On the robustness of ALMOSTR Bidimensional measure References Chapter Intro duction and preliminaries Intro duction The notion of eective pro cedure or algorithm was b orn in the early thirties building on the work of Church Godel Kleene Post and Turing Chur Chur Gode Klee Post Turi Turi They develop ed several formalizations of this concept such as calculus partial recursive functions and the Turing Machine formalism This was the base of Recursive Function Theory where recursive problems were dened as those that are solvable by an algorithm The construction of actual computers led to the consideration of feasibly solvable prob lems instead of recursive or theoretically solvable ones This distinction was related to the explosive growth of the exp onential function which implies that algorithms based on exhaustive searchmay b e infeasible in practice Therefore an increasing attention was paid in the sixties to the amount of computational resources used in the solution of a recursive problem Sp ecically the resources considered were mainly time and space With the work of Hartmanis Stearns and Lewis HartSt LewiStH SteaHaL Com plexity Theory started a division of recursive problems into complexity classes according to the amount of resources used in their resolution The computing mo del used here was the Turing Machine which corresp onds to a simple mathematical representation of a computer see Hop cUl for a complete description The problems that can be solved in time p olynomial in the length of the input are considered feasibly solvable and form the class denoted P But there exist many problems for which no p olynomial time algorithm is known many imp ortant ones among them have the prop erty of b eing easy to check that is once a solution is found it can be checked in p olynomial time that it is indeed a solution This leads to the denition of the class NP as the class of easytocheck prob lems there are many imp ortant problems in this class for instance those dealing with the satisfability of b o olean formulas or with the existence of a hamiltonian path in a graph and some practical op eration research problems such as the distribution of crews into planes It would be very interesting to know whether P and NP coincide In the seventies some techniques analogous to those in Recursive Function Theory for instance the concept of completeness Co ok Karp started to be develop ed and then used to attack the P NP problem This constitutes the b eginning of the eld of Structural Complexity versus which we develop next Structural Complexity describ es complexity classes using various typ es of resources in cluding time space nondeterministic time and space Bo olean circuit size and depth and alternating time and space We will not dene here all the mentioned resources let us just say that the word nondeterministic refers to the use of nondeterministic algorithms that INTRODUCTION are a generalization of the usual algorithms with the extra p ossibility of cho osing among several instructions that follow a given one and that the word alternating is indirectly related to the use of parallel algorithms in which several instructions can be run at the same time The problems considered are mainly decisional ones which are denoted as languages and we say that an algorithm recognizes a language when it solves the cor resp onding decisional problem The main op en problems in Structural Complexity have the form Is the class of languages that can be recognized with an amount f of resource included in the class of those recognized with an amount g of resource The ab ove mentioned P versus NP problem can b e formulated as Is the class of languages that can b e recognized with nondeterministic p olynomial time included in the class of the p olynomial time recognizable languages Other examples involve comparisons of p olynomial time P versus p olylogarithmic parallel time with p olynomial size hardware NC exp onential time E versus p olynomial size circuits Pp oly and p olynomial space PSPACE versus p olynomial time P The notions of oracle Turing Machine reduction and complete language are intro duced in order to compare the complexity of sp ecic languages An oracle Turing Machine is an ordinary Turing Machine equipp ed with direct access to a particular language A which is called oracle The oracle Turing Machine op erates as an ordinary one with the extra p ossibility of given a string q computing in a single step the answer to q A For each oracle A we can dene complexity classes according to the resources used to recognize a language when we can access oracle A This means that for each oracle A we have a particular computation universe where the solution of A is given for free A great eort was done to nd out which answers to op en problems of the form C D hold when translated into some such universe trying to get some light on the solution of the op en problem see BakeGiS for the rst work in this line Given a problem such as C D we say that it is nonrelativizable when there exist oracles A and B such that C D using oracle A and C D using oracle B that is the solution of the problem is dierent in the contexts of oracles A and B A nonrelativizable problem is considered dicult because most of the techniques used in Structural Complexity are indep endent of the oracle used Just a few new results have shown that nonrelativizable problems are not imp ossible to solve for instance Shamir has shown in Sham that PSPACE IP while there exist oracles for which the opp osite holds FortSi If we can recognize a language A with an oracle B this means that B is at least as hard to recognize as A since an algorithm for B would pro duce an algorithm for A This denes a partial preorder of languages denoted and called Turing reducibility with the meaning T that A B if A can be recognized using B as oracle Polynomial time reducibilities T app ear when considering
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