Algebraic Closures in Complexity Theory Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Bayerischen Julius-Maximilians-Universit¨at Wurzburg¨ vorgelegt von Elmar B¨ohler aus Kempten Wurzburg,¨ 2005 Eingereicht am: 26.9.2005 bei der Fakult¨at fur¨ Mathematik und Informatik 1. Gutachter: Prof. Dr. Klaus W. Wagner 2. Gutachter: Prof. Dr. Heribert Vollmer Tag der mundlic¨ hen Prufung:¨ 16.12.2005 3 Auf keinen Fall w¨are es mir m¨oglich gewesen diese Arbeit ohne die viele Hilfe anzufer- tigen, die freundliche Menschen mir zuteil werden ließen. Insbesondere gilt mein Dank meinem Doktorvater Klaus Wagner, der seinen Mitarbeitern nicht nur das ideale Forschung- sumfeld zu schaffen versteht, sondern mit seinen vielen guten Ideen dafur¨ sorgt, dass die Gruppe stets an spannenden neuen Problemen arbeiten kann. Ebenso gilt mein Dank Heribert Vollmer der mich immer auch in die Arbeit Hannoveraner Gruppe mit einge- bunden hat, wodurch mir eine gr¨oßere Breite in diesem (hoffentlich noch nicht letzten) Lernabschnitt meines Lebens gegeben wurde. Insofern hatte ich das Gluc¨ k eigentlich von zwei Doktorv¨atern betreut worden zu sein. Mein Dank gilt weiterhin Nadia Creignou mit der ich zusammenarbeiten durfte und die mich mehrfach in Marseille aufgenommen hat. Auch danke ich Edith Hemaspaandra die so freundlich war, mich an sch¨onen Arbeiten zu beteiligen. Besonders wichtig war mir immer die Diskussion mit meinen Kollegen Christian Glaßer, Daniel Meister, Steffen Reith und Stephen Travers. Besonderer Dank im Zusammenhang mit dieser Arbeit gebuhrt¨ Steffen, der sie teilweise, und Daniel, der sie ernsthafterweise ganz Korrektur gelesen hat! Schließlich m¨ochte ich mich bei all jenen bedanken, die mir mein Studium und meine Promotion m¨oglich gemacht haben. Insbesondere meine ich hier meine Tante Helga Wolf, die mich finanziell unterstutzt¨ hat; ohne sie w¨are mein Studium nicht m¨oglich gewesen. 4 It would not have been possible to finish this thesis without the help of many friendly people. I have to thank my supervisor Klaus Wagner especially, since he knows how to create a research-friendly environment and because his fruitful ideas lead the department to ever new and interesting problems. Also, I thank Heribert Vollmer who took care that I was integrated in the work of the group in Hannover, too, thus giving breadth to this learning epsisode of my live (which is, hopefully, not the last one). In this respect, I was lucky to have actually two supervisors for my doctorate. Furthermore, I want to thank Nadia Creignou, who I was allowed to work with and who hosted me several times in Marseille. I thank Edith Hemaspaandra because she let me work with her on beautiful papers. The discussion with my collegues Christian Glaßer, Daniel Meister, Steffen Reith and Stephen Travers was always of special importance to me. In connection with this thesis, I want to thank Steffen and Daniel particularly, since the former proofread half of the script and the latter seriously proofread the whole thing! Finally, I want to thank all those who enabled me to study and to graduate. I am especially grateful to my aunt Helga Wolf, who supported me financially; I would not have been able to study without her. Table of Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.1 Overview . 10 1.2 Publications . 11 2. Basics and Notations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 2.1 Sets, Relations, and Functions . 13 2.2 Algebraic Structures . 14 2.2.1 Closures . 14 2.2.2 Algebras, Sub-algebras, and Lattices . 15 2.3 Graphs, Trees, Boolean Functions, and Circuits . 17 2.4 Complexity-Theoretical Basics . 18 2.4.1 Turing Machines. 18 2.4.2 Complexity Classes . 20 2.4.3 Families of Boolean Circuits and Uniformity . 23 2.4.4 Reductions and Complete Sets . 25 3. Boolean Clones and Boolean Co-Clones : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 3.1 Clones and Co-Clones and their Galois Connection . 27 3.2 Clones of Boolean Functions . 31 3.3 Boolean Constraints, and Co-Clones of Boolean Relations . 35 3.4 Minimal Bases for Boolean Co-Clones . 41 4. Complexity Classifications for Problems on the Boolean Domain : : : : : 47 4.1 Satisfiability with Exceptions . 47 4.2 Frozen Variables . 50 4.3 Quantified Boolean Constraint Satisfaction with Bounded Alternations . 60 5. Clones in Structural Complexity Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 65 5.1 Identifying Complexity Clones . 65 5.2 An Example of a Hierarchy of Complexity Clones Below FP . 66 5.3 Finite Bases . 69 5.4 Dual-atoms for L(FP) . 71 5.4.1 Dual-Atoms of FP and Polynomial-Sized Circuits . 73 5.4.2 Time-Complexity Classes as Dual-Atoms of FP . 77 6 Table of Contents 6. The Generation Problem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83 6.1 Generation Problems for General Operations . 85 6.1.1 Length-Monotonic Polynomial-Time Operations . 86 6.1.2 Finiteness of Generated Sets: An Excursion . 89 6.1.3 Length-Monotonic Associative Polynomial-Time Operations . 91 6.2 Generation Problems for Polynomials . 95 6.2.1 The Main Case . 96 a b 6.2.2 GENs(x y c) is NP-complete . 104 6.3 The Generation Problem GEN(xc + ky) . 109 6.3.1 Notations . 111 6.3.2 NP-Hardness of Modified Sum-of-Subset Problems . 112 6.3.3 NP-Hardness of GEN(xc + ky) . 118 6.4 A Summary for the Generation Problem . 124 Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 127 Notations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 131 Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 137 1. Introduction There are two sorts of sciences: Deductive ones and inductive ones. The former, like biology and chemistry, study nature and try to derive rules that describe it as good as possible. The latter start from just a few basic laws, or objects and combine them in order to gain deeper cognitions and more elaborated tools. Computer science is an inductive discipline. Although modern computer systems, net- works, and software can be very complicated, all of them are built using a few basic objects which are then put together following some simple rules or operations. Take for example a functional programming language: In the basic package of such a language there are just a few important functions, and the only way one is allowed to combine them is basically the substitution of an argument of one function with another function. Nevertheless, most functional programming languages are provably as strong as the other popular computational models. As another example, take a computer-chip like a central processing unit (CPU). They have to be able to do numerous computations with large arguments and they control and synchronize most of the other components of a computer. These are large and difficult tasks, and to meet the demands, the CPU's have to be very complicated. How can you build such a complicated device? Of course, the answer is: You do not start with the whole device but with only a very small part of it. You could say, that the first step is to build a few gates (which can be seen as very simple processors themselves). Then you wire the gates together thus building a simple circuit which is still transparent. Take several such circuits to build an even larger one, and so on, until you finally arrive at the finished CPU. Naturally, this method works the other way round, too: We are not only able to build very complicated structures, we are also able to better control them. Take the chip example: If one looks at the design plans of a modern processor, one sees an ocean of gates that are all somehow linked together. But the designer of the processor will be able to point out small subsections, with only a few hundred gates, and he will be able to describe the function of that subsection to us. He can then go further and brake the subsection down to a sub-sub-section with only a handful of gates the function of which is comparatively easy to understand. By this divide and conquer method, we will be able to understand the whole processor in time. In algebra, closures are deeply related with the inductive principle. A closure is a set that is closed under a number of operations. We can visualize a set as a bag of objects and the operations as a toolkit containing tools to change or combine the objects in the bag. A 8 1. Introduction set S is closed under the operations in Ω if we cannot build new objects from the objects in S by application of the tools of Ω. With other words, if we apply a tool of Ω on, say, two objects of S the result is an object which is equal to one that is already in the bag. So, if S were not closed, you could build a closed set from S by applying the operations from Ω on S thus building a larger set, on which you apply the operations again, and so on, until you finally arrive at a set where new objects cannot be built any more, since all possible objects are there already (this could take infinitely long, though). In this thesis, the objects in our bags are functions and we are interested in building new functions from basic ones. The most straightforward way to do this, is to write down formulas with symbols that stand for the known functions. Take for example two func- tions f(x; y) and g(z): Then, if we substitute the argument x of f with g(z), we get the formula f(g(z); y) which may describe a new function that is neither equal to f nor to g. The operations, or the toolkit, we use to assemble new functions from old ones is called superposition. Sets of functions that are closed under superposition are called clones.