To my mother Maxine, who gave me a love of learning; to Susan, who is as happy and amazed as I am that The Book is finally completed; to Josh, Tim, and Teddy, who are impressed that their father is an Author; and to my late father George, who would have been proud.

This paper starts with the project of finding a large subclass of NP which exhibits a dichotomy. The approach is to find this subclass via syntactic prescriptions. While, the paper does not achieve this goal, it does isolate a class (of problems specified by) "Monotone Monadic SNP without inequality" which may exhibit this dichotomy. We justify the placing of all these restrictions by showing that classes obtained by using only two of the above three restrictions do not show this dichotomy, essentially using Ladner's Theorem. We then explore the structure of this class. We show all problems in this class reduce to the seemingly simpler class CSP. We divide CSP into subclasses and try to unify the collection of all known polytime algorithms for CSP problems and extract properties that make CSP problems NP-hard. This is where the second part of the title-"a study through Datalog and group theory"-comes in. We present conjectures about this class which would end in showing the dichotomy.

Two complexity measures for query languages are proposed. Data complexity is the complexity of evaluating a query in the language as a function of the size of the database, and expression complexity is the complexity of ewduating a query in the language as a function of the size of the expression defining the query. We study the data and expression complexity of logical langnages -relational calculus and its extensions by transitive closure, fixpoint and second order existential quantification -and algebraic languages -relational algebra and its extensions by bounded and unbounded looping. The pattern which will bc shown is that the expression complexity of the investigated languages is one exponential higher then their data complexity, and for both types of complexity we show completeness in some complexity class.

We investigate extensions of temporal logic by connectives de ned by nite automata on in nite words. We consider three di erent logics, corresponding to three di erent types of acceptance conditions (nite, looping and repeating) for the automata. It turns out, however, that these logics all have the same expressive power and that their decision problems are all PSPACE-complete. We also investigate connectives de ned by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem.

Of special interest in formal verification are safety properties, which assert that the system always stays within some allowed region. A computation that violates a general linear property reaches a bad cycle, which witnesses the violation of the property. Accordingly, current methods and tools for model checking of linear properties are based on a search for bad cycles. A symbolic implementation of such a search involves the calculation of a nested fixed-point expression over the system's state space, and is often very difficult. Every computation that violates a safety property has a finite prefix along which the property is violated. We use this fact in order to base model checking of safety properties on a search for finite bad prefixes. Such a search can be performed using a simple forward or backward symbolic reachability check. A naive methodology that is based on such a search involves a construction of an automaton (or a tableau) that is doubly exponential in the property. We present an analysis of safety properties that enables us to prevent the doubly-exponential blow up and to use the same automaton used for model checking of general properties, replacing the search for bad cycles by a search for bad prefixes.