We consider Hintikka et al.'s `independence-friendly first-order logic'. We apply it to a modal logic setting, defining a notion of `independent' modal logic, and we examine the associated fixpoint logics.

We show undecidability of hereditary history preserving simulation for finite asynchronous transition systems by a reduction from the halting problem of deterministic Turing machines. To make the proof more transparent we introduce an intermediate problem of deciding the winner in domino snake games. First we reduce the halting problem of deterministic Turing machines to domino snake games. Then we show how to model a domino snake game by a hereditary history simulation game on a pair of finite asynchronous transition systems.

We prove that the one-dimensional sandpile prediction problem is . The previously best known upper bound on the AC -scale was AC . We also prove that it is not in AC for any constant > 0.

We propose an extended modal mu-calculus to provide an `assembly language' for modal logics for real time, value-passing calculi, and other extended models of computation.

Contents 1 Introduction 2 2 Finite words 4 2.1 First-order and monadic second-order logics . . . . . . . . . . 4 2.2 Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Infinite words 10 3.1 Closure properties of #-automata . . . . . . . . . . . . . . . . 14 4 Infinite trees 21 4.1 Closure properties of tree automata . . . . . . . . . . . . . . . 23 4.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 The -calculus and alternating automata 28 5.1 Syntax and semantics of the -calculus . . . . . . . . . . . . . 28 5.2 Alternating automata . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 From the -calculus to alternating automata . . . . . . . . . . 32 LaBRI, Domaine Universitaire, btiment A30, 351, cours de la Libration, 33405 Talence Cedex, FRANCE. e-mail:igw@labri.fr; www: http://www.labri.fr/#igw These are notes for EFF Summer School, July 2001, with minor modifications. 5.4 From alternating automata to the -calculus . . . . . . . .

Inf-Datalog extends the usual least fixpoint semantics of Datalog with greatest fixpoint semantics: we defined inf-Datalog and characterized the expressive power of various fragments of inf-Datalog in [GFAA03]. In the present paper, we study the complexity of query evaluation on finite models for (various fragments of) inf-Datalog. We deduce a unified and elementary proof that global model-checking (i.e. computing all nodes satisfying a formula in a given structure) has 1. quadratic data complexity in time and linear program complexity in space for CTL and alternation-free modal µ-calculus, and 2. linearspace (data and program) complexities, linear-time program complexity and polynomial-time data complexity for Lµk (modal µ-calculus with fixed alternation-depth at most k).