This paper deals with logics of programs. The objective is to formalize a notion of program description, and to give both plausible (semantic) and effective (syntactic) criteria for the notion of truth of a description. A novel feature of this treatment is the development of the mathematics underlying Floyd-Hoare axiom systems independently of such systems. Other directions that such research might take are considered.

Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re

A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of modal µ-calculus. This result generalizes the standard approach in the literature where a finitary characterization is only provided for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-wellfounded sets.

Abstract. We study the strictness of the modal µ-calculus hierarchy over some restricted classes of transition systems. First, we show that the hierarchy is strict over reflexive frames. By proving the finite model theorem for reflexive systems the same results holds for finite models. Second, we prove that over transitive systems the hierarchy collapses to the alternation-free fragment. In order to do this the finite model theorem for transitive transition systems is also proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment. 1

this report were made possible mainly with the financial support of the Ford Foundation. Additional contributions were received from Cosmetics Oriflame Romania and the Open Society Institute. We owe much to John Robbins, who conceived and developed the idea of this project. We thank the following for their contributions, support and assistance: The main implementers of the project: Renate Weber, Nicole Watson, Roxana Tesiu, Gulhan Borubaeva, Aurelija Kuzmaite, Tanya Lokshina, Genoveva Tisheva, and all the local rapporteurs and host NGOs; IHF staff who provided editorial and administrative assistance: Brigitte Dufour, Paula TscherneLempi inen, Ursula Lindenberg, Joachim Frank, Maria Kolb, Natalia Lazareva, Rainer Tannenberger, Judith Vitt and David Theil; Friends in the diplomatic, foundation, and civil society community and others who helped: Irena Gross, Sylvia Hordosch, Evelyn Watson, Milos Uveric-Kostic, Agnes Horvthov, Liz Bonkow

Abstract. We show that the modal µ-calculus over GL collapses to the modal fragment by showing that the fixpoint formula is reached after two iterations and answer to a question posed by van Benthem in [vBe06]. Further, we introduce the modal µ ∼-calculus by allowing fixpoint constructors for any formula where the fixpoint variable appears guarded but not necessarily positive and show that this calculus over GL collapses to the modal fragment, too. The latter result allows us a new proof of the de Jongh, Sambin Theorem and provides a simple algorithm to construct the fixpoint formula. Keywords: Fixpoint, Modal µ-Calculus, Gödel-Löb Logic. 1

We present two sequent calculi for the modal µ-calculus over S5 and prove their completeness by using classical methods. One sequent calculus has an analytical cut rule and could be used for a decision procedure the other uses a modified version of the induction rule. We also provide a completeness theorem for Kozen’s Axiomatisation over S5 without using the completeness result established by Walukiewicz for the modal µ-calculus over arbitrary models.

In formal approaches to inductive learning, the ability to learn is understood as the ability to single out a correct hypothesis from a range of possibilities. Although most of the existing research focuses on the characteristics of the learner, in many paradigms the significance of the teacher’s abilities and strategies is in fact undeniable. Motivated by this observation, in this paper we highlight the interactive nature of learning by proposing a game-theoretical and logical approach. We consider learning as a sabotagetype game between Teacher and Learner, and present different variants based on the level of cooperativeness and the actions available to the players. We characterize the existence of a winning strategy in such games by formulas of Sabotage Modal Logic, analyzing also their complexity. Our work constitutes the first step towards a unified game-theoretical and logical approach to formal learning theory. 1

We provide a purely syntactical treatment of simultaneous fixpoints in the modal µ-calculus by proving directly in Kozen’s axiomatisation their properties as greatest and least fixpoints, that is, the fixpoint axiom and the induction rule. Further, we apply our result in order to get a completeness result for characteristic formulae of finite pointed transition systems. Keywords: Modal µ-calculus, proof theory, Kozen’s axiomatisation, simultaneous fixpoints

In formal approaches to inductive learning, the ability to learn is understood as the ability to single out a correct hypothesis from a range of possibilities. Although most of the existing research focuses on the characteristics of the learner, in many paradigms the significance of the teacher’s abilities and strategies is in fact undeniable. Motivated by this observation, in this paper we highlight the interactive nature of learning by proposing a game-theoretical and logical approach. We consider learning as a game, and present different levels of cooperativeness between the players. Then, we look at different variants of Sabotage Games as learning scenarios, expressing the conditions for learnability in Sabotage Modal Logic and analyzing their complexity. Our work constitutes the first step towards a unified game-theoretical and logical approach to formal learning theory. 1