Using automata to characterise fixed point temporal logics

This work examines propositional fixed point temporal and modal logics called mu-calculi and their relationship to automata on infinite strings and trees. We use correspondences between formulae and automata to explore definability in mu-calculi and their fragments, to provide normal forms for formulae, and to prove completeness of axiomatisations. The study of such methods for describing infinitary languages is of fundamental importance to the areas of computer science dealing with non-terminating computations, in particular to the specification and verification of concurrent and reactive systems. To emphasise the close relationship between formulae of mu-calculi and alternating automata, we introduce a new first recurrence acceptance condition for automata, checking intuitively whether the first infinitely often occurring state in a run is accepting. Alternating first recurrence automata can be identified with mu-calculus formulae, and ordinary, non-alternating first recurrence automata with formulae in a particular normal form, the strongly aconjunctive form. Automata with more traditional Büchi and Rabin acceptance conditions can be easily unwound to first recurrence automata, i.e. to mu-calculus formulae. In the other direction, we describe a powerset operation for automata that corresponds to fixpoints, allowing us to translate formulae inductively to ordinary Büchi and Rabin-automata. These translations give easy proofs of the facts that Rabin-automata, the full mu-calculus, its strongly aconjunctive fragment and the monadic second-order calculus of n successors SnS are all equiexpressive, that Büchi-automata, the fixpoint alternation class Pi_2 and the strongly aconjunctive fragment of Pi_2 are similarly related, and that the weak SnS and the fixpoint-alternation-free fragment of mu-calculus also coincide. As corollaries we obtain Rabin's complementation lemma and the powerful decidability result of SnS. We then describe a direct tableau decision method for modal and linear-time mu-calculi, based on the notion of definition trees. The tableaux can be interpreted as first recurrence automata, so the construction can also be viewed as a transformation to the strongly aconjunctive normal form. Finally, we present solutions to two open axiomatisation problems, for the linear-time mu-calculus and its extension with path quantifiers. Both completeness proofs are based on transforming formulae to normal forms inspired by automata. In extending the completeness result of the linear-time mu-calculus to the version with path quantifiers, the essential problem is capturing the limit closure property of paths in an axiomatisation. To this purpose, we introduce a new \exists\nu-induction inference rule

Liquid Democracy: An Analysis in Binary Aggregation and Diffusion

The paper proposes an analysis of liquid democracy (or, delegable proxy voting) from the perspective of binary aggregation and of binary diffusion models. We show how liquid democracy on binary issues can be embedded into the framework of binary aggregation with abstentions, enabling the transfer of known results about the latter---such as impossibility theorems---to the former. This embedding also sheds light on the relation between delegation cycles in liquid democracy and the probability of collective abstentions, as well as the issue of individual rationality in a delegable proxy voting setting. We then show how liquid democracy on binary issues can be modeled and analyzed also as a specific process of dynamics of binary opinions on networks. These processes---called Boolean DeGroot processes---are a special case of the DeGroot stochastic model of opinion diffusion. We establish the convergence conditions of such processes and show they provide some novel insights on how the effects of delegation cycles and individual rationality could be mitigated within liquid democracy. The study is a first attempt to provide theoretical foundations to the delgable proxy features of the liquid democracy voting system. Our analysis suggests recommendations on how the system may be modified to make it more resilient with respect to the handling of delegation cycles and of inconsistent majorities

The Best a Monitor Can Do

Existing notions of monitorability for branching-time properties are fairly restrictive. This, in turn, impacts the ability to incorporate prior knowledge about the system under scrutiny - which corresponds to a branching-time property - into the runtime analysis. We propose a definition of optimal monitors that verify the best monitorable under- or over-approximation of a specification, regardless of its monitorability status. Optimal monitors can be obtained for arbitrary branching-time properties by synthesising a sound and complete monitor for their strongest monitorable consequence. We show that the strongest monitorable consequence of specifications expressed in Hennessy-Milner logic with recursion is itself expressible in this logic, and present a procedure to find it. Our procedure enables prior knowledge to be optimally incorporated into runtime monitors

Strategic logics : complexity, completeness and expressivity

by transferring normative attributes from an agent to another. Such interactions are called delegation. Formal models of delegation and control were studied in, e.g., [189, 149, 191]. In this work, we consider the scenario where agents delegate control over propositions to other agents. The distinction between controllable and uncontrollable propositions stems from areas like discrete event systems and control theory, where, e.g., Boutilier [39] studied control in the context of deontic logic. Control and controllable propositions were also studied in [52, 66, 249, 248]. We now give an overview of the thesis. The main purpose of Chapter 2 is to introduce basic concepts and notation and to review relevant literature. The first section presents a brief survey on modal logic. Then, in sections 2.2, 2.3 and 2.4, we introduce epistemic, temporal and strategic modal logics and state known results that characterise their expressivity and computational complexity. In particular, we consider variants of ATL as extensions of branching-time logics. With such ATL-like logics we can describe dynamic multi-agent interactions. In Section 2.5, we discuss extensions of ATL with epistemic notions. Additionally, we suggest a framework for memory-bounded strategic reasoning. In particular, we introduce an epistemic variant of ATL that accounts for agents with limited memory resources as this case was neglected in the literature to date. In Chapter 3, we investigate the computational complexity of ATL and its epistemic extension ATEL. We show in detail how 'the complexity of the satisfiability problem for both logics can be settled at ExpTIME-complete. The part of the chapter about ATL is based on the paper 'ATL Satisfiability is Indeed ExpTIME-COmplete' by Walther, Lutz, Wolter and Wooldridge in the Journal of Logic and Computation, 2006 (265)' and the part about ATEL is based on the paper 'ATEL with Common and Distributed Knowledge is ExpTime-Complete' by Walther which was presented at the 4th Workshop on Methods for Modalities, Humbolt University, Berlin, December 1-2, 2005 [264]. In Chapter 4, we aim to extend the expressiveness of ATL without increasing its computational complexity. We introduce explicit names for strategies in the object language and extend modal operators with the possibility to bind agents to strategy names. In this way, we can fix the decisions of agents that possibly belong to several coalitions. By identifying the behaviqur of agents, we can reason about the effects of agents changing coalitions. Dynamic coalitions provide more flexibility to adapt abilities to a changing environment. We investigate the expressivity of the resulting logic ATLES and compare it to ATL and ATL*. Moreover, we formulate two model checking problems for ATLES and investigate their complexity as well as the complexity of the satisfiability problem for ATLES. Additionally, we present a complete axiomatisation. This chapter is based on the paper 'Alternating-time Temporal Logic with Explicit Strategies' by Walther, van der Hoek and Wooldridge which is going to presented at the 11th Conference on Theoretical Aspects of Rationality and Knowledge (TARK), Brussels, Belgium, June 25-27, 2007 [266]

Verification in the Hierarchical Development of Reactive Systems

In many approaches to the verification of reactive systems, operational semantics are used to model systems whereas specifications are expressed in temporal logics. Most approaches however fail to handle changes of the specification but assume, that the initial specification is indeed the intended one. Changing the specification thus necessitates to find an accordingly adapted system and to carry out the verification from scratch. During a systems life cycle however, changes of the requirements and resources necessitate repeated adaptations of specifications. We here propose a method that supports syntactic action refinement (in the process algebra TCSP and the Modal Mu-Calculus) and allows to automatically obtain (a priori) correct reactive systems by hierarchically adding details to the according specifications