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

. We present a framework for formal reasoning about the behaviour of distributed programs implementing open distributed systems (ODSs). The framework is based on the following key ingredients: a specification language based on the ¯-calculus, a hierarchical transitional semantics of the implementation language used, a judgment format allowing parametrised behavioural assertions, and a proof system for proving validity of such assertions which includes proof rules for property decomposition. This setting provides the expressive power for behavioural reasoning required by the complex open and dynamic nature of ODSs. The utility of the approach is illustrated on a prototypical ODS. 1 Introduction For a few years now, the Formal Design Techniques group at the Swedish Institute of Computer Science has pursued a programme aimed at enabling formal verification of complex open distributed systems (ODSs) through program code verification. While previous work by the group has been predominantly...

We provide a direct and modular translation from the temporal logics CTL, ETL, FCTL (CTL extended with the ability to express fairness) and the Modal µ-calculus to Monadic inf-Datalog with built-in predicates. We call it inf-Datalog because the semantics we provide is a little different from the conventional Datalog least fixed point semantics, in that some recursive rules (corresponding to least fixed points) are allowed to unfold only finitely many times, whereas others (corresponding to greatest fixed points) are allowed to unfold infinitely many times. We characterize

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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

We present an action theory with the power to represent recursive plans and the capability to reason about and synthesize recursive workflow control structures. In contrast with the software verification setting, reasoning does not take place solely over predefined data structures, and neither is there a process specification available in recursive form. Rather, specification takes the form of goals, and domain structure takes the form of a physical application setting containing objects. For this reason, well-founded induction is employed for its suitability for practical action domains where recursive structures must be described or inferred. Under this method, termination of the synthesized recursive workflow is a property that follows automatically. We show how a general workflow recursive construct is added to an action language that is then augmented with induction. This formalism is then transformed in a way amenable to automated reasoning. We demonstrate the method with a particular example specified in the theory, and then extracted from a proof constructed by the SNARK first-order theorem prover.