We develop a Logic in which the basic objects of concern are games, or equivalently, monotone predicate transforms. We give completeness and decision results and extend to certain kinds of many-person games. Applications to a cake cutting algorithm and to a protocol for exchanging secrets, are given. 1

The propositional µ-calculus as introduced by Kozen in [12] is considered. In that paper

We present a direct reduction of dynamic epistemic logic in the spirit of [4] to propositional dynamic logic (PDL) [17, 18] by program transformation. The program transformation approach associates with every update action a transformation on PDL programs. These transformations are then employed in reduction axioms for the update actions. It follows that the logic of public announcement, the logic of group announcements, the logic of secret message passing, and so on, can all be viewed as subsystems of PDL. Moreover, the program transformation approach can be used to generate the appropriate reduction axioms for these logics. Our direct reduction of dynamic epistemic logic to PDL was inspired by the reduction of dynamic epistemic logic to automata PDL of [13]. Our approach shows how the detour through automata can be avoided. 1

Dynamic logic, broadly conceived, is the logic that analyses change by decomposing actions into their basic building blocks and by describing the results of performing actions in given states of the world. The actions studied by dynamic logic can be of various kinds: actions on the memory state of a computer, actions of a moving robot in a closed world, interactions between cognitive agents performing given communication protocols, actions that change the common ground between speaker and hearer in a conversation, actions that change the contextually available referents in a conversation, and so on. In each of these application areas, dynamic logics can be used to model the states involved and the transitions that occur between them. Dynamic logic is a tool for both state description and action description. Formulae describe states, while actions or programs express state change. The levels of state descriptions and transition characterisations are connected by suitable operations that allow reasoning about pre- and postconditions of particular changes.

Guarded actions are changes with preconditions acting as a guard. Guarded action models are multimodal Kripke models with the valuations replaced by guarded actions. Call guarded action logic the result of adding product updates with guarded action models to PDL (propositional dynamic logic). We show that guarded action logic reduces to PDL. 1

ABSTRACT. We introduce quantificational modal operators as dynamic modalities with (extensions of) Henkin quantifiers as indices. The adoption of matrices of indices (with action identifiers, variables and/or quantified variables as entries) gives an expressive formalism which is here motivated with examples from the area of multi-agent systems. We study the formal properties of the resulting logic which, formally speaking, does not satisfy the normality condition. However, the logic admits a semantics in terms of (an extension of) Kripke structures. As a consequence, standard techniques for normal modal logic become available. We apply these to prove completeness and decidability, and to extend some standard frame results to this logic.

Dynamic First Order Logic results from interpreting quantification over a variable v as change of valuation over the v position, conjunction as sequential composition, disjunction as nondeterministic choice, and negation as (negated) test for continuation. We present a tableau style calculus for DFOL with explicit (simultaneous) substitution, prove its soundness and completeness, and point out its relevance for programming with dynamic first order logic, and for automatic program analysis.

Recent work on Knowledge Representation has brought new interest to Propositional Dynamic Logics (PDL's) by pointing out a tight correspondence between these logics and logics for representing structured knowledge (Description Logics, Terminological Languages). Nevertheless, this work has also made apparent the lack in the known PDL's of certain constructs needed to fully exploit the correspondence. These are constructs to locally (wrt single states) constrain the running of an atomic program or its converse (the running of the atomic program backward) to be deterministic, or to have a specified amount of nondeterminism only. In fact, we believe that PDL's can take advantage of this kind of constructs to model many interesting properties of actual computations. In this paper, we extend Converse PDL, first by including a construct for local determinism of simple programs (either an atomic program or the converse of an atomic program), and then by including constructs for graded nondeter...

We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded systems and cyber-physical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multi-agent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic

The -calculus integrates in a uniform and simple setting rst-order rewriting, -calculus and nondeterministic computations. Its abstraction mechanism is based on the rewrite rule formation and its main evaluation rule is based on matching modulo a theory T . In this rst part, the calculus is motivated and its syntax and evaluation rules for any theory T are presented. In the syntactic case, i.e. when T is the empty theory, we study its basic properties for the untyped case. We rst show how it uniformly encodes -calculus as well as rst-order rewriting derivations. Then we provide sucient conditions for ensuring conuence of the calculus. Keywords: rewriting, strategy, non-determinism, matching, rewriting-calculus, lambda-calculus, rule based language. 1