Krzysztof R. Apt

Verification of sequential and concurrent programs (2nd ed.)

This widely anticipated third edition provides a systematic exploration of one of the most common approaches to program verification, known as the assertional approach. This approach is applied to deterministic and nondeterministic sequential programs of varying complexity, together with both parallel and distributed concurrent programs. The expanded content also includes coverage of the verification of object-oriented programs. For each class of programs, the authors introduce proof systems for the verification of partial and total correctness, justified formally in corresponding soundness theorems. Case studies supplied throughout the book demonstrate the use of the proof systems and formally verify solutions to classical problems, such as producer/consumer and mutual exclusion. This modern update of a classic, reader-friendly textbook is perfect for an introductory course on program verification for advanced undergraduate or graduate students. Outlines for possible courses are suggested in the Preface.

Fairness in parallel programs: the transformational approach

Program transformations are proposed as a means of providing fair parallelism semantics for parallel programs with shared variables. The transformations are developed in two steps. First, abstract schedulers that implement the various fairness policies are introduced. These schedulers use random assignments z := ? to represent the unbounded nondeterminism induced by fairness. Concrete schedulers are derived by suitably refining the ?. The transformations are then obtained by embedding the abstract schedulers into the parallel programs. This embedding is proved correct on the basis of a simple transition semantics. Since the parallel structure of the original program is preserved, the transformations also provide a basis for syntax-directed proofs of total correctness under the fairness assumption. These proofs make use of infinite ordinals.

Diffusion in social networks with competing products

We introduce a new threshold model of social networks, in which the nodes influenced by their neighbours can adopt one out of several alternatives. We characterize the graphs for which adoption of a product by the whole network is possible (respectively necessary) and the ones for which a unique outcome is guaranteed. These characterizations directly yield polynomial time algorithms that allow us to determine whether a given social network satisfies one of the above properties. n nWe also study algorithmic questions for networkswithout unique outcomes.We show that the problem of computing the minimum possible spread of a product is NP-hard to approximate with an approximation ratio better than Ω(n), in contrast to the maximum spread, which is efficiently computable. We then move on to questions regarding the behavior of a node with respect to adopting some (resp. a given) product. We show that the problem of determining whether a given node has to adopt some (resp. a given) product in all final networks is co-NP-complete.

Choosing products in social networks

We study the consequences of adopting products by agents who form a social network. To this end we use the threshold model introduced in [1], in which the nodes influenced by their neighbours can adopt one out of several alternatives, and associate with each such social network a strategic game between the agents. The possibility of not choosing any product results in two special types of (pure) Nash equilibria. n nWe show that such games may have no Nash equilibrium and that determining the existence of a Nash equilibrium, also of a special type, is NP-complete. The situation changes when the underlying graph of the social network is a DAG, a simple cycle, or has no source nodes. For these three classes we determine the complexity of establishing whether a (special type of) Nash equilibrium exists. n nWe also clarify for these categories of games the status and the complexity of the finite improvement property (FIP). Further, we introduce a new property of the uniform FIP which is satisfied when the underlying graph is a simple cycle, but determining it is co-NP-hard in the general case and also when the underlying graph has no source nodes. The latter complexity results also hold for verifying the property of being a weakly acyclic game.

Social network games

htmlabstractOne of the natural objectives of the field of the social networks is to npredict agents’ behaviour. To better understand the spread of various nproducts through a social network [2] introduced a threshold model, in nwhich the nodes influenced by their neighbours can adopt one out of sev- neral alternatives. To analyze the consequences of such product adoption nwe associate here with each such social network a natural strategic game nbetween the agents. nIn these games the payoff of each player weakly increases when more nplayers choose his strategy, which is exactly opposite to the congestion ngames. The possibility of not choosing any product results in two special ntypes of (pure) Nash equilibria. nWe show that such games may have no Nash equilibrium and that ndetermining an existence of a Nash equilibrium, also of a special type, nis NP-complete. This implies the same result for a more general class nof games, namely polymatrix games. The situation changes when the nunderlying graph of the social network is a DAG, a simple cycle, or, more ngenerally, has no source nodes. For these three classes we determine the ncomplexity of an existence of (a special type of) Nash equilibria. nWe also clarify for these categories of games the status and the com- nplexity of the finite best response property (FBRP) and the finite im- nprovement property (FIP). Further, we introduce a new property of the nuniform FIP which is satisfied when the underlying graph is a simple cy- ncle, but determining it is co-NP-hard in the general case and also when nthe underlying graph has no source nodes. The latter complexity results nalso hold for the property of being a weakly acyclic game. A preliminary nversion of this paper appeared as [19]

Epistemic protocols for distributed gossiping

Gossip protocols aim at arriving, by means of point-to-point or group communications, at a situation in which all the agents know each others secrets. We consider distributed gossip protocols which are expressed by means of epistemic logic. We provide an operational semantics of such protocols and set up an appropriate framework to argue about their correctness. Then we analyze specific protocols for complete graphs and for directed rings.

Social Networks with Competing Products

We introduce a new threshold model of social networks, in which the nodes influenced by their neighbours can adopt one out of several alternatives. We characterize social networks for which adoption of a product by the whole network is possible respectively necessary and the ones for which a unique outcome is guaranteed. These characterizations directly yield polynomial time algorithms that allow us to determine whether a given social network satisfies one of the above properties. We also study algorithmic questions for networks without unique outcomes. We show that the problem of determining whether a final network exists in which all nodes adopted some product is NP-complete. In turn, we also resolve the complexity of the problems of determining whether a given node adopts some respectively, a given product in some respectively, all networks. Further, we show that the problem of computing the minimum possible spread of a product is NP-hard to approximate with an approximation ratio better than Ωn, in contrast to the maximum spread, which is efficiently computable. Finally, we clarify that some of the above problems can be solved in polynomial time when there are only two products.

The Role of Monotonicity in the Epistemic Analysis of Strategic Games

It is well-known that in finite strategic games true common belief (or common knowledge) of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies. We establish a general theorem that deals with monotonic rationality notions and arbitrary strategic games and allows to strengthen the above result to arbitrary games, other rationality notions, and transfinite iterations of the elimination process. We also clarify what conclusions one can draw for the customary dominance notions that are not monotonic. The main tool is Tarski’s Fixpoint Theorem.

Coordination Games on Directed Graphs

We study natural strategic games on directed graphs, which capture the idea of coordination in the absence of globally common strategies. We show that these games do not need to have a pure Nash equilibrium and that the problem of determining their existence is NP-complete. The same holds for strong equilibria. We also exhibit some classes of games for which strong equilibria exist and prove that a strong equilibrium can then be found in linear time.

Paradoxes in social networks with multiple products

We show that various paradoxes can arise in a natural class of social networks. They demonstrate that more services or products may have adverse consequences for all members of the network and conversely that restricting the number of choices may be beneficial for every member of the network. These phenomena have been confirmed by a number of empirical studies. In our analysis we use a simple threshold model of social networks introduced in Apt and Markakis (2011), and more fully in Apt and Markakis (2014). In this model the agents, influenced by their neighbours, can adopt one out of several alternatives. We identify and analyze here four types of paradoxes that can arise in these networks. These paradoxes shed light on possible inefficiencies arising when one modifies the sets of products available to the agents forming a social network or the network structure. One of the paradoxes corresponds to the well-known Braess paradox in congestion games and shows that by adding more choices to a node, the network may end up in a situation that is worse for everybody. We exhibit a dual version of this, according to which removing a product available to an agent can eventually make everybody better off. The other paradoxes that we identify show that by adding or removing a product from the choice set of an agent may lead to permanent instability. Finally, we also identify conditions under which some of these paradoxes cannot arise.