# Generalising the Dining Philosophers Problem: Competitive Dynamic Resource Allocation in Multi-agent Systems

## Abstract

We consider a new generalisation of the Dining Philosophers problem with a set of agents and a set of resource units which can be accessed by them according to a fixed graph of accessibility between agents and resources. Each agent needs to accumulate a certain (fixed for the agent) number of accessible resource units to accomplish its task, and once it is accomplished the agent releases all resources and starts accumulating them again. All this happens in succession of discrete ‘rounds’ and yields a concurrent game model of ‘dynamic resource allocation’. We use the Alternating time Temporal Logic (ATL) to specify important properties, such as goal achievability, fairness, deadlock, starvation, etc. These can be formally verified using the efficient model checking algorithm for ATL. However, the sizes of the resulting explicit concurrent game models are generally exponential both in the number of resources and the number of agents, which makes the ATL model checking procedure generally intractable on such models, especially when the number of resources is large. That is why we also develop an abstract representation of the dynamic resource allocation models and develop a symbolic version of the model checking procedure for ATL. That symbolic procedure reduces the time complexity of model checking to polynomial in the number of resources, though it can take a worst-case double exponential time in the number of agents.

## Keywords

Dining philosophers games Dynamic resource allocation Alternating time temporal logic ATL Symbolic model checking## Notes

### Acknowledgements

The work of Valentin Goranko and Riccardo De Masellis was supported by a research grant 2015-04388 of the Swedish Research Council, which also partly funded a working visit of Nils Timm to Stockholm. Valentin Goranko was also partly supported by a visiting professorship grant by the University of Pretoria.

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