Service discovery becomes a key to accelerating the evolution of web services as the number of services is expected to increase dramatically. Foregoing work on service discovery is primarily based on the interfaces of services through the use of ontology. Ongoing work targets at service behavior, with not only individual message exchanges being captured, but also constraints between these message exchanges. In this paper, we propose a formal approach to expressing the service and query behavior using ?-calculus for service matchmaking. The resulting ?-calculus expressions of services and queries are precise in defining single operations involving message exchanges as well as execution sequence between operations. Based on the formalizations, service matchmaking between a service query and a service description is reasoned through the capability of ?-calculus. Expressing service behavior using ?-calculus is expected to be a promising way to realize intelligent service discovery.