Are there any out of box API (webservice) exposed from Appian for external syste

Currently, there are no out of the box web services that would provide this functionality.

What are you trying to achieve?

The reason why I am asking is that there are many integration options that you can explore, and some custom plugins that are available on Forum, but it is hard to give you better direction without more information.

One of the use case is of auto completion of task in Appian Process when the required action is performed in the external system. The external system in this case may need to call some function to complete the task so that the process can proceed.

If the task is performed by an external system, you should not have a user input task in Appian. Consider using a call to web service that can send the data to your external system and receive the response back (if its done immediately)
On the other hand, if you have a user input task that can be done both as a user input task in Appian as well as by an external system, you can expose an Appian process as a web service and use this WS from the external system to notify Appian of completion of task. You need an exception on your user input task based on Messaging to allow the task to be skipped if the job is completed externally first.

You are essentially passing data between external system and Appian and you act on that data in the target system. There are several integration options, including JMS, file transfer and process models called as web services with the keyword being "no customization".