As I understand it, SOAP and REST are just protocols for communication.
When a web service is written, can it be accessed through either SOAP or REST or Both, or is it specific to the web service?
Is WSDL an API specification for the web service, and is it independent of whether you use SOAP or REST?
To answer this question we need to know what is it WSDL, SOAP and REST.
1) The Web Services Description Language (WSDL) is an XML-based interface definition language that is used for describing the functionality offered by a web service.
2) SOAP (originally Simple Object Access Protocol) is a protocol specification for exchanging structured information in the implementation of web services in computer networks.
3) WSDL is often used in combination with SOAP and an XML Schema to provide Web services over the Internet. A client program connecting to a Web service can read the WSDL file to determine what operations are available on the server. Any special datatypes used are embedded in the WSDL file in the form of XML Schema. The client can then use SOAP to actually call one of the operations listed in the WSDL file using for example XML over HTTP.
4) Representational state transfer (REST) or RESTful web services is a way of providing interoperability between computer systems on the Internet. REST-compliant Web services allow requesting systems to access and manipulate textual representations of Web resources using a uniform and predefined set of stateless operations. Other forms of Web service exist, which expose their own arbitrary sets of operations such as WSDL and SOAP.
When a web service is written, can it be accessed through either SOAP or REST or Both, or is it specific to the web service?
It depends on implementation of web service
Is WSDL an API specification for the web service, and is it independent of whether you use SOAP or REST?
Normally you chose to use WSDL and SOAP or REST or something else like json-rpc