Even in the Democratic Peopl'es Republic of Massachusetts, dogs worrying livestock can be shot.
Florida Law:
767.03. Good defense for killing dog
In any action for damages or of a criminal prosecution against any person for killing or injuring a dog, satisfactory proof that said dog had been or was killing any animal included in the definitions of "domestic animal" and "livestock" as provided by s. 585.01 shall constitute a good defense to either of such actions.
767.01. Dog owner's liability for damages to persons, domestic animals, or livestock
Owners of dogs shall be liable for any damage done by their dogs to a person or to any animal included in the definitions of "domestic animal" and "livestock" as provided by s. 585.01.