SELECT docid from xml_access 
                                       WHERE (
                                                (
                                                 lower(principal_name) = 'uid=ARC,o=lter,dc=ecoinformatics,dc=org' 
                                                 AND perm_type = 'allow' 
                                                 AND (permission='4' OR permission='5' OR permission='6' OR permission='7')
                                                )
                                                OR 
                                                (
                                                     lower(principal_name) = 'public' 
                                                     AND perm_type = 'allow' 
                                                     AND (permission='4' OR permission='5' OR permission='6' OR permission='7')
                                                )
                                             ) 



(21987 rows)

Modification:

SELECT docid from xml_access 
                                       WHERE 
                                                (
                                                 (lower(principal_name) = 'uid=ARC,o=lter,dc=ecoinformatics,dc=org' OR lower(principal_name) = 'public')
                                                 AND perm_type = 'allow' 
                                                 AND (permission>3)
                                                )
(21987 rows)