Steve Martin will compose the music for the Public Theater’s upcoming Shakespeare in the Park production of “As You Like It,” penning a bluegrass score for the Bard comedy.Helmer Daniel Sullivan’s staging updates the action to the 1840s in the American South. Renee Elise Goldsberry (“Good People”), Omar Metwally (“Sixteen Wounded”) and Macintyre Dixon join previously announced topliner Lily Rabe in the cast, with Goldsberry tapped to play Celia, the cousin of Rabe’s character, Rosalind. Sullivan has become a regular helmer of Shakespeare in the Park offerings. His 2009 staging of “Twelfth Night,” which starred Anne Hathaway, incorporated live music from the band Hem. The Public also has set dates for the Shakespeare in the Park program’s two offerings at the outdoor Delacorte Theater in Central Park. “As You Like It” runs June 5-30, while the revival of tuner “Into the Woods” is skedded to run July 23-Aug. 25. Further casting remains to be announced for both productions.