A programmable execution unit (42) of a graphics processor includes a functional unit (50) that is operable to execute instructions (51). The output of the functional unit (50) can both be written to a register file (46) and fed back directly as an input to the functional unit by means of a feedback circuit (52). Correspondingly, an instruction that is to be executed by the functional unit (50) can select as its inputs either the fed-back output (52) from the execution of the previous instruction, or inputs from the registers (46). A register access descriptor (54) between each instruction in a group of instructions (53) specifies the registers whose values will be available on the register ports that the functional unit will read when executing the instruction, and the register address where the result of the execution of the instruction will be written to. The programmable execution unit (42) executes group of instructions (53) that are to be executed atomically.