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Pneumatically-actuated soft robots have advantages over traditional rigid robots in many applications. In particular, their flexible bodies and gentle air-powered movements make them more suitable for use around humans and other objects that could be injured or damaged by traditional robots. However, existing systems for controlling soft robots currently require dedicated electromechanical hardware (usually solenoid valves) to maintain the actuation state (expanded or contracted) of each independent actuator. When combined with power, computation, and sensing components, this control hardware adds considerable cost, size, and power demands to the robot, thereby limiting the feasibility of soft robots in many important application areas. In this work, we introduce a pneumatic memory that uses air (not electricity) to set and maintain the states of large numbers of soft robotic actuators without dedicated electromechanical hardware. These pneumatic logic circuits use normally-closed microfluidic valves as transistor-like elements; this enables our circuits to support more complex computational functions than those built from normally-open valves. We demonstrate an eight-bit nonvolatile random-access pneumatic memory (RAM) that can maintain the states of multiple actuators, control both individual actuators and multiple actuators simultaneously using a pneumatic version of time division multiplexing (TDM), and set actuators to any intermediate position using a pneumatic version of analog-to-digital conversion. We perform proof-of-concept experimental testing of our pneumatic RAM by using it to control soft robotic hands playing individual notes, chords, and songs on a piano keyboard. By dramatically reducing the amount of hardware required to control multiple independent actuators in pneumatic soft robots, our pneumatic RAM can accelerate the spread of soft robotic technologies to a wide range of important application areas.