(A) A trial started with the illumination of a light-emitting diode (LED), after which the monkeys had to grasp and to hold a touch bar for some time. This triggered the presentation of a sequence of two tones, S1 and S2, separated by a delay. The monkeys were required to either release the bar within a specific interval after S2 (go response) or to continue holding the bar and thus not to release the bar within this interval (no-go response). Bar release caused the LED to turn off. The monkeys were rewarded with a small amount of water for correct go responses and correct no-go responses. (B) Each tone could have a frequency of either 3 kHz (orange boxes) or 1 kHz (blue boxes), resulting in the four possible sequences 3–3, 3–1, 1–3, and 1–1. In Task 1, the go response was required for the sequence 3–3 and the no-go response for the other sequences. In Task 2, the go response was required for the sequence 1–1 and the no-go response for the other sequences. The monkeys were cued which task to perform in a trial, with a green LED located to the monkey’s right for Task 1 and with a red LED located to the monkey’s left for Task 2. (C–D) Sensorimotor associations of S1s and S2s in Task 1 (C) and Task 2 (D) from the perspective of an ideal observer. At the start of a given trial, the observer could not know which motor response would be required to obtain a reward, and identification of the frequency of S1 was a necessary first step towards resolving this uncertainty. However, in Task 1, when S1 was 3 kHz, identification of S1 was not sufficient and S1 therefore could not signal the required motor response (S1-uncertain). The sufficient information was provided by S2, with the go response signaled by S2 when it was 3 kHz (S2-go) and the no-go response when it was 1 kHz (S2-no-go). When S1 was 1 kHz, identification of S1 was both necessary and sufficient for resolving the uncertainty and therefore S1 could signal the required motor response (S1-no-go). In this case, identification of S2 was not necessary for resolving the uncertainty and S2 was therefore not needed to signal the motor response (S2-nil). In Task 2, in trials where S1 was 1 kHz, S1 could not signal the required motor response (S1-uncertain) and S2 signaled the go response when it was also 1 kHz (S2-go) and the no-go response when it was 3 kHz (S2-no-go). In trials where S1 was 3 kHz, S1 could signal the required motor response (S1-no-go) and S2 was not needed to signal the motor response (S2-nil).