In computing, triple modular redundancy, sometimes called triple-mode redundancy,[1] (TMR) is a fault-tolerant form of N-modular redundancy, in which three systems perform a process and that result is processed by a majority-voting system to produce a single output. If any one of the three systems fails, the other two systems can correct and mask the fault. If the voter fails then the complete system will fail. However, in a good TMR system the voter is much more reliable than the other TMR components. Alternatively, if there is another stage of TMR logic following the current one (for example, in systems such as the Saturn Launch Vehicle Digital Computer), then three voters are used – one for each copy of the next stage of logic.

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Triple Modular Redundancy. Three identical logic circuits (logic gates) are used to compute the specified Boolean function. The set of data at the input of the first circuit are identical to the input of the second and third gates.

In computing, triple modular redundancy, sometimes called triple-mode redundancy,[1] (TMR) is a fault-tolerant form of N-modular redundancy, in which three systems perform a process and that result is processed by a majority-voting system to produce a single output. If any one of the three systems fails, the other two systems can correct and mask the fault. If the voter fails then the complete system will fail. However, in a good TMR system the voter is much more reliable than the other TMR components. Alternatively, if there is another stage of TMR logic following the current one (for example, in systems such as the Saturn Launch Vehicle Digital Computer), then three voters are used – one for each copy of the next stage of logic.

The TMR concept can be applied to many forms of redundancy, such as software redundancy in the form of N-version programming.

Some ECC memory uses triple modular redundancy hardware (rather than the more common Hamming code), because triple modular redundancy hardware is faster than Hamming error correction hardware.[2] Space satellite systems often use TMR,[3] [4][5] although satellite RAM usually uses Hamming error correction.[6]

To utilize triple modular redundancy, a ship must have at least three chronometers. At one time, the cost of three sufficiently accurate chronometers was more than the cost of a smaller merchant vessel.[7] Some vessels carried more than three chronometers – for example, the HMS Beagle carried 22 chronometers.[8] Some communication systems use N-modular redundancy as a simple form of forward error correction. For example, 5-modular redundancy communication systems (such as FlexRay) use the majority of 5 samples – if any 2 of the 5 results are erroneous, the other 3 results can correct and mask the fault.

## The majority gate

In TMR, three identical logic circuits (logic gates) are used to compute the same set of specified Boolean function. If there are no circuit failures, the outputs of the three circuits are identical. But due to circuit failures, the outputs of the three circuits may be different. A majority gate is used to decide which of the circuits’ outputs is the correct output. The majority gate output is 1 if two or more of the inputs of the majority gate are 1; output is 0 if two or more of the majority gate’s inputs are 0. The majority gate is a simple AND–OR circuit: if the inputs to the majority gate are denoted by x, y and z, then the output of the majority gate is

$xy \lor yz \lor xz$

Thus, the majority gate is the carry output of a full adder, i.e., the majority gate is a voting machine.[9]

## TMR operation

Assuming the Boolean function computed by the three identical logic gates has value 1, then: (a) if no circuit has failed, all three circuits produce an output of value 1, and the majority gate output has value 1. (b) if one circuit fails and produces an output of 0, while the other two are working correctly and produce an output of 1, the majority gate output is 1, i.e., it still has the correct value. And similarly for the case when the Boolean function computed by the three identical circuits has value 0. Thus, the majority gate output is guaranteed to be correct as long as no more than one of the three identical logic circuits has failed.[9]

The majority gate itself could fail. However, in contrast to the relatively complicated Boolean functions computed in triplicate by the TMR system, the majority gate is a simple circuit, thus its probability of failure is significantly smaller than that of each of the three circuits computing the Boolean function.[9]

## Triple modular redundancy in popular culture

• The three pre-cogs in Minority Report lead to a conviction, even when one is in the minority.
• To rule out that a single win was "a fluke", some competitions use a two out of three falls match. This isn't true TMR, however, because the three falls are not independent of each other – each competitor knows who has most falls at any point in the competition, which influences their future actions.
• In Arthur C. Clarke's science fiction novel Rendezvous with Rama, the Ramans make heavy use of triple redundancy.