example, a weaker type of integrity would be for the decision value to equal a value that some correct process proposed – not necessarily all of them. There is also a condition known as validity in the literature which refers to the property that a message sent by a process must be delivered. A protocol that can correctly guarantee consensus amongst n processes of which at most t fail is said to be t-resilient. In evaluating the performance of consensus protocols two factors of interest are running time and message complexity. Running time is given in Big O notation in the number of rounds of message exchange as a function of some input parameters (typically the number of processes and/or the size of the input domain). Message complexity refers to the amount of message traffic that is generated by the protocol. Other factors may include memory usage and the size of messages. Models of computation Varying models of computation may define a "consensus problem". Some models may deal with fully connected graphs, while others may deal with rings and trees. In some models message authentication is allowed, whereas in others processes are completely anonymous. Shared memory models in which processes communicate by accessing objects in shared memory are also an important area of research. Communication channels with direct or transferable authentication In most models of communication protocol participants communicate through authenticated channels. This means that messages are not anonymous, and receivers know the source of every message they receive. Some models assume a stronger, transferable form of authentication, where each message is signed by the sender, so that a receiver knows not just the immediate source of every message, but the participant that initially created the message. This stronger type of authentication is achieved by digital signatures, and when this stronger form of authentication is available, protocols can tolerate a larger number of faults. The two different authentication models are often called oral communication and written communication models. In an oral communication model, the immediate source of information is known, whereas in stronger, written communication models, every step along the receiver learns not just the immediate source of the message, but the communication history of the message. Inputs and outputs of consensus In the most traditional single-value consensus protocols such as Paxos, cooperating nodes agree on a single value such as an integer, which may be of variable size so