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An ef cient algorithm for computing the set of attributes functionally determined by is useful not only for testing whether is a superkey, but also for several other tasks, as we will see later in this section Let be a set of attributes We call the set of all attributes functionally determined by under a set F of functional dependencies the closure of under F; we denote it by + Figure 77 shows an algorithm, written in pseudocode, to compute + The input is a set F of functional dependencies and the set of attributes The output is stored in the variable result To illustrate how the algorithm works, we shall use it to compute (AG)+ with the functional dependencies de ned in Section 732 We start with result = AG The rst time that we execute the while loop to test each functional dependency, we nd that A B causes us to include B in result To see this fact, we observe that A B is in F, A result (which is AG), so result := result B A C causes result to become ABCG CG H causes result to become ABCGH CG I causes result to become ABCGHI The second time that we execute the while loop, no new attributes are added to result, and the algorithm terminates Let us see why the algorithm of Figure 77 is correct The rst step is correct, since always holds (by the re exivity rule) We claim that, for any subset of result, Since we start the while loop with result being true, we can add to result only if result and But then result by the re exivity rule, so by transitivity Another application of transitivity shows that (using and ) The union rule implies that result , so functionally determines any new result generated in the while loop Thus, any attribute returned by the algorithm is in + It is easy to see that the algorithm nds all + If there is an attribute in + that is not yet in result, then there must be a functional dependency for which result, and at least one attribute in is not in result It turns out that, in the worst case, this algorithm may take an amount of time quadratic in the size of F There is a faster (although slightly more complex) algorithm that runs in time linear in the size of F; that algorithm is presented as part of Exercise 714 result := ; while (changes to result) do for each functional dependency in F do begin if result then result := result ; end Figure 77 An algorithm to compute + , the closure of under F.

You may be asking what this has to do with a stateful firewall like the security appliances First, understand that when connections are being set up, traffic flows in two directions through the appliance Assume that you have a user on the inside of your network who initiates a TCP connection to a device on the outside of your network Because TCP has a defined set of rules for setting up a connection, it is easy for the appliance to understand what is happening in the connection setup process In other words, it is easy for the appliance to inspect this traffic As I discussed in 1, a stateful firewall keeps track of the state of a connection In this example, the appliance sees the outgoing SYN and realizes that this is a setup request from an inside user Because it is a stateful firewall, the appliance will add an entry in its connection (state) table so that the SYN/ACK from the destination will be permitted back in, and the inside user will be able to complete the connection with the final ACK The appliance will then permit traffic to flow back and forth between these two machines for only this connection (unless the inside user opens another connection to this destination) Likewise, TCP goes through a well-defined process when tearing down a connection When the appliance sees the tear-down process (the FIN and FIN/ACK or an RST), the appliance knows that the connection is being terminated and will remove the connection from its state table Therefore, once the entry is removed from the appliance state table, if the outside destination device tries to send traffic through the appliance using the old connection parameters, the appliance will drop the traffic

The join operation requires that the system test pairs of tuples to see whether they satisfy the join condition; if they do, the system adds the pair to the join output Parallel join algorithms attempt to split the pairs to be tested over several processors Each processor then computes part of the join locally Then, the system collects the results from each processor to produce the nal result

Because it is a stateful firewall, the appliance drops, by default, all new inbound TCP connections that try to enter your network To allow this traffic, you will have to explicitly permit the TCP connection types that you want SECURITY ALERT! By default the appliances deny all traffic flows that originate from a lowersecurity-level interface and that are trying to reach a higher-security-level interface One problem that TCP has, however, is that it is very predictable, which sometimes plays into the hands of attackers For instance, an attacker might attempt to send a flood of TCP SYNs to an internal device, pretending to try to set up TCP connections The real intention of the attacker, however, is not to complete the three-way handshake for each