§12.2 – Finite State Machines with Output

Remember the general picture of a computer as being a transition function T:S×I→S×O? If the state set S is finite (not infinite), we call this system a finite state machine. If the domain S×I is reasonably small, then we can specify T explicitly by writing out its complete graph. However, this is practical only for machines that have a very small information capacity.

Size of FSMs The information capacity of an FSM is C = I[S] = log |S|. Thus, if we represent a machine having an information capacity of C bits as an FSM, then its state transition graph will have |S| = 2C nodes. E.g. suppose your desktop computer has a 512MB memory, and 60GB hard drive. Its information capacity, including the hard drive and memory (and ignoring the CPU’s internal state), is then roughly ~512×223 + 60×233 = 519,691,042,816 b. How many states would be needed to write out the machine’s entire transition function graph? 2519,691,042,816 = A number having >1.7 trillion decimal digits!

One Problem with FSMs as Models The FSM diagram of a reasonably-sized computer is more than astronomically huge. Yet, we are able to design and build these computers using only a modest amount of industrial resources. Why is this possible? Answer: A real computer has regularities in its transition function that are not captured if we just write out its FSM transition function explicitly. I.e., a transition function can have a small, simple, regular description, even if its domain is enormous.

Other Problems with FSM Model It ignores many important physical realities: How is the transition function’s structure to be encoded in physical hardware? How much hardware complexity is required to do this? How close in physical space is one bit’s worth of the machine’s information capacity to another? How long does it take to communicate information from one part of the machine to another? How much energy gets dissipated to heat when the machine updates its state? How fast can the heat be removed, and how much does this impact the machine’s performance? Let’s consider a basic example.

Vending Machine Example Suppose a certain vending machine accepts nickels, dimes, and quarters. If >30¢ is deposited, change is immediately returned. If the “coke” button is pressed, the machine drops a coke. Can then accept a new payment. Ignore any other buttons, bills, out of change, etc.

Modeling the Machine Input symbol set: I = {nickel, dime, quarter, button} We could add “nothing” or  as an additional input symbol if we want. Representing “no input at a given time.” Output symbol set: O = {, 5¢, 10¢, 15¢, 20¢, 25¢, coke}. State set: S = {0, 5, 10, 15, 20, 25, 30}. Representing how much money has been taken.

Transition Function Table Old state Input New state Output n 5  d 10 q 25 b 15 30 Old state Input New state Output 10 n 15  d 20 q 30 5¢ b 25 10¢ 5

Transition Function Table cont. Old state Input New state Output 20 n 25  d 30 q 15¢ b 5¢ 20¢ Old state Input New state Output 30 n 5¢ d 10¢ q 25¢ b coke

Another Format: State Table Each entry shows new state, output symbol Old state Input Symbol n d q b 5, 10, 25, 0, 5 15, 30, 10 20, 30,5¢ 15 30,10¢ 20 30,15¢ 25 30,20¢ 30 30,25¢ 0,coke

Directed-Graph State Diagram As you can see, these can get kind of busy. q,5¢ d,5¢ q q q,20¢ d d d n n n n n n 5 10 15 20 25 30 n,5¢ b b b b b b d,10¢ q,15¢ q,25¢ b,coke q,10¢

Formalizing FSMs Just like the general transition-function definition from earlier, but with the output function separated from the transition function, and with the various sets added in, along with an initial state. A finite-state machine M=(S, I, O, f, g, s0) S is the state set. I is the alphabet (vocabulary) of input symbols O is the alphabet (vocabulary) of output symbols f is the state transition function g is the output function s0 is the initial state. Our transition function from before is T = (f,g).

Construct a state table for the finite-state machine in Fig. 3. Find the output string for the input 101011 Answer: 001000