13-Nov-1513-Nov-1513-Nov-15 State Machines

What is a state machine? A state machine is a different way of thinking about computation A state machine has some number of states, and transitions between those states Transitions occur because of inputs A “pure” state machine only knows which state it is in—it has no other memory or knowledge This is the kind of state machine you learn about in your math classes When you program a state machine, you don’t have that restriction 2

State machine I/O State machines are designed to respond to a sequence of inputs, such as The individual characters in a string A series of external events State machines may produce output (often as a result of transitions) Alternatively, the only “result” of a state machine may be the state it ends up in 3

Example I: Even or odd The following machine determines whether the number of A s in a string is even or odd Circles represent states; arrows represent transitions 4 Inputs are the characters of a string The “output” is the resultant state The double circle represents a “final” (accepting) state A evenodd start A anything but A

Error states Again, a state machine is a way of doing certain kinds of computations The input is a sequence of values (typically, a String) Some inputs may be illegal (for example, syntax errors in a program) A state machine is used to recognize certain kinds of inputs We say the machine succeeds if it recognizes its input, otherwise it fails Some states may be marked as final states (they are drawn with concentric circles) A state machine succeeds if: It is in a final state when it reaches the end of its input A state machine fails if: It encounters an input for which it has no defined transition It reaches the end of its input, but is not in a final state State machines may have a error state with the following characteristics: The error state is not a final state An unexpected input will cause a transition to the error state All subsequent inputs cause the state machine to remain in the error state 5

Nondeterministic state machines There are two ways in which a state machine may be nondeterministic: There may be two or more transitions from a state for the same input—either arrow may be followed There may be an “empty transition” (denoted by the Greek letter ε) from a state—the transition might or might not be taken A nondeterministic machine is said to accept (recognize) an input if there is any way for it to get to a final state, given that input 6

Simplifying drawings I State machines can get pretty complicated The formal, mathematical definition of a state machine requires it to have a transition from every state for every possible input To satisfy this requirement, we often need an error state, so we can have transitions for illegal (unrecognized) inputs When we draw a state machine, we don’t need to draw the error state--we can just assume it’s there The error state is still part of the machine Any input without a transition on our drawing is assumed to go to the error state Another simplification: Use * to indicate “all other characters” This is a convention when drawing the machine—it does not mean we look for an asterisk in the input 7

8 Example II: Nested parenthesis The following example tests whether parentheses are properly nested (up to 3 deep) How can we extend this machine to handle arbitrarily deep nesting? start ) ( ) ( ) ( OK Error ) **** ( *

9 Nested parentheses II Question: How can we use a state machine to check parenthesis nesting to any depth? Answer: We can’t (with a finite number of states) We need to count how deep we are into a parenthesis nest: 1, 2, 3,..., 821,... The only memory a state machine has is which state it is in However, if we aren’t required to use a pure state machine, we can add memory (such as a counter) and other features

10 Nested parentheses III This machine is based on a state machine, but it obviously is not just a state machine OK ( do count=1 ) and count==1 do count=0 ( do count++ ) and count>1 do count-- start

11 The states of a Thread A Thread is an object that represents a single flow of execution through a program A Thread’s lifetime can be described by a state machine ready waiting runningdead start

12 Example: Making numbers bold In HTML, you indicate boldface by surrounding the characters with... Suppose we want to make all the integers bold in an HTML page—we can write a state machine to do this NORMALNUMBER digit output digit nondigit output nondigit *: output * end of input output start digit output digit end

13 State machines in Java In a state machine, you can have transitions from any state to any other state This is difficult to implement with Java’s loops and if statements The trick is to make the “state” a variable, and to embed a switch ( state ) statement inside a loop Each case is responsible for resetting the “state” variable as needed to represent transitions

14 Outline of the bold program void run() { int state = NORMAL; for (int i = 0; i < testString.length(); i++) { char ch = testString.charAt(i); switch (state) { case NORMAL: { not inside a number } case NUMBER: { inside a number } } if (state == NUMBER) result.append(" ");

15 The two states case NORMAL: if (Character.isDigit(ch)) { result.append(" " + ch); state = NUMBER; break; } else { result.append(ch); } break; case NUMBER: if (!Character.isDigit(ch)) { result.append(" " + ch); state = NORMAL; break; } else { result.append(ch); } break;

16 Conclusions A state machine is a good model for a number of problems You can think of the problem in terms of a state machine but not actually do it that way You can implement the problem as a state machine (e.g. making integers bold) Best done as a switch inside some kind of loop Pure state machines have some severe limitations Java lets you do all kinds of additional tests and actions; you can ignore these limitations

17 The End