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FINITE AUTOMATA

Regular expressions are convenient for specifying lexical tokens, but we need a formalism that can be implemented as a computer program. For this we can use finite automata (N.B. the singular of automata is automaton). A finite automaton has a finite set of states; edges lead from one state to another, and each edge is labeled with a symbol. One state is the start state, and certain of the states are distinguished as final states. Image 2.3 shows some finite automata. We number the states just for convenience in discussion. The start state is numbered 1 in each case. An edge labeled with several characters is shorthand for many parallel edges; so in the ID machine there are really 26 edges each leading from state 1 to 2, each labeled by a different letter.

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Image 2.3: Finite automata for lexical tokens. The states are indicated by circles; final states are indicated by double circles. The start state has an arrow coming in from nowhere. An edge labeled with several characters is shorthand for many parallel edges.

In a deterministic finite automaton (DFA), no two edges leaving from the same state are labeled with the same symbol. A DFA accepts or rejects a string as follows. Starting in the start state, for each character in the input string the automaton follows exactly one edge to get to the next state. The edge must be labeled with the input character. After making n transitions for an n-character string, if the automaton is in a final state, then it accepts the string. If it is not in a final state, or if at some point there was no appropriately labeled edge to follow, it rejects. The language recognized by an automaton is the set of strings that it accepts. For example, it is clear that any string in the language recognized by automaton ID must begin with a letter. Any single letter leads to state 2, which is final; so a single-letter string is accepted. From state 2, any letter or digit leads back to state 2, so a letter followed by any number of letters and digits is also accepted. In fact, the machines shown in Image 2.3 accept the same languages as the regular expressions of Image 2.2. These are six separate automata; how can they be combined into a single machine that can serve as a lexical analyzer? We will study formal ways of doing this in the next section, but here we will just do it ad hoc: Image 2.4 shows such a machine. Each final state must be labeled with the token-type that it accepts. State 2 in this machine has aspects of state 2 of the IF machine and state 2 of the ID machine; since the latter is final, then the combined state must be final. State 3 is like state 3 of the IF machine and state 2 of the ID machine; because these are both final we use rule priority to disambiguate - we label state 3 with IF because we want this token to be recognized as a reserved word, not an identifier.

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Image 2.4: Combined finite automaton.

We can encode this machine as a transition matrix: a two-dimensional array (a vector of vectors), subscripted by state number and input character. There will be a "dead" state (state 0) that loops to itself on all characters; we use this to encode the absence of an edge.

int edges[][] = { /* ...012...-...e f g h i j... */
/* state 0 */ {0,0,...0,0,0...0...0,0,0,0,0,0...},
/* state 1 */ {0,0,...7,7,7...9...4,4,4,4,2,4...},
/* state 2 */ {0,0,...4,4,4...0...4,3,4,4,4,4...},
/* state 3 */ {0,0,...4,4,4...0...4,4,4,4,4,4...},
/* state 4 */ {0,0,...4,4,4...0...4,4,4,4,4,4...},
/* state 5 */ {0,0,...6,6,6...0...0,0,0,0,0,0...},
/* state 6 */ {0,0,...6,6,6...0...0,0,0,0,0,0...},
/* state 7 */ {0,0,...7,7,7...0...0,0,0,0,0,0...},
/* state 8 */ {0,0,...8,8,8...0...0,0,0,0,0,0...},
 et cetera
}


There must also be a "finality" array, mapping state numbers to actions - final state 2 maps to action ID, and so on.

RECOGNIZING THE LONGEST MATCH

It is easy to see how to use this table to recognize whether to accept or reject a string, but the job of a lexical analyzer is to find the longest match, the longest initial substring of the input that is a valid token. While interpreting transitions, the lexer must keep track of the longest match seen so far, and the position of that match. Keeping track of the longest match just means remembering the last time the automaton was in a final state with two variables, Last-Final (the state number of the most recent final state encountered) and Input-Position-at-Last-Final. Every time a final state is entered, the lexer updates these variables; when a dead state (a nonfinal state with no output transitions) is reached, the variables tell what token was matched, and where it ended. Image 2.5 shows the operation of a lexical analyzer that recognizes longest matches; note that the current input position may be far beyond the most recent position at which the recognizer was in a final state.

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Image 2.5: The automaton of Image 2.4 recognizes several tokens. The symbol | indicates the input position at each successive call to the lexical analyzer, the symbol ⊥ indicates the current position of the automaton, and ⊺ indicates the most recent position in which the recognizer was in a final state.
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