Thompson's construction

In computer science, Thompson's construction is an algorithm for transforming a regular expression into an equivalent nondeterministic finite automaton (NFA). This NFA can be used to match strings against the regular expression.

Regular expression and nondeterministic finite automaton are two abstract representation of formal languages. While regular expressions are used e.g. to describe advanced search patterns in "find and replace"-like operations of text processing utilities, the NFA format is better suited for execution on a computer. Hence, this algorithm is of practical interest, since it can be considered as a compiler from regular expression to NFA. On a more theoretical point of view, this algorithm is a part of the proof that they both accept exactly the same languages, that is, the regular languages.

A thus obtained automaton can be made deterministic by the powerset construction and then be minimized to get an optimal automaton corresponding to the given regular expression, but it may also be used directly.

The algorithm

The algorithm works recursively by splitting an expression into its constituent subexpressions, from which the NFA will be constructed using a set of rules.^[1] More precisely, from a regular expression E, the obtained automaton A with the transition function δ respects the following properties:

• A has exactly one initial state q₀, which is not accessible from any other state. That is, for any state q and any letter a, $\delta(q,a)$ does not contain q₀.
• A has exactly one final state q_f, which is not co-accessible from any other state. That is, for any letter a, $\delta(q_f,a)=\emptyset$.
• Let c be the number of concatenation of the regular expression E and let s be the number of symbols apart from parentheses — that is, |, *, a and ε. Then, the number of states of A is 2s − c (linear in the size of E).
• The number of transitions leaving any state is at most two.
• Since an NFA of m states and at most e transitions from each state can match a string of length n in time O(emn), a Thompson NFA can do pattern matching in linear time, assuming a fixed-size alphabet.^[2]

Rules

The following rules are depicted according to Aho et al. (1986),^[3] p. 122. N(s) and N(t) is the NFA of the subexpression s and t, respectively.

The empty-expression ε is converted to

A symbol a of the input alphabet is converted to

The union expression s|t is converted to

State q goes via ε either to the initial state of N(s) or N(t). Their final states become intermediate states of the whole NFA and merge via two ε-transitions into the final state of the NFA.

The concatenation expression st is converted to

The initial state of N(s) is the initial state of the whole NFA. The final state of N(s) becomes the initial state of N(t). The final state of N(t) is the final state of the whole NFA.

The Kleene star expression s^* is converted to

An ε-transition connects initial and final state of the NFA with the sub-NFA N(s) in between. Another ε-transition from the inner final to the inner initial state of N(s) allows for repetition of expression s according to the star operator.

• The parenthesized expression (s) is converted to N(s) itself.

Example

Two examples are now given, a small informal one with the result, and a bigger with a step by step application of the algorithm.

Small Example

The picture show the result of Thompson's construction on (ε|a*b). The pink oval corresponds to a, the teal oval corresponds to a*, the green oval corresponds to b, the orange oval corresponds to a*b and the blue oval corresponds to ε.

Application of the algorithm

NFA obtained from regular expression (0|(1(01*(00)*0)*1)*)*

As an example, the picture shows the result of Thompson's construction algorithm on the regular expression (0|(1(01*(00)*0)*1)*)* that denotes the set of binary numbers that are multiples of 3: { ε, "0", "00", "11", "000", "011", "110", "0000", "0011", "0110", "1001", "1100", "1111", "00000", ... }.

The upper right part shows the logical structure (syntax tree) of the expression, with "." denoting concatenation (assumed to have variable arity); subexpressions are named a-q for reference purposes. The left part shows the nondeterministic finite automaton resulting from Thompson's algorithm, with the entry and exit state of each subexpression colored in magenta and cyan, respectively. An ε as transition label is omitted for clarity — unlabelled transitions are in fact ε transitions. The entry and exit state corresponding to the root expression q is the start and accept state of the automaton, respectively.

The algorithm's steps are as follows:

q:	start converting Kleene star expression									(0|(1(01*(00)*0)*1)*)*
	b:	start converting union expression								0|(1(01*(00)*0)*1)*
		a:	convert symbol							0
		p:	start converting Kleene star expression							(1(01*(00)*0)*1)*
			d:	start converting concatenation expression						1(01*(00)*0)*1
				c:	convert symbol					1
				n:	start converting Kleene star expression					(01*(00)*0)*
					f:	start converting concatenation expression				01*(00)*0
						e:	convert symbol			0
						h:	start converting Kleene star expression			1*
							g:	convert symbol		1
						h:	finished converting Kleene star expression			1*
						l:	start converting Kleene star expression			(00)*
							j:	start converting concatenation expression		00
								i:	convert symbol	0
								k:	convert symbol	0
							j:	finished converting concatenation expression		00
						l:	finished converting Kleene star expression			(00)*
						m:	convert symbol			0
					f:	finished converting concatenation expression				01*(00)*0
				n:	finished converting Kleene star expression					(01*(00)*0)*
				o:	convert symbol					1
			d:	finished converting concatenation expression						1(01*(00)*0)*1
		p:	finished converting Kleene star expression							(1(01*(00)*0)*1)*
	b:	finished converting union expression								0|(1(01*(00)*0)*1)*
q:	finished converting Kleene star expression									(0|(1(01*(00)*0)*1)*)*

An equivalent minimal deterministic automaton is shown below.

Relation to other algorithms

Thompson's is one of several algorithms for constructing NFAs from regular expressions;^[4] an earlier algorithm was given by McNaughton and Yamada.^[5] Converse to Thompson's construction, Kleene's algorithm transforms a finite automaton into a regular expression.

Glushkov's construction algorithm is similar to Thompson's construction, once the ε-transitions are removed.

References

1. Ken Thompson (Jun 1968). "Programming Techniques: Regular expression search algorithm". Communications of the ACM 11 (6): 419–422. doi:10.1145/363347.363387.
2. Xing, Guangming. "Minimized Thompson NFA" (PDF).
3. Alfred V. Aho, Ravi Sethi, Jeffrey Ullman: Compilers: Principles, Techniques and Tools. Addison Wesley, 1986
4. Watson, Bruce W. (1995). A taxonomy of finite automata construction algorithms (PDF) (Technical report). Eindhoven University of Technology. Computing Science Report 93/43.
5. R. McNaughton, H. Yamada (Mar 1960). "Regular Expressions and State Graphs for Automata". IEEE Trans. on Electronic Computers 9 (1): 39–47. doi:10.1109/TEC.1960.5221603.