Octic reciprocity

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Thompson's Construction Algorithm (TCA) derives a nondeterministic finite automaton (NFA) from any regular expression by splitting it into its constituent subexpressions, from which the NFA will be constructed using a set of rules.

It was invented by Ken Thompson.

Rules

The following rules are depicted according to Aho et. al (1986), p. 122. $N(s)$ and $N(t)$ are NFAs of the subexpressions $s$ and $t$ respectively.

The expression $\varepsilon$ 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 $\varepsilon$ 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 $\varepsilon$-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 $\varepsilon$-transition connects initial and final state of the NFA with the sub-NFA $N(s)$ in between. Another $\varepsilon$-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.

References

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