3.7.24. Automaton with same input symbol

A constraint for which the catalogue provides an automaton belonging to the following category:

  • Symbols of the alphabet are split in two categories: neutral ones and non-neutral ones.

  • Non-neutral symbols correspond to symbols occurring on transitions between two distinct states, while neutral symbols correspond to all the other symbols of the alphabet.

  • Self-loops labelled by a neutral symbol do not modify any counter.

  • Ignoring transitions labelled by neutral symbols, every state has its incoming transitions labelled by the same non-neutral symbol.

  • Ignoring transitions labelled by neutral symbols, outgoing transitions of a state are not labelled by the symbol associated with its incoming non-loop transitions.

For such automata we define the semantics of a state s as the regular expression associated with the language fragment obtained from entering state s to just before leaving state s.

Figure 3.7.6. Semantics of the states of the automaton of the πšŸπšŠπš•πš•πšŽπš’ constraint
ctrs/preface-146-tikz

As an example, consider the πšŸπšŠπš•πš•πšŽπš’ constraint and its automaton depicted by FigureΒ 3.7.6. The alphabet π’œ corresponds to the set of symbols {<,=,>} from which < and > are non-neutral symbols (i.e.,Β the symbols associated with the transitions between states s and u), and = is a neutral symbol. First there is no counter modification on all self-loops. If we remove the self-loops carrying the neutral symbol = we have that:

  • All incoming transitions in state s are labelled by the non-symbol <, and all outgoing transitions from state s are not labelled by <.

  • All incoming transitions in state u are labelled by the non-symbol >, and all outgoing transitions from state u are not labelled by >.

The corresponding state semantics is given by the upper-leftmost box.