Nicola Cotumaccio

Nicola Cotumaccio

Over the years, bisimulations have emerged as a pervasive paradigm, finding applications in numerous areas, including concurrency theory, model checking, automata theory, logic, programming languages and category theory. In this paper, we establish a connection between bisimulations and data compression. More precisely, we study the relationship between bisimulations and Wheeler automata (Alanko et al., SODA 2020), a class of automata that has received considerable attention in recent years. The standard notion of bisimulation is not appropriate, so we introduce Wheeler bisimulations, that is, bisimulations that respect the convex structure of the considered Wheeler automata. We show that Wheeler bisimilarity induces a unique minimal Wheeler NFA (analogously to standard bisimulations). In particular, in the deterministic case, we retrieve the minimal Wheeler deterministic automaton of a given language. We also show that the minimal Wheeler NFA induced by Wheeler bisimulations can be built in linear time. This is in contrast with standard bisimulations, for which the corresponding minimal NFA can be built in $ O(m \log n) $ time (where $ m $ is the number of edges and $ n $ is the number of states) by adapting Paige-Tarjan partition refinement algorithm. Compared to previous state-reduction techniques, our bisimulation-induced construction is the first for which (i) we obtain a canonical Wheeler NFA and (ii) the resulting Wheeler NFA can be built in linear time.

Giovanna D'Agostino, Nicola Cotumaccio, Alberto Policriti et al.

The states of a deterministic finite automaton A can be identified with collections of words in Pf(L(A)) -- the set of prefixes of words belonging to the regular language accepted by A. But words can be ordered and among the many possible orders a very natural one is the co-lexicographic one. Such naturalness stems from the fact that it suggests a transfer of the order from words to the automaton's states. In a number of papers automata admitting a total ordering of states coherent with the ordering of the set of words reaching them have been proposed. Such class of ordered automata -- the Wheeler automata -- turned out to be efficiently stored/searched using an index. Unfortunately not all automata can be totally ordered as previously outlined. However, automata can always be partially ordered and an intrinsic measure of their complexity can be defined and effectively determined, as the minimum width of one of their admissible partial orders. As shown in previous works, this new concept of width of an automaton has useful consequences in the fields of graph compression, indexing data structures, and automata theory. In this paper we prove that a canonical, minimum-width, partially-ordered automaton accepting a language L -- dubbed the Hasse automaton H of L -- can be exhibited. H provides, in a precise sense, the best possible way to (partially) order the states of any automaton accepting L, as long as we want to maintain an operational link with the (co-lexicographic) order of Pf(L(A)). Using H we prove that the width of the language can be effectively computed from the minimum automaton recognizing the language. Finally, we explore the relationship between two (often conflicting) objectives: minimizing the width and minimizing the number of states of an automaton.