A General Approach to State Complexity of Operations: Formalization and Limitations

Abstract

The state complexity of the result of a regular operation is often positively correlated with the number of distinct transformations induced by letters in the minimal deterministic finite automaton of the input languages. That is, more transformations in the inputs means higher state complexity in the output. When this correlation holds, the state complexity of a unary operation can be maximized using languages in which there is one letter corresponding to each possible transformation; for operations of higher arity, we can use m-tuples of languages in which there is one letter corresponding to each possible m-tuple of transformations. In this way, a small set of languages can be used as witnesses for many common regular operations, eliminating the need to search for witnesses -though at the expense of using very large alphabets. We formalize this approach and examine its limitations. We define a class of "uniform" operations for which this approach works; the class is closed under composition and includes common operations such as star, concatenation, reversal, union, and complement. Our main result is that the worst-case state complexity of a uniform operation can be determined by considering a finite set of witnesses, and this set depends only on the arity of the operation and the state complexities of the inputs.

Algebraic and Combinatorial Tools for State Complexity : Application to the Star-Xor Problem

“…for every regular language L and every 1-uniform morphism φ, ⊗(φ −1 (L)) = φ −1 (⊗(L)). For example, it is proven in [8] that the Kleene star and the reverse are 1-uniform. Suppose now that ⊗ is a unary 1-uniform operation.…”

“…Witnesses with small alphabets are indeed favoured in this area of research when they can be found, as evidenced by several studies ( [5,6]). This approach has been described independently by Caron et al in [3] as the monster approach and by Davies in [8] as the OLPA (One Letter Per Action) approach but was implicitly present in older papers like [2,10].…”

A combinatorial approach for the state complexity of the Shuffle product

“…Next, to reach any valid state in the most easier way, it is relevant to consider automata K and L having a maximum of transitions. This is the idea of monsters detailed in [4,6] and formalized in the next section. Brzozowski et al implicitly use this notion to prove the result for any n when m ≤ 5.…”

“…In some cases, the classical method has to be enhanced by an algebraic approach consisting in building a witness for a certain class of rational operations by searching in a set of automata with as many transition functions as possible. This method has the advantage of being applied to a large class of operations and has been described independently by Caron et al in [4] as the monster approach and by Davies in [6] as the OLPA approach but was implicitly present in older papers like [2], [7].…”

New tools for state complexity

“…The authors learned that Sylvie Davies has independently and in the same time obtained some of the results presented in this paper. Her work can be found in [7].…”