Xiaoxue Piao

Operational state complexity of nested word automata

We introduce techniques to prove lower bounds for the number of states needed by finite automata operating on nested words. We study the state complexity of Boolean operations and obtain lower bounds that are tight within an additive constant. The results for union and complementation differ from corresponding bounds for ordinary finite automata. For reversal and concatenation, we establish lower bounds that are of a different order than the worst-case bounds for ordinary finite automata.

State complexity of kleene-star operations on trees

The concatenation of trees can be defined either as a sequential or a parallel operation, and the corresponding iterated operation gives an extension of Kleene-star to tree languages. Since the sequential tree concatenation is not associative, we get two essentially different iterated sequential concatenation operations that we call the bottom-up star and top-down star operation, respectively. We establish that the worst-case state complexity of bottom-up star is

Descriptional Complexity of Input-Driven Pushdown Automata

(n + \frac{3}{2}) \cdot 2^{n-1}

State trade-offs in unranked tree automata

. The bound differs by an order of magnitude from the corresponding result for string languages. The state complexity of top-down star is similar as in the string case. The iteration of the parallel concatenation has to be defined slightly differently in order to yield a regularity preserving operation.

State complexity of the concatenation of regular tree languages

It is known that a nondeterministic input-driven pushdown automaton (IDPDA) can be determinized. Alur and Madhusudan (“Adding nesting structure to words”, J.ACM 56(3), 2009) showed that a deterministic IDPDA simulating a nondeterministic IDPDA with n states and stack symbols may need, in the worst case, \(2^{\Omega(n^2)}\) states. In their construction, the equivalent deterministic IDPDA does, in fact, not need to use the stack. This paper considers the size blow-up of determinization in more detail, and gives a lower bound construction, that is tight within a multiplicative constant, with respect to the size of the nondeterministic automaton both for the number of states and the number of stack symbols. The paper also surveys the recent results on operational state complexity of IDPDAs, and on the cost of converting a nondeterministic automaton to an unambiguous one, and an unambiguous automaton to a deterministic one.

A prediction-based adaptive grouping differential evolution algorithm for constrained numerical optimization

A common definition of tree automata operating on unranked trees uses a set of vertical states that define the bottom-up computation, and the transitions on vertical states are determined by so called horizontal languages recognized by finite automata on strings. It is known that, in this model, a deterministic tree automaton with the smallest total number of states (that is, vertical states and states used for automata to define the horizontal languages) does not need to be unique nor have the smallest possible number of vertical states. We consider the question by how much we can reduce the total number states by introducing additional vertical states. We give an upper bound for the state trade-off for deterministic tree automata where the horizontal languages are defined by DFAs (deterministic finite automata). Also, we give a lower bound construction that reduces the number of horizontal states, roughly, from 4n to 8n by doubling the number of vertical states. The lower bound is close to the worst-case upper bound in the case where the number of vertical states is multiplied by a constant. We show that deterministic tree automata where the horizontal languages are specified by NFAs (nondeterministic finite automata) can have no trade-offs between the numbers of vertical states and horizontal states, respectively. We study corresponding trade-offs also for nondeterministic tree automata.

Operational State Complexity of Deterministic Unranked Tree Automata

We consider the state complexity of basic concatenation operations for regular tree languages. We show that the sequential (respectively, parallel) concatenation of tree languages recognized by deterministic bottom-up automata with m and n states can be recognized by an automaton with (n+1)@?(m@?2^n+2^n^-^1)-1 (respectively, m@?2^n+2^n^-^1-1) states, and establish matching state complexity lower bounds. The bound for sequential concatenation of tree languages differs by an order of magnitude from the corresponding bound for regular string languages.

Lower bounds for the size of deterministic unranked tree automata

In this paper, a new adaptive grouping differential evolution (AGDE) algorithm is proposed to improve the optimization performance by implementing a prediction strategy of the constraints for constrained optimization problems. It is unnecessary to calculate the constraint values when dealing with the constraints in this method. The constraints are handled after a simple prediction according to the Lipschitz condition. When the constraints are very complex, the load arisen from the calculation of the constraint values is reduced dramatically and the feasibility of the solutions remains with great probability. In AGDE algorithm, the population is dynamically grouped to three subpopulations with respective newly-designed mutation strategy. Meanwhile, the mutation factor and crossover probability are adopted associated with the evolutionary process according to the information of the entire population. Both of the above improvements not only increase the diversity of population and speed up the convergence, but also reduce the complexity of the parameter selection. Four sets of comparative experiments are carried out to evaluate the feasibility and effectiveness of the proposed method that deals with the constraints.

State complexity of projection and quotient on unranked trees

We consider the state complexity of basic operations on tree languages recognized by deterministic unranked tree automata. For the operations of union and intersection the upper and lower bounds of both weakly and strongly deterministic tree automata are obtained. For tree concatenation we establish a tight upper bound that is of a different order than the known state complexity of concatenation of regular string languages. We show that (n+1) ( (m+1)2^n-2^(n-1) )-1 vertical states are sufficient, and necessary in the worst case, to recognize the concatenation of tree languages recognized by (strongly or weakly) deterministic automata with, respectively, m and n vertical states.

State Complexity of Kleene-Star Operations on Regular Tree Languages

Tree automata operating on unranked trees use regular languages, called horizontal languages, to define the transitions of the vertical states that define the bottom-up computation of the automaton. It is well known that the deterministic tree automaton with smallest total number of states, that is, number of vertical states and number of states used to define the horizontal languages, is not unique and it is hard to establish lower bounds for the total number of states. By relying on existing bounds for the size of unambiguous finite automata, we give a lower bound for the size blow-up of determinizing a nondeterministic unranked tree automaton. The lower bound improves the earlier known lower bound that was based on an ad hoc construction.