Yuli Ye

Syntactic complexity of ideal and closed languages

The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn-1 is a tight upper bound on the complexity of right ideals and prefix-closed languages, and that there exist left ideals and suffix-closed languages of syntactic complexity nn-1 + n - 1, and two-sided ideals and factor-closed languages of syntactic complexity nn-2 + (n - 2)2n-2 + 1.

Syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that n^n^-^2 is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be (n-1)^n^-^2+(n-2), (n-1)^n^-^3+(n-2)^n^-^3+(n-3)2^n^-^3, and (n-1)^n^-^3+(n-3)2^n^-^3+1, respectively, and exhibit languages with these syntactic complexities.

Syntactic Complexity of Regular Ideals

The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn−1, nn−1 + n − 1, and nn−2 + (n − 2)2n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced.

COVERING OF TRANSIENT SIMULATION OF FEEDBACK-FREE CIRCUITS BY BINARY ANALYSIS

Transient simulation of a gate circuit is an efficient method of counting signal changes occurring during a transition of the circuit. It is known that this simulation covers the results of classical binary analysis, in the sense that all signal changes appearing in binary analysis are also predicted by the simulation. For feedback-free circuits of 1- and 2-input gates, it had been shown that the converse also holds, if wire delays are taken into account. In this paper we generalize this result. First, we prove that, for any feedback-free circuit N of arbitrary gates, there exists an expanded circuit, constructed by adding a number of delays to each wire of N, such that binary analysis of covers transient simulation of N. For this result, the number of delays added to a wire is obtained from the transient simulation. Our second result involves adding only one delay per wire, which leads to the singular circuit of N. This result is restricted to circuits consisting only of gates realizing functions from the set , functions obtained by complementing any number of inputs and/or the output of a function from , and FORKS. The numbers of inputs of the AND, OR and XOR gates are arbitrary, and all functions of two variables are included. We show that binary analysis of such a circuit covers transient simulation of N. We also show that this result cannot be extended to arbitrary gates, if we allow only a constant number of delays per wire.

ON THE COMPLEXITY OF THE EVALUATION OF TRANSIENT EXTENSIONS OF BOOLEAN FUNCTIONS

Electronic version of an article published as International Journal of Foundations of Computer Science, 23(01), 2012, 21–35. http://dx.doi.org/10.1142/S0129054112400023

Syntactic complexity of Prefix-, Suffix-, and Bifix-free regular languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of prefix-, suffix-, and bifix-free regular languages. We prove that nn-2 is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix- and bifix-free regular languages, and conjecture tight upper bounds on their size.

On the Complexity of the Evaluation of Transient Extensions of Boolean Functions

Transient algebra is a multi-valued algebra for hazard detection in gate circuits. Sequences of alternating 0s and 1s, called transients, represent signal values, and gates are modeled by extensions of boolean functions to transients. Formulas for computing the output transient of a gate from the input transients are known for NOT, AND, OR} and XOR gates and their complements, but, in general, even the problem of deciding whether the length of the output transient exceeds a given bound is NP-complete. We propose a method of evaluating extensions of general boolean functions. We introduce and study a class of functions with the following property: Instead of evaluating an extension of a boolean function on a given set of transients, it is possible to get the same value by using transients derived from the given ones, but having length at most 3. We prove that all functions of three variables, as well as certain other functions, have this property, and can be efficiently evaluated.