Ruqian Lu

An Optimized Majority Logic Synthesis Methodology for Quantum-Dot Cellular Automata

Quantum-dot cellular automata (QCA) has been widely considered as a replacement candidate for complementary metal-oxide semiconductor (CMOS). The fundamental logic device in QCA is the majority gate. In this paper, we propose an efficient methodology for majority logic synthesis of arbitrary Boolean functions. We prove that our method provides a minimal majority expression and an optimal QCA layout for any given three-variable Boolean function. In order to obtain high-quality decomposed Boolean networks, we introduce a new decomposition scheme that can decompose all Boolean networks efficiently. Furthermore, our method removes all the redundancies that are produced in the process of converting a decomposed network into a majority network. In existing methods, however, these redundancies are not considered. We have built a majority logic synthesis tool based on our method and several existing logic synthesis tools. Experiments with 40 multiple-output benchmarks indicate that, compared to existing methods, 37 benchmarks are optimized by our method, up to 31.6%, 78.2%, 75.5%, and 83.3% reduction in level count, gate count, gate input count, and inverter count, respectively, is possible with the average being 4.7%, 14.5%, 13.3%, and 26.4%, respectively. We have also implemented the QCA layouts of 10 benchmarks by using our method. Results indicate that, compared to existing methods, up to 33.3%, 76.7%, and 75.5% reduction in delay, cell count, and area, respectively, is possible with the average being 8.1%, 28.9%, and 29.0%, respectively.

The use of tail inequalities on the probable computational time of randomized search heuristics

For the purpose of analyzing the time cost of evolutionary algorithms (EAs) or other types of randomized search heuristics (RSHs) with certain requirements on the probability of obtaining a target solution, this paper proposes a new index, called the probable computational time (PCT), which complements expected running time analysis. Using simple tail inequalities, such as Markovs inequality and Chebyshevs inequality, we also provide basic properties of PCT, explicitly exhibiting the general relationships between the expected running time and the PCT. To present deeper estimations of the PCT for specific RSHs and problems, we demonstrate a new inequality that is based on the general form of the Chernoff inequality and previous methods such as fitness-based partitions and potential functions, which have been used to analyze the expected running time of RSHs. The precondition of the new inequality is that the total running time can be described as the sum of a linear combination of some independent geometrically distributed variables and a constant term. The new inequality always provides meaningful upper bounds for the PCT under such circumstances. Some applications of the new inequality for simple EAs, ant colony optimization (ACO) and particle swarm optimization (PSO) algorithms on simple pseudo-Boolean functions are illustrated in this paper.

Counter designs in quantum-dot cellular automata

In this paper, we present a new method for the design of an n-bit synchronous binary up counter in quantum-dot cellular automata (QCA). This method is based on the JK flip-flop which almost always produces the simplest combinational logic in traditional sequential circuits. We implement a new QCA architecture for the JK flip-flop. Compared to the existing QCA JK flip-flop, the majority gate count, cell count, clock cycle count, and area of our QCA JK flip-flop are reduced by 57.1%, 88.1%, 55.6%, and 92.0%, respectively. Based on our QCA JK flip-flop, a method of extending state cells is proposed to design the QCA layout of the n-bit counter, such that all the clock cycle counts between any two state cells become 1. This feature can ensure that one count just takes one clock cycle, in existing methods, however, one count needs to take n−1 clock cycles. Comparisons indicate that, by applying our method, the hardware requirements (i.e., complexity and area) for QCA n-bit counter can be greatly reduced.

A theory of computation based on unsharp quantum logic: Finite state automata and pushdown automata

When generalizing the projection-valued measurements to the positive operator-valued measurements, the notion of the quantum logic generalizes from the sharp quantum logic to the unsharp quantum logic. It is known that: (i) the distributive law is one of the main differences between the sharp quantum logic and the boolean logic, and the block or the center of the sharp quantum structures are boolean algebras; (ii) the unsharp quantum logic does not satisfy the non-contradiction law, which forces the block or the center of unsharp quantum structures to be multiple valued algebras, rather than boolean algebras. Multiple valued algebras, as special quantum structures, are the algebraic semantics of multiple valued logic. Interestingly, we recently discovered that the difference between some unsharp quantum structures and multiple valued algebras is also some kind of distributive law. Choosing an orthomodular lattice (an algebraic model of a sharp quantum logic) to be the truth valued lattice, Ying et al. have systematically developed automata theory based on sharp quantum logic. In this paper, choosing a lattice ordered quantum multiple valued algebra E (an extended lattice ordered effect algebra E, respectively) to be the truth valued lattice, we also systematically develop an automata theory based on unsharp quantum logic. We introduce E-valued finite-state automata and E-valued pushdown automata in the framework of unsharp quantum logic. We study the classes of languages accepted by these automata and re-examine their various properties in the framework of unsharp quantum logic. The study includes the equivalence between finite-state automata and regular expressions, as well as the equivalence between pushdown automata and context-free grammars. It is also demonstrated that the universal validity of some important properties (such as some closure properties of languages and Kleene theorem etc.) depends heavily on the aforementioned distributive law. More precisely, when the underlying model degenerates into an MV algebra, then all the counterparts of properties in classical automata are valid. This is the main difference between automata theory based on unsharp quantum logic and automata theory based on sharp quantum logic.

Automata theory based on unsharp quantum logic

By studying two unsharp quantum structures, namely extended lattice ordered effect algebras and lattice ordered QMV algebras, we obtain some characteristic theorems of MV algebras. We go on to discuss automata theory based on these two unsharp quantum structures. In particular, we prove that an extended lattice ordered effect algebra (or a lattice ordered QMV algebra) is an MV algebra if and only if a certain kind of distributive law holds for the sum operation. We introduce the notions of (quantum) finite automata based on these two unsharp quantum structures, and discuss closure properties of languages and the subset construction of automata. We show that the universal validity of some important properties (such as sum, concatenation and subset constructions) depend heavily on the above distributive law. These generalise results about automata theory based on sharp quantum logic.

Semirings and pseudo MV algebras

In this paper, we describe the relationships between pseudo MV algebras and semirings. We also give definitions of automata on lattice ordered semirings, prove that the family of K-Languages is closed under union, and discuss the conditions for the closedness of families of K-languages under intersection, generalized intersection and reversal operations.

Automata theory based on lattice-ordered semirings

In this paper, definitions of

Weak QMV algebras and some ring-like structures

{mathcal{K}}