To simplify the process for identifying 12 types of symmetric variables in the canonical OR-coincidence(COC) algebra system, we propose a new symmetry detection algorithm based on OR-NXOR expansion. By analyzing the relationships between the coefficient matrices of sub-functions and the order coefficient subset matrices based on OR-NXOR expansion around two arbitrary logical variables, the constraint conditions of the order coefficient subset matrices are revealed for 12 types of symmetric variables. Based on the proposed constraints, the algorithm is realized by judging the order characteristic square value matrices. The proposed method avoids the transformation process from OR-NXOR expansion to AND-OR-NOT expansion, or to AND-XOR expansion, and solves the problem of completeness in the dj-map method. The application results show that, compared with traditional methods, the new algorithm is an optimal detection method in terms of applicability of the number of logical variables, detection type, and complexity of the identification process. The algorithm has been implemented in C language and tested on MCNC91 benchmarks. Experimental results show that the proposed algorithm is convenient and efficient.
To simplify the process for identifying 12 types of symmetric variables in Boolean functions, we propose a new symmetry detection algorithm based on minterm expansion or the truth table. First, the order eigenvalue matrix based on a truth table is defined according to the symmetry definition of a logic variable. By analyzing the constraint conditions of the order eigenvalue matrix for 12 types of symmetric variables, an algorithm is proposed for identifying symmetric variables of the Boolean function. This algorithm can be applied to identify the symmetric variables of Boolean functions with or without don't-care terms. The proposed method avoids the restriction by the number of logic variables of the graphical method, spectral coefficient methods, and AND-XOR expansion coefficient methods, and solves the problem of completeness in the fast computation method. The algorithm has been implemented in C language and tested on MCNC91 benchmarks. The application results show that, compared with the traditional methods, the new algorithm is an optimal detection method in terms of the applicability of the number of logic variables, the Boolean function including don't-care terms, detection type, and complexity of the identification process.
Compared with complementary metal–oxide semiconductor(CMOS), the resonant tunneling device(RTD) has better performances; it is the most promising candidate for next-generation integrated circuit devices. The universal logic gate is an important unit circuit because of its powerful logic function, but there are few function synthesis algorithms that can implement an n-variable logical function by RTD-based universal logic gates. In this paper, we propose a new concept, i.e., the truth value matrix. With it a novel disjunctive decomposition algorithm can be used to decompose an arbitrary n-variable logical function into three-variable subset functions. On this basis, a novel function synthesis algorithm is proposed, which can implement arbitrary n-variable logical functions by RTD-based universal threshold logic gates(UTLGs), RTD-based three-variable XOR gates(XOR3s), and RTD-based three-variable universal logic gate(ULG3s). When this proposed function synthesis algorithm is used to implement an n-variable logical function, if the function is a directly disjunctive decomposition one, the circuit structure will be very simple, and if the function is a non-directly disjunctive decomposition one, the circuit structure will be simpler than when using only UTLGs or ULG3s. The proposed function synthesis algorithm is straightforward to program, and with this algorithm it is convenient to implement an arbitrary n-variable logical function by RTD-based universal logic gates.
