Abstract: Formal verification is proposed to ensure the
correctness of the design and make functional verification more
efficient. As cache plays a vital role in the design of System on Chip
(SoC), and cache with Memory Management Unit (MMU) and cache
memory unit makes the state space too large for simulation to verify,
then a formal verification is presented for such system design. In the
paper, a formal model checking verification flow is suggested and a
new cache memory model which is called “exhaustive search model”
is proposed. Instead of using large size ram to denote the whole cache
memory, exhaustive search model employs just two cache blocks. For
cache system contains data cache (Dcache) and instruction cache
(Icache), Dcache memory model and Icache memory model are
established separately using the same mechanism. At last, the novel
model is employed to the verification of a cache which is module of a
custom-built SoC system that has been applied in practical, and the
result shows that the cache system is verified correctly using the
exhaustive search model, and it makes the verification much more
manageable and flexible.
Abstract: Non-negative matrix factorization (NMF) is a useful computational method to find basis information of multivariate nonnegative data. A popular approach to solve the NMF problem is the multiplicative update (MU) algorithm. But, it has some defects. So the columnwisely alternating gradient (cAG) algorithm was proposed. In this paper, we analyze convergence of the cAG algorithm and show advantages over the MU algorithm. The stability of the equilibrium point is used to prove the convergence of the cAG algorithm. A classic model is used to obtain the equilibrium point and the invariant sets are constructed to guarantee the integrity of the stability. Finally, the convergence conditions of the cAG algorithm are obtained, which help reducing the evaluation time and is confirmed in the experiments. By using the same method, the MU algorithm has zero divisor and is convergent at zero has been verified. In addition, the convergence conditions of the MU algorithm at zero are similar to that of the cAG algorithm at non-zero. However, it is meaningless to discuss the convergence at zero, which is not always the result that we want for NMF. Thus, we theoretically illustrate the advantages of the cAG algorithm.