Efficient deterministic finite automata split-minimization derived from Brzozowski's algorithm

Minimization of deterministic finite automata is a classic problem in Computer Science which is still studied nowadays. In this paper, we relate the different split-minimization methods proposed to date, or to be proposed, and the algorithm due to Brzozowski which has been usually set aside in any classification of DFA minimization algorithms. In our work, we first propose a polynomial minimization method derived from a paper by Champarnaud et al. We also show how the consideration of some efficiency improvements on this algorithm lead to obtain an algorithm similar to Hopcroft s classic algorithm. The results obtained lead us to propose a characterization of the set of possible splitters.García Gómez, P.; López Rodríguez, D.; Vázquez-De-Parga Andrade, M. (2014). Efficient deterministic finite automata split-minimization derived from Brzozowski's algorithm. International Journal of Foundations of Computer Science. 25(6):679-696. doi:10.1142/S0129054114500282S679696256Vázquez de Parga, M., García, P., & López, D. (2013). A polynomial double reversal minimization algorithm for deterministic finite automata. Theoretical Computer Science, 487, 17-22. doi:10.1016/j.tcs.2013.03.005Courcelle, B., Niwinski, D., & Podelski, A. (1991). A geometrical view of the determinization and minimization of finite-state automata. Mathematical Systems Theory, 24(1), 117-146. doi:10.1007/bf02090394POLÁK, L. (2005). MINIMALIZATIONS OF NFA USING THE UNIVERSAL AUTOMATON. International Journal of Foundations of Computer Science, 16(05), 999-1010. doi:10.1142/s0129054105003431Gries, D. (1973). Describing an algorithm by Hopcroft. Acta Informatica, 2(2). doi:10.1007/bf00264025Blum, N. (1996). An O(n log n) implementation of the standard method for minimizing n-state finite automata. Information Processing Letters, 57(2), 65-69. doi:10.1016/0020-0190(95)00199-9Knuutila, T. (2001). Re-describing an algorithm by Hopcroft. Theoretical Computer Science, 250(1-2), 333-363. doi:10.1016/s0304-3975(99)00150-

A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)

We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). This improvement is achieved by applying a new modern Lenstra-type algorithm, finding optimal ellipsoid roundings, and considering sparse encodings of polynomials. For the bounded case, our algorithm attains a time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, s l^{O(1)} d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input.Comment: 28 pages, 10 figure