The purpose of present work is to introduce and study two minimal realizations for a given fuzzy behaviour in bicategory-theoretic setup. One such realization is based on Myhill Nerode’s theory, while the other realization is based on derivative of the given fuzzy behaviour. It is shown here that there exists a unique 2-cell in bicategory of crisp deterministic fuzzy machines between both the realizations.
PierceB.C., Basic Category Theory for Computer Scientists, The MIT Press, Cambridge, 1991.
2.
QiuD., Automata theory based on complete residuated latticevalued logic(I), Science in China44 (2001), 419–429.
3.
QiuD., Automata theory based on complete residuated latticevalued logic(II), Science in China45 (2002), 442–452.
4.
SantosE.S., Max-product machines, Journal of Mathematical Analysis and Applications37 (1972), 677–686.
5.
LeiH. and LiY.M., Minimization of states in automata theory based on finite lattice-ordered monoids, Information Sciences177 (2007), 1413–1421.
6.
KumbhojkarH.V. and ChaudhriS.R., On proper fuzzification of fuzzy finite state machines, International Journal of Fuzzy Mathematics4 (2008), 1019–1027.
7.
XingH., QiuD., LiuF. and FanZ., Equivalence in automata theory based on complete residuated lattice-valued logic, Fuzzy Sets and Systems158 (2007), 1407–1422.
8.
XingH. and QiuD., Automata theory based on complete residuated lattice-valued logic: A categorical approach, Fuzzy Sets and Systems160 (2009), 2416–2428.
9.
GoguenJ.A., L-fuzzy sets, Journal of Mathematical Analysis and Applications18 (1967), 145–174.
10.
GoguenJ.A., Minimal realization of machines in closed categories, Bulletin of American Mathematical Society78 (1972), 777–783.
11.
AdamekJ. and TrnkovaV., Automata and Algebras in Categories, Kluwer, 1990.
12.
BénabouJ., Introduction to bicategories, Reports of the Midwest Category Seminar, Lecture Notes in Mathematics47 (1967), 1–77.
13.
IgnjatovićJ., ĆirićM. and BogdanovićS., Determinization of fuzzy automata with membership values in complete residuated lattices, Information Sciences178 (2008), 164–180.
14.
IgnjatovićJ., ĆirićM., BogdanovićS. and PetkovićT., Myhill- Nerode type theory for fuzzy languages and automata, Fuzzy Sets and Systems161 (2010), 1288–1324.
15.
MočkořJ., A category of fuzzy automata, International Journal of General Systems20 (1991), 73–82.
16.
MočkořJ., Fuzzy and non-deterministic automata, Soft Computing3 (1999), 221–226.
17.
MočkořJ., Semigroup homomorphisms and fuzzy automata, Soft Computing6 (2002), 423–427.
18.
MordesonJ.N. and MalikD.S., Fuzzy Automata and Languages: Theory and Applications, Chapman and Hall/CRC. London/Boca Raton, 2000.
19.
AbolpourK. and ZahediM.M., Isomorphism between two BLgeneral fuzzy automata, Soft Computing16 (2012), 103–118.
20.
PeevaK.G., Behaviour, reduction and minimization of finite L-automata28 (1988), 171–181.
21.
PeevaK., Finite L-fuzzy machines, Fuzzy Sets and Systems141 (2004), 415–437.
22.
PeevaK. and ZaharievZ.l., Computing behavior of finite fuzzy machines-Algorithm and its application to reduction and minimization178 (2008), 4152–4165.
23.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
24.
ArbibM.A. and ManesE.G., Machines in a category: An expository introduction, SIAM Review16 (1974), 163–192.
25.
ArbibM.A. and ManesE.G., Basic concepts of category theory applicable to computation and control, Proc First International Symposium AMherst MA, Lecture Notes in Computer Science25 (1975), 2–41.
26.
ArbibM.A. and ManesE.G., A categorist’s view of automata and systems, Proc First International Symposium AMherst MA, Lecture Notes in Computer Science25 (1975), 62–78.
27.
BarrM. and WellsC., Category Theory for Computing Science, Prentice-Hall International (UK) Limited, Englewood Cliffs, NJ, 1996.
28.
DoostfatemehM. and KremerS.C., New Directions in Fuzzy Automata38 (2005), 175–214.
29.
BasakN.C. and GuptaA., On quotient machines of a fuzzy automaton and the minimal machine, Fuzzy Sets and Systems125 (2002), 223–229.
30.
BělohlávekR., Determinism and fuzzy automata, Information Sciences143 (2002), 205–209.
31.
RosebrughR., SabadiniN. and WaltersR.F.C., Minimal realization in bicategories of automata, Mathematical Structures in Computer Science8 (1998), 93–116.
32.
EilenbergS., Automata, Languages and Machines: A, Academic Press, New York, 1974.
33.
TiwariS.P., YadavV.K. and SinghA.K., Construction of a minimal realization and monoid for a fuzzy language: A categorical approach, Journal of Applied Mathematics and Computing47 (2014), 401–416.
34.
PetkovićT., Congruences and homomorphisms of fuzzy automata, Fuzzy Sets and Systems157 (2006), 444–458.
35.
TrnkováV., Automata and categories, Lecture Notes in Computer Science32 (1975), 160–166.
36.
TrnkováV., Relational automata in a category and their languages, Lecture Notes in Computer Science56 (1977), 340–355.
37.
ChengW. and MoZ., Minimization algorithm of fuzzy finite automata, Fuzzy Sets and Systems141 (2004), 439–448.
38.
WeeW.G., On generalizations of adaptive algorithm and application of the fuzzy sets concept to pattern classification, Ph. D. Thesis, Purdue University, Lafayette, IN, 1967.
39.
LihuaW. and QiuD., Automata theory based on complete residuated lattice-valued logic: Reduction and minimization, Fuzzy Sets and Systems161 (2010), 1635–1656.
40.
KimY.H., KimJ.G. and ChoS.J., Products of T-generalized state machines and T-generalized transformation semigroups, Fuzzy Sets and Systems93 (1998), 87–97.
41.
LiY. and PedryczW., Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids, Fuzzy Sets and Systems156 (2005), 68–92.
42.
LiY. and PedryczW., The equivalence between fuzzy Mealy and fuzzy Moore machines, Soft Computing10 (2006), 953–959.
43.
LiY. and PedryczW., Minimization of lattice finite automata and its application to the decomposition of lattice languages, Fuzzy Sets and Systems158 (2007), 1423–1436.
44.
LiY.M., A categorical approach to lattice-valued fuzzy automata, Fuzzy Sets and Systems156 (2006), 855–864.
45.
JančićZ. and ĆirićM, Brzozowski type determinization for fuzzy automata, Fuzzy Sets and Systems249 (2014), 73–82.