Two types of simulations and four types of bisimulations for weighted finite automata over the complete max-plus semiring we define as solutions of particular systems of matrix inequations. We ...provide a procedure that either decides that there is a simulation or bisimulation of a given type between two automata, and outputs the greatest one, or decides that no simulation or bisimulation of that type exists. The procedure is iterative and does not have to end in a finite number of steps. Certain conditions under which this procedure must terminate in a finite number of steps are described in a slightly more general context in Stamenković et al. (Discrete Event Dynamic Systems, 32:1–25,
2022
). We also propose a modification of this procedure which, in case there is no simulation or bisimulation of a given type between two max-plus automata, detects this in finitely many steps and faster than the original procedure. In the same case, that modification also finds a natural number such that containment or equivalence is valid for all input words of length less than that number. For max-plus automata with non-negative weights, we point out the differences that occur when the above mentioned procedure is applied over the complete max-plus semiring, and when it is applied over its non-negative part with minus infinity added.
In this paper we introduce and study weakly linear systems, i.e. systems consisting of matrix inequalities Eqs.
17
–
20
, over the max-plus quantale which is also known as complete max-plus algebra. ...We prove the existence of the greatest solution contained in a given matrix
X
0
, and present a procedure for its computation. In the case of weakly linear systems consisting of finitely many matrix inequalities, when all finite elements of matrices
X
0
,
A
s
and
B
s
,
s
∈
I
are integers, rationals or particular irrationals and a finite solution exists, the procedure finishes in a finite number of steps. If in that case an arbitrary finite solution is given, a lower bound on the number of computational steps is calculated. Otherwise, we use our algorithm to compute approximations to finite solutions.
In this paper we consider different types of ranks of fuzzy matrices over residuated lattices. We investigate relations between ranks and prove that row rank, column rank and Schein rank of ...idempotent fuzzy matrices are equal. In particular, ranks and corresponding decompositions of fuzzy matrices representing fuzzy quasi-orders are studied in detail. We show that fuzzy matrix decomposition by ranks can be used in the state reduction of fuzzy automata. Moreover, we prove that using rank decomposition of fuzzy matrices improves results of any state reduction method based on merging indistinguishable states of fuzzy automata.
In our recent paper we have established close relationships between state reduction of a fuzzy automaton and resolution of a particular system of fuzzy relation equations. In that paper we have also ...studied reductions by means of those solutions which are fuzzy equivalences. In this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasi-orders. Generally, fuzzy quasi-orders and fuzzy equivalences are equally good in the state reduction, but we show that right and left invariant fuzzy quasi-orders give better reductions than right and left invariant fuzzy equivalences. We also show that alternate reductions by means of fuzzy quasi-orders give better results than alternate reductions by means of fuzzy equivalences. Furthermore we study a more general type of fuzzy quasi-orders, weakly right and left invariant ones, and we show that they are closely related to determinization of fuzzy automatons. We also demonstrate some applications of weakly left invariant fuzzy quasi-orders in conflict analysis of fuzzy discrete event systems.
Li and Pedrycz have proved fundamental results that provide different equivalent ways to represent fuzzy languages with membership values in a lattice-ordered monoid, and generalize the well-known ...results of the classical theory of formal languages. In particular, they have shown that a fuzzy language over an integral lattice-ordered monoid can be represented by a fuzzy regular expression if and only if it can be recognized by a fuzzy finite automaton. However, they did not give any efficient method for constructing an equivalent fuzzy finite automaton from a given fuzzy regular expression. In this paper we provide such an efficient method. Transforming scalars appearing in a fuzzy regular expression αinto letters of the new extended alphabet, we convert the fuzzy regular expression αto an ordinary regular expression αR. Then, starting from an arbitrary nondeterministic finite automaton A that recognizes the language ‖αR‖ represented by the regular expression αR, we construct fuzzy finite automata Aα and Aαr with the same or even less number of states than the automaton A, which recognize the fuzzy language ‖α‖represented by the fuzzy regular expression α. The starting nondeterministic finite automaton A can be obtained from αR using any of the well-known constructions for converting regular expressions to nondeterministic finite automata, such as Glushkov–McNaughton–Yamada's position automaton, Brzozowski's derivative automaton, Antimirov's partial derivative automaton, or Ilie–Yu's follow automaton.
We show that the state reduction problem for fuzzy automata is related to the problem of finding a solution to a particular system of fuzzy relation equations in the set of all fuzzy equivalences on ...its set of states. This system may consist of infinitely many equations, and finding its non-trivial solutions may be a very difficult task. For that reason we aim our attention to some instances of this system which consist of finitely many equations and are easier to solve. First, we study right invariant fuzzy equivalences, and their duals, the left invariant ones. We prove that each fuzzy automaton possesses the greatest right (resp. left) invariant fuzzy equivalence, which provides the best reduction by means of fuzzy equivalences of this type, and we give an effective procedure for computing this fuzzy equivalence, which works if the underlying structure of truth values is a locally finite residuated lattice. Moreover, we show that even better reductions can be achieved alternating reductions by means of right and left invariant fuzzy equivalences. We also study strongly right and left invariant fuzzy equivalences, which give worse reductions than right and left invariant ones, but whose computing is much easier. We give an effective procedure for computing the greatest strongly right (resp. left) invariant fuzzy equivalence, which is applicable to fuzzy automata over an arbitrary complete residuated lattice.
The aim of this study was to determine: (1) differences in anthropometric characteristics and agility between different functional classes of wheelchair basketball players and (2) the relationship ...between anthropometric characteristics and agility with the functional classification of wheelchair basketball players. The sample of participants consisted of 40 wheelchair basketball players, aged 33.9 ± 11.2 years. Anthropometric characteristics (longitudinal and circular dimensions, as well as skinfold thickness) were assessed, and agility was measured using the modified T-test and Figure-of-Eight test. The results of the one-way analysis of variance showed significant differences with very large effects between players of different functional classes in body mass, sitting height, and sitting reach height, while significant differences with large effects were recorded in the agility assessment tests: the T-test and the Figure-of-Eight test. Also, the results of the correlation analysis indicate that there are significant moderate positive correlations of sitting height and reaching height with functional classification.
Cilјevi ovog istraživanja su bili da se utvrde: (1) razlike u antropometrijskim karakteristikama i agilnosti između različitih funkcionalnih klasa košarkaša u kolicima i (2) povezanost između ...antropometrijskih karakteristika i agilnosti sa funkcionalnom klasifikacijom košarkaša u kolicima. Uzorak ispitanika činilo je 40 košarkaša u kolicima, uzrasta 33.9 ± 11.2 godina. Procenjivane su antropometrijske karakteristike (longitudionalne i cirkularne dimenzije, kožni nabori), a za procenu agilnosti korišćeni su modifikovani T-test i osmica test. Rezultati jednosmerne analize varijanse ukazuju na postojanje značajnih razlika sa veoma velikim efektima između igrača različitih funkcionalnih klasa kod telesne mase, sedeće visine i sedeće dohvatne visine, dok su značajne razlike sa velikim efektima zabeležene kod testova za procenu agilnosti: T-testa i osmice testa. Takođe, rezultati korelacione analize ukazuju na to da postoje značajne umerene pozitivne korelacije sedeće visine i dohvatne visine sa funkcionalnom klasifikacijom.
PDF
