FUZZY RELATION EQUATIONS AND FUZZY AUTOMATA

Transcription

1 FUZZY RELATION EQUATIONS AND FUZZY AUTOMATA Miroslav Ćirić 1, Jelena Ignjatović 1 and Nada Damljanović 2 1 University of Niš, Faculty of Sciences and Mathematics Višegradska 33, Niš, Serbia miroslav.ciric@pmf.edu.rs, jelena.ignjatovic@pmf.edu.rs 2 University of Kragujevac, Technical faculty in Čačak Svetog Save 65, Čačak, Serbia nada@tfc.kg.ac.rs Research supported by Ministry of Education and Science, Republic of Serbia, Grant No Liptovský Ján ( ) FSTA 2012 Nada Damljanović 1 / 22

2 Fuzzy relation equations and inequalities Employed in the study of fuzzy automata: in the study of simulation, bisimulation and equivalence of fuzzy automata, in the reduction of the number of states of fuzzy automata, in the study of subsystems of fuzzy transition systems... These applications have led to the introduction of certain new types of fuzzy relation equations and inequalities, which have been investigated from the general point of view. Liptovský Ján ( ) FSTA 2012 Nada Damljanović 2 / 22

3 Basic concepts Fuzzy relations L = (L,,,,, 0, 1) complete residuated lattice ϕ : A B L fuzzy relation between sets A and B R(A, B) lattice of fuzzy relations between A and B ϕ : A A L fuzzy relation on a set A R(A) lattice of fuzzy relations on A ϕ ψ composition of fuzzy relations ϕ R(A, B) and ψ R(B, C) (ϕ ψ)(a, c) = b B ϕ(a, b) ψ(b, c) (a A, c C) ϕ 1 R(B, A) inverse of a fuzzy relation ϕ R(A, B) ϕ 1 (b, a) def = ϕ(a, b) (a A, b B) Liptovský Ján ( ) FSTA 2012 Nada Damljanović 3 / 22

4 Basic concepts L complete residuated lattice, X alphabet Fuzzy automaton over L and X a quadruple A = (A, δ A, σ A, τ A ) A (non-empty) set of states δ A : A X A L fuzzy transition function σ A : A L fuzzy set of initial states τ A : A L fuzzy set terminal states X the free monoid over the alphabet X δ A can be uniquely extended up to a mapping δ A fuzzy transition relation on A determined by u X : A X A L δ A u (a, b) = δ A (a, u, b), for all a, b A (1) δ A uv = δ A u δ A v, for all u, v X (2) Liptovský Ján ( ) FSTA 2012 Nada Damljanović 4 / 22

5 Bisimulations Origin of bisimulation concurrency theory R. Milner, 1980, D. Park, 1981 modal logic (Kripke systems) J. van Benthem, 1976 set theory M. Forti, F. Honsell, 1983 Role of bisimulation equivalence between two transition systems, reduction of the number of states Recent research M. Ćirić, J. Ignjatović, N. Damljanović, M. Bašić (FSS, 2012) Bisimulations between fuzzy automata M. Ćirić, J. Ignjatović, I. Jančić, N. Damljanović (FSS, to appear) Computation of the greatest simulations and bisimulations M. Ćirić, J. Ignjatović, M. Bašić, I. Jančić (Comput. Math. Appl., to appear) Bisimulations between nondeterministic automata Liptovský Ján ( ) FSTA 2012 Nada Damljanović 5 / 22

6 Simulations between fuzzy automata A = (A, δ A, σ A, τ A ), B = (B, δ B, σ B, τ B ) fuzzy automata ϕ R(A, B) non-empty fuzzy relation ϕ is a forward simulation if σ A σ B ϕ 1, (fs-1) ϕ 1 δ A x δ B x ϕ 1, for every x X, (fs-2) ϕ 1 τ A τ B, (fs-3) ϕ is a backward simulation if τ A ϕ τ B, (bs-1) δ A x ϕ ϕ δ B x, for every x X, (bs-2) σ A ϕ σ B. (bs-3) Liptovský Ján ( ) FSTA 2012 Nada Damljanović 6 / 22

7 Simulations between fuzzy automata σ A a 0 ϕ b 0 σ B A x 1 x k.... x 1 x k B a k b k x k+1 x k+1 a k+1 b k+1 x k+2 x n.... x k+2 x n τ A a n b n Figure 1: Forward and backward simulation τ B Liptovský Ján ( ) FSTA 2012 Nada Damljanović 7 / 22

8 Bisimulations between fuzzy automata Homotypic bisimulations ϕ is a forward bisimulation if both ϕ and ϕ 1 are forward simulations σ A σ B ϕ 1, σ B σ A ϕ, (fb-1) ϕ 1 δ A x δ B x ϕ 1, ϕ δ B x δ A x ϕ, for every x X, (fb-2) ϕ 1 τ A τ B, ϕ τ B τ A, (fb-3) ϕ is a backward bisimulation, if both ϕ and ϕ 1 are backward simulations τ A ϕ τ B, τ B ϕ 1 τ A, (bb-1) δ A x ϕ ϕ δ B x, δ B x ϕ 1 ϕ 1 δ A x, for every x X, (bb-2) σ A ϕ σ B, σ B ϕ 1 σ A. (bb-3) Liptovský Ján ( ) FSTA 2012 Nada Damljanović 8 / 22

9 Bisimulations between fuzzy automata Heterotypic bisimulations ϕ is a forward-backward bisimulation if ϕ is a forward simulation and ϕ 1 is a backward simulation σ A σ B ϕ 1, τ B ϕ 1 τ A, (fbb-1) ϕ 1 δ A x = δ B x ϕ 1, for every x X, (fbb-2) σ B ϕ 1 σ A, ϕ 1 τ A τ B, (fbb-3) ϕ is a backward-forward bisimulation if ϕ is a backward simulation and ϕ 1 is a forward simulation σ B σ A ϕ, τ A ϕ τ B, (bfb-1) δ A x ϕ = ϕ δ B x, for every x X, (bfb-2) σ A ϕ σ B, ϕ τ B τ A. (bfb-3) Simulations languague inclusion Bisimulations languague equivalence Simulations and bisimulations compatibility with transitions, initial and terminal states Liptovský Ján ( ) FSTA 2012 Nada Damljanović 9 / 22

10 Bisimulations between fuzzy automata Heterotypic bisimulations ϕ is a forward-backward bisimulation if ϕ is a forward simulation and ϕ 1 is a backward simulation σ A σ B ϕ 1, τ B ϕ 1 τ A, (fbb-1) ϕ 1 δ A x = δ B x ϕ 1, for every x X, (fbb-2) σ B ϕ 1 σ A, ϕ 1 τ A τ B, (fbb-3) ϕ is a backward-forward bisimulation if ϕ is a backward simulation and ϕ 1 is a forward simulation σ B σ A ϕ, τ A ϕ τ B, (bfb-1) δ A x ϕ = ϕ δ B x, for every x X, (bfb-2) σ A ϕ σ B, ϕ τ B τ A. (bfb-3) Simulations languague inclusion Bisimulations languague equivalence Simulations and bisimulations compatibility with transitions, initial and terminal states Liptovský Ján ( ) FSTA 2012 Nada Damljanović 9 / 22

11 Computation of the greatest simulations and bisimulations Let A = (A, δ A, σ A, τ A ) and B = (B, δ B, σ B, τ B ) be fuzzy automata. We define fuzzy relations ψ w R(A, B), for w {fs, bs, fb, bb, fbb, bfb}, in the following way: ψ fs = τ A τ B, ψ bs = σ A σ B, ψ fb = (τ A τ B ) (τ A τ B ) = τ A τ B, ψ bb = (σ A σ B ) (σ A σ B ) = σ A σ B, ψ fbb = (τ A τ B ) (σ A σ B ), ψ bfb = (σ A σ B ) (τ A τ B ). Liptovský Ján ( ) FSTA 2012 Nada Damljanović 10 / 22

12 Computation of the greatest simulations and bisimulations Moreover, we define functions φ w : R(A, B) R(A, B), for w {fs, bs, fb, bb, fbb, bfb}, as follows: φ fs (α) = x X[(δ B x α 1 )\δ A x ] 1, φ bs (α) = x X(α δ B x )/δ A x, φ fb (α) = x X[(δ B x α 1 )\δ A x ] 1 [(δ A x α)\δ B x ] = φ fs (α) [φ fs (α 1 )] 1, φ bb (α) = x X[(α δ B x )/δ A x ] [(α 1 δ A x )/δ B x ] 1 = φ bs (α) [φ bs (α 1 )] 1, φ fbb (α) = x X[(δ B x α 1 )\δ A x ] 1 [(α 1 δ A x )/δ B x ] 1 = φ fs (α) [φ bs (α 1 )] 1, φ bfb (α) = x X[(α δ B x )/δ A x ] [(δ A x α)\δ B x ] = φ bs (α) [φ fs (α 1 )] 1, for any α R(A, B). Notice that in the expression φ w (α 1 ) (w {fs, bs}) we denote by φ w a function from R(B, A) into itself. Liptovský Ján ( ) FSTA 2012 Nada Damljanović 11 / 22

13 Computation of the greatest simulations and bisimulations Theorem Let A = (A, δ A, σ A, τ A ) and B = (B, δ B, σ B, τ B ) be fuzzy automata and w {fs, bs, fb, bb, fbb, bfb}. A fuzzy relation ϕ R(A, B) satisfies conditions (w-2) and (w-3) if and only if it satisfies ϕ φ w (ϕ), ϕ ψ w. (3) Boils down on computation of the greatest post-fixed point of the function φ w Liptovský Ján ( ) FSTA 2012 Nada Damljanović 12 / 22

14 Computation of the greatest simulations and bisimulations Theorem Let A = (A, δ A, σ A, τ A ) and B = (B, δ B, σ B, τ B ) be fuzzy automata, let w {fs, bs, fb, bb, fbb, bfb}, and let a sequence {ϕ k } k N of fuzzy relations from R(A, B) be defined by ϕ 1 = ψ w, ϕ k+1 = ϕ k φ w (ϕ k ), for each k N. (4) If Im(ψ w ) x X ( Im(δA x ) Im(δ B x ) ) is a finite subalgebra of L, then the following is true: (a) the sequence {ϕ k } k N is finite and descending, and there is the least natural number k such that ϕ k = ϕ k+1 ; (b) ϕ k is the greatest fuzzy relation in R(A, B) which satisfies (w-2) and (w-3); (c) if ϕ k satisfies (w-1), then it is the greatest fuzzy relation in R(A, B) which satisfies (w-1), (w-2) and (w-3); (d) if ϕ k does not satisfy (w-1), then there is no any fuzzy relation in R(A, B) which satisfies (w-1), (w-2) and (w-3). Liptovský Ján ( ) FSTA 2012 Nada Damljanović 13 / 22

15 Computation of the greatest simulations and bisimulations Theorem Let A = (A, δ A, σ A, τ A ) and B = (B, δ B, σ B, τ B ) be fuzzy automata, let w {fs, bs, fb, bb, fbb, bfb}, let {ϕ k } k N be a sequence of fuzzy relations from R(A, B) defined by (4), and let ϕ = k N ϕ k. (5) If L is a complete residuated lattice satisfying the following conditions: x ( ) y i = (x y i ) and x ( ) y i = (x y i ), (6) i I i I i I for all x L and {y i } i I L. Then the following is true: (a) ϕ is the greatest fuzzy relation in R(A, B) which satisfies (w-2) and (w-3); (b) if ϕ satisfies (w-1), then it is the greatest fuzzy relation in R(A, B) which satisfies (w-1), (w-2) and (w-3); (c) if ϕ does not satisfy (w-1), then there is no any fuzzy relation in R(A, B) which satisfies (w-1), (w-2) and (w-3). Liptovský Ján ( ) FSTA 2012 Nada Damljanović 14 / 22 i I

16 Bisimulation equivalence relations E fuzzy equivalence relation on A = (A, δ A, σ A, τ A ) ( A E, E 1 E, E E E) E is forward bisimulation on A, E δ A x δ A x E, for any x X, and E τ A τ A, E δ A x E = δ A x E, for any x X, and E τ A = τ A. Any fuzzy automaton has the greatest forward and backward simulations, which are fuzzy quasi-orders, and the greatest forward and backward bisimulations, which are fuzzy equivalence relations. State reduction Replace a fuzzy automaton by an equivalent fuzzy automaton with fewer states equivalent recognize the same fuzzy language M. Ćirić, A. Stamenković, J. Ignjatović, T. Petković (LNCS 2007; JCSS 2010) reduction by fuzzy equivalences factor fuzzy automata A. Stamenković, M. Ćirić, J. Ignjatović (IS, to appear) reduction by fuzzy quasi-orders afterset (foreset) fuzzy automata Liptovský Ján ( ) FSTA 2012 Nada Damljanović 15 / 22

17 Uniform forward bisimulations R R(A, B) fuzzy relation F. Klawonn, 2000 partial fuzzy function R R 1 R R M. Ćirić, J. Ignjatović, S. Bogdanović, 2009 uniform fuzzy relation complete, surjective, partial fuzzy function R R 1 R = R (also R 1 R R 1 = R 1 ) uniform fuzzy relation fuzzy equivalence between two sets Theorem Let A = (A, δ A, σ A, τ A ) and B = (B, δ B, σ B, τ B ) be fuzzy automata and let ϕ R(A, B) be a uniform fuzzy relation. Then ϕ is a forward bisimulation if and only if the following hold: σ A ϕ ϕ 1 = σ B ϕ 1, σ A ϕ = σ B ϕ 1 ϕ, (7) δ A x ϕ ϕ 1 = ϕ δ B x ϕ 1, ϕ 1 δ A x ϕ = δ B x ϕ 1 ϕ, x X, (8) τ A = ϕ τ B, ϕ 1 τ A = τ B. (9) Liptovský Ján ( ) FSTA 2012 Nada Damljanović 16 / 22

18 Uniform forward bisimulations Fuzzy equivalence relations on A and B induced by ϕ R(A, B). Kernel of ϕ : E ϕ A(a 1, a 2 ) = b B ϕ(a 1, b) ϕ(a 2, b), for all a 1, a 2 A (10) Co-kernel of ϕ : E ϕ B (b 1, b 2 ) = ϕ(a, b 1 ) ϕ(a, b 2 ), for all b 1, b 2 B a A (11) Theorem Let fuzzy automata A = (A, σ A, X, δ A, τ A ) and B = (B, σ B, X, δ B, τ B ) be UFB-equivalent. Then there exists a uniform forward bisimulation ϕ between A and B whose kernel E ϕ A is the greatest forward bisimulation on A and the co-kernel Eϕ B is the greatest forward bisimulation on B. Moreover, ϕ is the greatest forward bisimulation between A and B. Liptovský Ján ( ) FSTA 2012 Nada Damljanović 17 / 22

19 Uniform forward bisimulations Theorem Let A = (A, δ A, σ A, τ A ) and B = (B, δ B, σ B, τ B ) be fuzzy automata, and let E and F be the greatest forward bisimulations on A and B, respectively. Then A and B are UFB-equivalent if and only if there exists an isomorphism φ : A/E B/F such that for all a 1, a 2 A. Ẽ(E a1, E a2 ) = F(φ(E a1 ), φ(e a2 )), (12) Liptovský Ján ( ) FSTA 2012 Nada Damljanović 18 / 22

20 Weakly linear systems Weakly linear systems {V i } i I R(A), {W i } i I R(B) families of fuzzy relations Z R(A, B) given fuzzy relation U unknown fuzzy relation in R(A, B) (1) U 1 V i W i U 1 (i I) U Z (2) V i U U W i (i I) U Z combinations of (1) and (2) (for U and U 1 ) (3) U 1 V i W i U 1 U W i V i U (i I) U Z (4) V i U U W i W i U 1 U 1 V i (i I) U Z (5) V i U = U W i (i I) U Z (6) U 1 V i = W i U 1 (i I) U Z Liptovský Ján ( ) FSTA 2012 Nada Damljanović 19 / 22

21 Weakly linear systems A = B homogeneous weakly linear sistems J. Ignjatović, M. Ćirić, S. Bogdanović (FSS, 2010) The systems (1) (6) have the greatest solutions. If Z is a fuzzy quasi-order, then the greatest solutions to (1) (3) are fuzzy quasi-orders. If Z is a fuzzy equivalence, then the greatest solutions to (4) (6) are fuzzy equivalences. A B heterogeneous weakly linear sistems J. Ignjatović, M. Ćirić, N. Damljanović, I. Jančić (FSS, 2012) All systems (1) (6) have the greatest solutions (they may be empty). If Z is a partial fuzzy function, then the greatest solutions to (3) and (4) are partial fuzzy functions. Liptovský Ján ( ) FSTA 2012 Nada Damljanović 20 / 22

22 Computation of the greatest solutions φ (1) (R) def = [(W i R 1 )\V i ] 1 i I φ (2) (R) def = i I(R W i )/V i φ (3) (R) def = i I[(W i R 1 )\V i ] 1 [(V i R)\W i ] = φ (1) (R) φ (1) (R 1 ) φ (4) (R) def = i I[(R W i )/V i ] [(R 1 V i )/W i ] 1 = φ (2) (R) φ (2) (R 1 ) φ (5) (R) def = i I[(R W i )/V i ] [(V i R)\W i ] = φ (2) (R) φ (1) (R 1 ) φ (6) (R) def = i I[(W i R 1 )\V i ] 1 [(R 1 V i )/W i ] 1 = φ (1) (R) φ (2) (R 1 ) (Z/V)(a, b) = a A (Z\W)(a, b) = b B V(a, a) Z(a, b) (a A, b B) W(b, b ) Z(a, b ) (a A, b B) Liptovský Ján ( ) FSTA 2012 Nada Damljanović 21 / 22

23 Meaning of uniform solutions The greatest solutions to (3) and (4) partial fuzzy functions (R R 1 R R) Question: When the greatest solution is a uniform fuzzy relation? If there is a uniform solution, then the greatest solution is uniform uniform solution some kind of equivalence between fuzzy relational systems (A, {V i } i I ) and (B, {W i } i I ) H uniform solution H determines an isomorphism between factor fuzzy relational systems of (A, {V i } i I ) and (B, {W i } i I ) w.r.t. fuzzy equivalences H H 1 and H 1 H Liptovský Ján ( ) FSTA 2012 Nada Damljanović 22 / 22