Algorithms for Computing Complete Deterministic Fuzzy Automata via Invariant Fuzzy Quasi-Orders

Transcription

1 Faculty of Sciences and Mathematics Department of Computer Science University of Niš, Serbia Algorithms for Computing Complete Deterministic Fuzzy Automata via Invariant Fuzzy Quasi-Orders Stefan Stanimirović joint work with Miroslav Ćirić and Jelena Ignjatović WATA 2018, Leipzig, Germany May 26, 2018

2 Content 1 1. Introduction 2. Preliminaries 3. A determinization algorithm for fuzzy automata 4. A canonization algorithm for fuzzy automata

3 Introduction Determinization of automata 2 Determinization is a procedure of finding a language equivalent deterministic finite automaton (DFA, for short) for a given nondeterministic finite automaton (NFA, for short) It is a fundamental problem in automata theory Contrary to NFAs, for any DFA there is a trivial linear time, constant space, online algorithm to check whether an input sequence is accepted or not For this reason, in many applications it pays to convert NFAs into DFAs Particularly significant determinization methods are those that provide a minimal DFA equivalent to the original NFA, called canonization methods. For every NFA there exists a DFA that is language equivalent The well-known accessible subset construction is simple, but exponential algorithm for determinization a state in the determinized automaton is a subset of states of the input automaton

4 Introduction Determinization of automata 3 x, y NFA: y x x, y a 0 a 1 a 2 a 3 x y y {a 0} {a 0, a 1} DFA: x y x y y x x {a 0, a 3} {a 0, a 2} {a 0, a 1, a 3}

5 Introduction Determinization of automata 4 ϕ D ε = [ ] ϕ D x = [ ] ϕ D [ ] x 2 = Figure (a): A - a fuzzy automaton over X = {x} and the product (Gougen) structure Figure (b): Minimal complete deterministic fuzzy automaton (CDFA) of A Unlike NFAs, which can always be determinized, not all fuzzy finite automata FFAs, can be determinized Some well-known algorithms cannot compute CDFA, even if it exists. Another problem: How to find a minimal CDFA equivalent to A? Note: In general, a minimal CDFA is not unique (there may be several equivalent CDFAs with the same number of states, but with different membership values in their transitions, fuzzy initial and fuzzy terminal states).

6 Our motivation and goal 5 To provide improvements of the determinization and canonization methods for fuzzy automata via factorization of fuzzy states previously studied in J. R. G. de Mendívil, J. R. Garitagoitia, Determinization of fuzzy automata via factorization of fuzzy states, Information Sciences, 283 (2014) J. R. G. de Mendívil, Conditions for minimal fuzzy deterministic finite automata via Brzozowski s procedure, IEEE Transactions on Fuzzy Systems (2017) Our methods are based on the usage of the right and left invariant fuzzy quasi-orders - originally used to model the indistinguishability of states of FFAs and employed in the state reduction of fuzzy automata A. Stamenković, M. Ćirić, J. Ignjatović, Reduction of fuzzy automata by means of fuzzy quasi-orders, Information Sciences 275 (2014)

7 Our motivation and goal Continuation 6 Our methods basically combine the determinization and the state reduction problems, similarly as in Z. Jančić, I. Micić, J. Ignjatović, M. Ćirić, Further improvement of determinization methods for fuzzy finite automata, Fuzzy Sets and Systems 301(2016) Our methods always result in a smaller or equal CDFA compared to the one obtained by the factorization of fuzzy states; It can even result in a finite CDFA in some cases when the factorization of fuzzy states results in an infinite CDFA;

8 Fuzzy sets and relations 7 Complete residuated lattices Structure of truth values (complete) residuated lattice L = (L,,,,, 0, 1) (L,,, 0, 1) - (complete) lattice with the least element 0 and the greatest element 1, (L,, 1) - commutative monoid with the unit 1, adjunction property: for every x, y, z L, x y z x y z.

9 Fuzzy sets and relations (cont.) 8 Examples of complete residuated lattices L = [0, 1], = max, = min, and Product (Goguen) operations: a b = a b, a b = { 1, a b b/a, otherwise Gödel operations: Łukasiewicz operations: a b = min{a, b}, a b = { 1, a b b, otherwise a b = max{a + b 1, 0}, a b = min{1 a + b, 1},

10 Fuzzy sets and relations (cont.) 9 Fuzzy sets and fuzzy relations fuzzy subset of A function α : A L L A all fuzzy subsets of A (fuzzy vectors of length A with entries in L, A-finite) fuzzy relation between A and B function ϕ : A B L L A B all fuzzy relations between A and B (fuzzy matrices of type A B with entries in L, A,B-finite) Fuzzy relations ϕ L A B, φ L B C Composition: ϕ φ L A C - the matrix product Fuzzy relation ϕ L A B and fuzzy sets α L A, β L B Composition: α ϕ L B, ϕ β L A - the vector-matrix products Fuzzy sets: α 1, α 2 L A Composition: α 1 α 2 L - the scalar (dot) product

11 Fuzzy automata and languages 10 Fuzzy automata L complete residuated lattice, X finite alphabet, X free monoid, ε X empty word A = (A, σ, δ, τ) fuzzy automaton A set of states (not necessarily finite) σ L A fuzzy set of initial states, τ L A fuzzy set of terminal (final) states, δ L A X A fuzzy transition function δ {δ x} x X δ x L A A, δ x(a, b) = δ(a, x, b) L product structure A = {a 1, a 2, a 3}, σ = [1 0 0], τ = [0 0 1] 1, δ x = , δ y =

12 Fuzzy automata and languages (cont.) Fuzzy languages fuzzy language f : X L fuzzy subset of the free monoid X fuzzy language recognized by A [[A]](u) = σ δ u τ (degree to which A accepts the word u) 11 CDFAs Let A = (A, σ, δ, τ) be a fuzzy automaton over L and X. Then A is called: complete, if ( x X)( a A)( b A)δ x(a, b) > 0 deterministic, if (!a A)σ(a) > 0 ( x X)( a, b 1, b 2 A)δ x(a, b 1) > 0 and δ x(a, b 2) > 0 b 1 = b 2 If A is simultaneously complete and deterministic, then it is called complete deterministic fuzzy automaton (CDFA for short).

13 First concept: Factorization of fuzzy states 12 Factorization of A Let A be a fuzzy automation over X and L. A factorization of automation A, or a factorization of A, is an ordered pair D = (f, g) of functions g : L A L and f : L A L A that satisfy the following properties σ = g(σ) f(σ), for every σ L A ; g( ) = 1; Factorizations of fuzzy states The idea of factorizations was firstly introduced by D. Kirsten, L. Mäurer, On the determinization of weighted automata, J. Automata, Lang. Combin. 10 (2005) in the determinization algorithm for weighted finite automata (WFAs) over semirings.

14 Factorization of fuzzy states (cont.) 13 Trivial factorization The factorization D N = (f N, g N) of A, where g N(σ) = 1 and f N(σ) = σ for every σ L A is called the trivial factorization. Mohri s factorization Mohri s factorization: the ordered pair D M = (f M, g M), where f M : L A L A and g M : L A L, for every σ L A and a A by { 1, σ = g M(σ) =, and fm(σ)(a) = gm(σ) σ(a); a A σ(a), otherwise Maximal factorization A factorization D = (f, g) of A is called maximal if g(a σ) = a g(σ), f(a σ) = f(σ), for all a L and σ L A such that a σ. Maximal factorizations play a crutial role in the determinization of fuzzy automata.

15 Second concept: (Weakly) right and left invariant fuzzy quasi-orders 14 Weakly right and left invariant fuzzy relations A = (A, σ, δ, τ) - fuzzy automaton, ϕ - reflexive fuzzy relation on A. Then ϕ is: weakly right invariant, if ϕ τ u = τ u, for each u X. weakly left invariant, if σ u ϕ = σ u, for each u X. Right and left invariant fuzzy relations A = (A, σ, δ, τ) - fuzzy automaton, ϕ - fuzzy quasi-order on A. Then ϕ is: right invariant, if ϕ δ x ϕ = δ x ϕ, for each x X, ϕ τ = τ, left invariant, if ϕ δ x ϕ = ϕ δ x, for each x X, σ ϕ = σ. ϕ is right (left) invariant ϕ is weakly right (left) invariant

16 A determinization algorithm for fuzzy automata 15 Construction of family of fuzzy sets A = (A, σ, δ, τ) fuzzy automaton; X = {x 1, x 2,..., x m} alphabet; ϕ fuzzy relation on A; We define a family {ϕ D u} u X of fuzzy sets over A inductively by Construction of CDFA ϕ D ε = f(σ ϕ), Set A D ϕ = {ϕ D u u X }; ϕ D ux = f(ϕ D u δ x ϕ), for every u X and x X. Define fuzzy sets { δϕ D : A D ϕ X A D ϕ L, σϕ D : A D ϕ L and τϕ D : A D ϕ L with δϕ(α, D g(ϕ D u δ x ϕ), if α = ϕ D u and β = ϕ D ux, u X x, β) = ; 0, otherwise { σϕ(α) D g(σ ϕ), if α = ϕ D ε = 0, otherwise ; τ ϕ(α) D = α τ, for every α A D ϕ.

17 A determinization algorithm for fuzzy automata Example ϕ = ϕ ri = σ ϕ = [1 0 0] = g(σ ϕ) f(σ ϕ) = 1 [100] = 1 ϕ D ε ϕ D ε δ x ϕ = [ ] = g(ϕ D ε δ x ϕ) f(ϕ D ε δ x ϕ) = 0.5 [0.610] = 0.5 ϕ D x ϕ D ε δ y ϕ = [0 0 0] = g(ϕ D ε δ y ϕ) f(ϕ D ε δ y ϕ) = 1 [000] = 1 ϕ D y ϕ D x δ x ϕ = [ ] = g(ϕ D x δ x ϕ) f(ϕ D x δ x ϕ) = 0.3 [111] = 0.3 ϕ D x 2

18 A determinization algorithm for fuzzy automata Example (cont.) 17

19 A determinization algorithm for fuzzy automata (cont.) 18 Lemma Let A = (A, σ, δ, τ) be a fuzzy automaton over X, D = (f, g) be a factorization of A, and ϕ a fuzzy relation on A. Then A D ϕ = (A D ϕ, σ D ϕ, δ D ϕ, τ D ϕ) is a CDFA. Theorem Let A = (A, σ, δ, τ) be a fuzzy automaton over X, D = (f, g) a factorization of A, ϕ a reflexive weakly right invariant fuzzy relation on A, and let A D ϕ = (A D ϕ, σ D ϕ, δ D ϕ, τ D ϕ). Then [[A]] = [[A D ϕ]]. Theorem Let A = (A, σ, δ, τ) be a fuzzy automaton over X, D = (f, g) a maximal factorization of A, and let ϕ and φ be right invariant fuzzy quasi-orders on A. Moreover, let A D ϕ = (A D ϕ, σ D ϕ, δ D ϕ, τ D ϕ) and A D φ = (A D φ, σ D φ, δ D φ, τ D φ) be the corresponding CDFAs of A. If ϕ φ holds, then A D φ A D ϕ follows.

20 Example 1 19 Consider a fuzzy automation A over the alphabet X = {x, y} and the product (Gougen) structure { 1, a b a b = a b, a b = b/a, otherwise with the transition graph shown in the next Figure.

21 Example 1 (cont.) 20 The transition tree (a) and the transition graph (b) of the CDFA A D of the fuzzy automaton A:

22 Example 1 (cont.) On the other hand, the greatest right invariant fuzzy quasi-order on A is given by ϕ = ϕ ri = The transition tree (a) and the transition graph (b) of the CDFA A D ϕ of the fuzzy automaton A:

23 Example 2 22 Let A be an automaton over the one-element alphabet X = {x} and the product structure given by the transition graph shown in Figure a). The Nerode automaton of A and the CDFA A D of A have infinitely many states. The greatest right invariant fuzzy quasi-order is equal to ϕ = ϕ ri = The resulting CDFA A D ϕ is shown in Figure b).

24 A canonization algorithm for fuzzy automata 23 If the previous algorithm yields the finite CDFA, then it is not necessarily a minimal CDFA equivalent to the starting fuzzy automaton A Previous method can be adapted to a double-reverse canonization method for fuzzy finite automata over complete residuated lattices The idea of constructing the double-reverse canonization algorithm is not new in the literature: Brzozowski s original idea was to construct, for a given nondeterministic automaton A, the minimal deterministic automaton d(r(d(r(a)))) equivalent to A, where by r(a) we mean the reverse, and by d(a) we mean the determinization process (for example, the well-known subset construction) We adapt the Brzozowski s procedure and perform twice the construction of the CDFA that recognizes the reverse fuzzy language of the starting fuzzy automaton

25 A canonization algorithm for fuzzy automata (cont.) Construction of family of fuzzy sets A = (A, σ, δ, τ) fuzzy automaton; X = {x 1, x 2,..., x m} alphabet; ϕ fuzzy relation on A; We define a family { ϕ D u} u X of fuzzy sets over A inductively by 24 ϕ D ε = f(ϕ τ), ϕ D xu = f(ϕ δ x ϕ D u). Construction of CDFA equivalent to A Set à D ϕ = { ϕ D u} u X Define fuzzy sets { δ ϕ D : à D ϕ X à D ϕ L by, σ ϕ D : à D ϕ L and τ ϕ D : à D ϕ L with δ ϕ(α, D g(ϕ δ x ϕ D u), α = ϕ D u and β = ϕ D xu, u X x, β) = ; 0, otherwise { σ ϕ(α) D g(ϕ τ), α = ϕ D ε = 0, otherwise, τ ϕ(α) D = σ α, for every α à D ϕ.

26 A canonization algorithm for fuzzy automata (cont.) 25 Lemma Let A = (A, σ, δ, τ) be a fuzzy automaton over X, D = (f, g) be a factorization of A, and ϕ a fuzzy relation on A. Then à D ϕ = (à D ϕ, σ D ϕ, δ D ϕ, τ D ϕ) is a CDFA. Theorem Let A = (A, σ, δ, τ) be a fuzzy automaton over X, D = (f, g) a factorization of A, ϕ a reflexive weakly left invariant fuzzy relation on A, and let à D ϕ = (à D ϕ, σ D ϕ, δ D ϕ, τ D ϕ). Then [[A]] = [[à D ϕ]]. Theorem Let A = (A, σ, δ, τ) be a fuzzy automaton over X, D = (f, g) a maximal factorization of A, and let ϕ and φ be left invariant fuzzy quasi-orders on A. Let à D ϕ = (à D ϕ, σ D ϕ, δ D ϕ, τ D ϕ) and à D φ = (à D φ, σ D φ, δ D φ, τ D φ) be the corresponding CDFAs equivalent to A. If ϕ φ holds, then à D φ à D ϕ follows.

27 A canonization algorithm for fuzzy automata (cont.) 26 Let A = (A, σ, δ, τ) be a fuzzy automaton over X and L, D = (f, g) a maximal factorization of A, and ϕ a left invariant fuzzy quasi-order on A. By constructing the fuzzy automaton à D ϕ = (à D ϕ, σ ϕ, D δ ϕ, D τ ϕ), D we obtain CDFA that is equivalent to the reverse fuzzy automaton A of the initial fuzzy automaton A. Given the maximal factorization D 1 = (f 1, g 1) of à D ϕ and the left invariant fuzzy quasi-order ϕ 1 on à D ϕ, we ask if the fuzzy automaton (à D ϕ) ϕ 1 is equivalent to A: The obtained fuzzy automaton is a minimal CDFA equivalent to the starting fuzzy automaton, Fuzzy relation ϕ 1 does not play a role in the second reverse process and can be replaced by the identity relation. D 1

28 A canonization algorithm for fuzzy automata (cont.) 27 Theorem Let A = (A, σ, δ, τ) be a fuzzy automaton over X and L, D = (f, g) a maximal factorization of A, ϕ a left invariant fuzzy quasi-order on A, and D 1 = (f 1, g 1) a maximal factorization of A D ϕ. Define a fuzzy automaton B as B = (Ã D ϕ)d 1. Then B is a minimal CDFA equivalent to A.

29 Example 1 28 Consider a fuzzy automation A over the alphabet X = {x, y} and the product (Gougen) structure { 1, a b a b = a b, a b = b/a, otherwise with the transition graph shown in the next Figure.

30 Example 1 (cont.) The reverse CDFA à D is infinite, and part of its transition graph is shown in Figure (a). On the other hand, we have that the greatest left invariant fuzzy quasi-order is equal to ϕ = ϕ li = The automaton à D ϕ is finite, and its transition graph is shown in Figure (b). 29

31 Example 1 (cont.) 30 The transition graph of the fuzzy automaton B = (Ã D ϕ)d 1 is shown in the following figure.

32 Example 2 Let A be an automaton over the alphabet X = {x, y} and the product (Gougen) structure with the transition graph shown in Figure (a). The greatest right invariant and the greatest left invariant fuzzy quasi-orders on A are respectively equal to ϕ = ϕ ri = , ϕ li = The Brzozowski fuzzy automaton B cannot be obtained by means of the Algorithm. On the other hand, the resulting CDFA A D ϕ is shown in Figure (b). a) b)

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