Fundamental Problems of Fuzzy Automata Theory

Transcription

1 Fundamental Problems of Fuzzy Automata Theory Jelena Ignjatović Department of Computer Science Faculty of Sciences University of Niš, Serbia ARISTOTLE UNIVERSITY OF THESSALONIKI April 2016, Thessaloniki, Greece 1 Jelena Ignjatović General Problems of Fuzzy Automata Theory

2 Fuzzy sets and fuzzy logic Fuzzy logic logic of graded truth or intermediate truth provides a way to express subtle nuances in reasoning successful in modeling uncertainty Structures of truth values has to be ordered the ordering need not be linear operations for modeling logical operations triangular norms and conorms on the real unit interval [0, 1] Gödel, Łukasiewicz and product structure nonlinear structures: complete residuated lattices (incl Brouwerian lattices, Heyting algebras, MV-algebras, BL-algebras, etc) quantales, lattice-ordered monoids, etc 2 Jelena Ignjatović General Problems of Fuzzy Automata Theory

3 Fuzzy vs classical logic Classical Fuzzy logics with Fuzzy logics with Boolean logic linearly ordered more general structures structures of truth values of truth values (not necessarily linearly ordered) two-element structures on [0, 1] residuated lattices, Boolean algebra determined by t-norms quantales, etc 3 Jelena Ignjatović General Problems of Fuzzy Automata Theory

4 Complete residuated lattices Complete residuated lattice a tuplel=(l,,,,, 0, 1) such that (L1) (L,,, 0, 1) is a complete lattice with the least element 0 and the greatest element 1, (L2) (L,, 1) is a commutative monoid with the unit 1, (L3) and satisfy the residuation property: for all x, y, z L, x y z x y z special cases, on [0, 1] with x y=min(x, y) and x y=max(x, y): { x y=max(x+y 1, 0), Łukasiewicz structure: x y=min(1 x+y, 1) { x y=min(x, y), Gödel structure: x y=1 if x y and = y otherwise { x y=x y), product structure: x y=1 if x y and = y x otherwise 4 Jelena Ignjatović General Problems of Fuzzy Automata Theory

5 Fuzzy sets and fuzzy relations Fuzzy sets fuzzy subset of a set A is a functionα:a L equality:α=β α(a)=β(a), for each a A inclusion:α β α(a) β(a), for each a A union and intersection ( ) α i (a)= α i (a), i I i I ( ) α i (a)= α i (a) if A is finite, with A =n, thenαis an n-dimensional fuzzy vector L A the set of all fuzzy subsets of A Fuzzy relations fuzzy relation between sets A and B is a fuzzy subset of A B, ie, a function R : A B L if A and B are finite, A =m and B =n, then R is a m n fuzzy matrix L A B the set of all fuzzy relations between A and B i I i I 5 Jelena Ignjatović General Problems of Fuzzy Automata Theory

6 Fuzzy equivalences and fuzzy quasi-orders a fuzzy relation R L A A is reflexive, if R(a, a)=1, for all a A; symmetric, if R(a, b)=r(b, a), for all a, b A; transitive, if R(a, b) R(b, c) R(a, c), for all a, b, c A Fuzzy equivalence reflexive, symmetric and transitive fuzzy relation for a fuzzy equivalence E on A and a A, a fuzzy subset E a L A defined by E a (b)=e(a, b) is an equivalence class of E determined by a Fuzzy quasi-order reflexive and transitive fuzzy relation for a fuzzy quasi-order Q on A and a A, a fuzzy subset Q a L A defined by Q a (b)=q(a, b) is an afterset of E determined by a 6 Jelena Ignjatović General Problems of Fuzzy Automata Theory

7 Compositions Composition of fuzzy relations (matrix product) for R L A B and S L B C, the composition R S L A C is defined by (R S)(a, c)= R(a, b) S(b, c) Composition of a fuzzy set and a fuzzy relation (vector-matrix products) for R L A B,α L A andβ L B, the compositionsα R L B and R β L A are defined by (α R)(b)= α(a) R(a, b), (R β)(a)= R(a, b) β(b) a A Composition of fuzzy sets (scalar product, dot product) forα,β L A, the compositionα β Lis defined by α β= α(a) β(a) b B a A b B 7 Jelena Ignjatović General Problems of Fuzzy Automata Theory

8 Residuals right residual of T by S (for S L A B, T L A C ): S\T L B C is given by (S\T)(b, c)= S(a, b) T(a, c) a A left residual of T by S (for S L B C, T L A C ): T/S L A B is given by (T/S)(a, b)= S(b, c) T(a, c) residuation property: S T U T S\U S U/T S\U=max{X L B C S X U}, U/T=max{X L A B X S U} other kinds of residuals: fuzzy set by scalar fuzzy set by fuzzy set scalar by fuzzy set fuzzy set by fuzzy relation c C 8 Jelena Ignjatović General Problems of Fuzzy Automata Theory

9 Historical overview Concept of fuzzy automata natural generalization of the concept of non-deterministic automata Močkoř, Bělohlávek, Li and Pedrycz Močkoř fuzzy automata represented as nested systems of non-deterministic automata Bělohlávek deterministic automata with fuzzy sets of final states represented as nested systems of deterministic automata Li and Pedrycz fuzzy automata represented as automata with fuzzy transition relations taking membership values in a lattice ordered monoid Nondeterministic automaton quantuplea=(a, X,δ,σ,τ) A set of states, δ A X A X input alphabet (δ x A A) transition relation (a, x, b) δ (a, b) δ x, for all a, b A, x X σ A, τ A sets of initial and terminal states 9 Jelena Ignjatović General Problems of Fuzzy Automata Theory

10 Nondeterministic automata Transition relations, sets of initial and terminal states represented by Boolean matrices and vectors: δ x = 0 1 1, δ y= 0 0 1, σ=[ ] 0, τ= x, y x x a 0 x y y x a 1 a 2 y Extended transition relations the family{δ u } u X A A defined by δ ε = A, δ ux =δ u δ x (δ uv =δ u δ v ), u, v X, x X X monoid of words over X,ε empty word 10 Jelena Ignjatović General Problems of Fuzzy Automata Theory

11 Nondeterministic automata Successful path start in an initial, terminate in a final state u X is accepted (recognized) word (label of successful path); (σ δ u ) τ Language recognized bya [[A]] X set of all words accepted bya u [[A]] ( a 1, a 2 A) a 1 σ (a 1, a 2 ) δ u a 2 τ [[A]]={u X (σ δ u ) τ }={u X σ δ u τ=1} automataaandbare equivalent if [[A]]=[[B]] 11 Jelena Ignjatović General Problems of Fuzzy Automata Theory

12 Fuzzy automata Fuzzy automaton quantuplea=(a, X,δ,σ,τ) A set of states, X input alphabet δ : A X A L fuzzy transition function σ : A L,τ : A L fuzzy sets of initial and terminal states Fuzzy transition relations the family{δ x } x X L A A defined byδ x (a, b)=δ(a, x, b) the family{δ u } u X L A A defined by δ ε = A, δ ux =δ u δ x (δ uv =δ u δ v ), u, v X, x X Fuzzy language recognized bya [[A]] L X is defined by [[A]](u)=σ δ u τ, for u X fuzzy automataaandbare equivalent if [[A]]=[[B]] 12 Jelena Ignjatović General Problems of Fuzzy Automata Theory

13 Determinization Crisp-deterministic fuzzy automaton (CDFA) δ is a crisp function of A X into A, ie, anyδ u is a crisp function on A σ is a crisp singleton σ={a 0 } τ is a fuzzy set and [[A]](u)=τ(δ u (a 0 )) Fuzzy Seets and Systems, 2010 (Ignjatović, Ćirić, Bogdanović, Petković) extensive study of crisp-deterministic fuzzy automata Determinization problem practical applications of automata mainly require deterministic automata Determinization problem: construct a crisp-deterministic fuzzy automaton equivalent to A crisp noneterministic case: possible exponential growth of the number of states during the determinization fuzzy case: the determinization may even yield an infinite automaton the need for determinization methods that would mitigate the potential enormous growth of the number of states 13 Jelena Ignjatović General Problems of Fuzzy Automata Theory

14 Our determinization methods Fuzzy Seets and Systems, 2008 Nerode automaton ofa Accessible Subset Construction (ASC) A σ = (A σ,σ ε, X,δ σ,τ σ ) A σ ={σ u u X } σ u =σ δ u, for u X (equivalentlyσ ux =σ u δ x ) initial state σ ε =σ, transition function δ σ (σ u, x)=σ ux fuzzy set of final states τ σ (σ u )=σ u τ we build the transition tree of the new CDFA smaller automata than Bělohlávek (INS, 2002), Li and Pedrycz (FSS, 2005) x/0, y/0 a 0 1 x/05, y/1 x/0, y/0 x/1, y/05 a 1 σ e 1 x y 05 1 σ x σ y y x, y x 14 Jelena Ignjatović General Problems of Fuzzy Automata Theory

15 Right and left invariant fuzzy quasi-orders/equivalences Right invariant fuzzy quasi-orders/equivalences fuzzy quasi-orders/equivalences which are solutions to system R τ τ, R δ x δ x R (x X) Left invariant fuzzy quasi-orders/equivalences fuzzy quasi-orders/equivalences which are solutions to system σ R σ, δ x R R δ x (x X) Information Sciences, 2011 (Jančić, Ignjatović, Ćirić) Fuzzy Sets and Systems, 2016 (Jančić, Micić, Ignjatović, Ćirić) Simultaneous determinization and state reduction Q right invariant fuzzy quasi-order A Q = (A Q, Q ε, X,δ Q,τ Q ) A Q ={Q u u X } states Q ε =σ Q, Q ux = Q u δ x Q transitions δ Q (Q u, x)=q ux terminal states τ Q (Q u )=Q u τ 15 Jelena Ignjatović General Problems of Fuzzy Automata Theory

16 Our determinization methods we build the transition tree of the new CDFA in polinomial time Q left invariant fuzzy quasi-order A Q = (A Q, Q ε, X,δ Q,τ Q ) A Q ={Q u u X } states Q ε = Q τ, Q xu = Q δ x Q u transitions δ Q (Q u, x)=q xu terminal states τ Q (Q u )=σ Q u x a 3 a 1 y x, y y x x a 2 y 16 Jelena Ignjatović General Problems of Fuzzy Automata Theory

17 Our determinization methods x, y σ x 2 y σ xyx y x x, y y σ x 2 x x σ x x y σ xy y σ ε x y σ y Q x 2 x y x Q x x y x, y Q xy y x Q ε y Q y A few more methods for determinization of fuzzy automata Determinization of fuzzy automata by means of the degrees of language inclusion Determinization of fuzzy automata using simulations 17 Jelena Ignjatović General Problems of Fuzzy Automata Theory

18 Determinisation and state reduction Further improvement: children automaton related to deterministion using transition sets Q the greatest right invariant fuzzy quasi-order A c Q = (Ac Q, Qc ε, X,δc Q,τc Q ) X={x 1,, x m } finite alphabet A c Q ={Qc u u X } states Q c u = (Q ux 1,, Q uxm, Q u τ) transitions δ c Q (Qc u, x) Q c ux terminal states τ c Q (Qc u)=q u τ 18 Jelena Ignjatović General Problems of Fuzzy Automata Theory

19 Our canonization methods Canonization problem Construction a minimal crisp-deterministic fuzzy automaton equivalent toa Fuzzy Sets and Systems, 2014 (Jančić, Ćirić) Brzozowski type determinization algorithm fuzzy version reversion ASC reversion ASC minimal CDFA canonization by means of the degrees of language inclusion generally faster canonization algorithm 19 Jelena Ignjatović General Problems of Fuzzy Automata Theory

20 State reduction State reduction problem many constructions used in practice cause enormous growth in a number of states determinization Discrete Event Systems: modular approach lead to parallel compositions with large number of states State reduction problem: reduce or minimize the number of states ofa the state minimization of fuzzy automata is computationally hard Practical state reduction problem: replaceaby an equivalent automaton with as small as possible number of states, which need not be minimal but must be effectively computable 20 Jelena Ignjatović General Problems of Fuzzy Automata Theory

21 Factor automata Afterset automata Factor automatona/e ofawrt E E fuzzy equivalence on the set of states A A/E=(A/E, X, T E,σ E,τ E ) A/E={E a a A} factor set or quotient set states σ E (E a )=(σ E)(a) transitions δ E x (E a, E b )=(E δ x E)(a, b) terminal states τ E (E a )=(E τ)(a) Afterset automatona/q a fuzzy equivalence is replaced by a fuzzy quasi-order, and classes by aftersets Fuzzy language recognized bya/q [[A/R]](e)=σ R τ [[A/R]](u)=σ R δ x1 R δ x2 R R δ xn R τ, u=x 1 x 2 x n X 21 Jelena Ignjatović General Problems of Fuzzy Automata Theory

22 The general system The equivalence ofa/q anda The general system A/Q is equivalent toaif and only if Q is a solution to the system σ τ=σ R τ, σ δ x1 δ x2 δ xn τ=σ R δ x1 R δ x2 R R δ xn R τ x 1, x 2,, x n X (R is an unknown taking values in L A A ) The general system it may be hard for solving it may consist of infinitely many equations we have to find as possible greater solutions (greater solutions provide better reductions) in the general case, there is no the greatest solution Instances of the general system systems whose any solution is a solution to the general system we need instances consisting of finitely many equations or inequalities which have the greatest solution 22 Jelena Ignjatović General Problems of Fuzzy Automata Theory

23 Our main results Journal of Computer and System Sciences, 2010 (Ćirić, Stamenković, Ignjatović, Petković) methods for computing the greatest right and left invariant fuzzy equivalences fuzzy equivalences provide better reductions than crisp equivalences alternate reductions by right and left invariant fuzzy equivalences provide even better results fuzzy automaton reduced by the greatest RIFE can not be reduced again by a RIFE, but can by a LIFE, and vice versa Information Sciences, 2014 (Stamenković, Ćirić, Ignjatović) methods for computing the greatest right and left invariant fuzzy quasi-orders fuzzy quasi-orders provide better reductions than fuzzy equivalences alternate reductions by right and left invariant fuzzy quasi-orders applications to FDES conflict analysis of FDES 23 Jelena Ignjatović General Problems of Fuzzy Automata Theory

24 Reduction of fuzzy automata x/0, y/0 x/0, y/1 2 x/0, y/1 x/1, y/1 x/0, y/1 x/1, y/1 x/1, y/1 1 R 1 R 2 x/0, y/0 x/0, y/0 x/0, y/0 x/0, y/1 x/0, y/1 3 x/0, y/0 24 Jelena Ignjatović General Problems of Fuzzy Automata Theory

25 Simulation, bisimulation, equivalence Bisimulation relations originated independently in computer science, modal logic and set theory concurrency theory a means for testing behavioural equivalence among processes, for reduction of the state-space of processes applications: program verification, model checking, functional languages, object-oriented languages, databases, compiler optimization,etc Our contribution we defined two types of simulations for fuzzy automata forward, backward four types of bisimulations forward, backward, forward-backward, backward-forward 25 Jelena Ignjatović General Problems of Fuzzy Automata Theory

26 Equivalent fuzzy automata 1 1 a 1 x/1, y/05 x/1, y/06 x/06, y/ x/07, y/1 x/07, y/1 a 0 b 0 b 1 1 x/03, y/06 x/05, y/06 x/04, y/02 x/04, y/07 x/06, y/07 x/03, y/04 1 x/06, y/07 05 a 2 x/1, y/ Jelena Ignjatović General Problems of Fuzzy Automata Theory

27 Simulation, bisimulation, equivalence Two main roles of bisimulations fuzzy automata theory model the equivalence between states of two different fuzzy automata, reduce the number of states Fuzzy automata A=(A,δ A,σ A,τ A ),B=(B,δ B,σ B,τ B ) ϕ : A B L Forward simulations σ A σ B ϕ 1 ϕ 1 δ A x δb x ϕ 1 ϕ 1 τ A τ B Backward simulations σ A ϕ σ B δ A x ϕ ϕ δb x τ A ϕ τ B 27 Jelena Ignjatović General Problems of Fuzzy Automata Theory

28 Forward and backward simulations ϕ σ A a 0 b 0 σ B A x 1 x k x 1 x k B a k b k x k+1 x k+1 x k+2 x n a k+1 b k+1 x k+2 x n τ A a n b n τ B 28 Jelena Ignjatović General Problems of Fuzzy Automata Theory

29 Forward and backward simulations A σ A a 0 x 1 x k a k x k+1 ϕ forward simulation σ B b 0 x 1 B x k b k x k+1 x k+2 x n a k+1 b k+1 x k+2 x n τ A a n b n τ B 28 Jelena Ignjatović General Problems of Fuzzy Automata Theory

30 Forward and backward simulations arbitrary successful run A σ A a 0 x 1 x k a k x k+1 ϕ forward simulation σ B b 0 x 1 B x k b k x k+1 x k+2 x n a k+1 b k+1 x k+2 x n τ A a n b n τ B 28 Jelena Ignjatović General Problems of Fuzzy Automata Theory

31 Forward and backward simulations arbitrary successful run A σ A a 0 x 1 x k a k x k+1 δ A x k+1 ϕ forward simulation σ B b 0 x 1 B x k b k x k+1 x k+2 x n a k+1 b k+1 x k+2 x n τ A a n b n τ B 28 Jelena Ignjatović General Problems of Fuzzy Automata Theory

32 Forward and backward simulations arbitrary successful run A σ A a 0 x 1 x k a k x k+1 x k+2 x n a k+1 δ A x k+1 ϕ forward simulation σ B b 0 x 1 B x k building b k a successful run which simulates x k+1 the original run b k+1 x k+2 x n τ A a n b n τ B 28 Jelena Ignjatović General Problems of Fuzzy Automata Theory

33 Forward and backward simulations arbitrary successful run A σ A a 0 x 1 x k a k x k+1 x k+2 x n a k+1 ϕ δ A x k+1 δ B x k+1 forward simulation σ B b 0 x 1 B x k building b k a successful run which simulates x k+1 the original run b k+1 x k+2 x n τ A a n b n τ B 28 Jelena Ignjatović General Problems of Fuzzy Automata Theory

34 Our main results Fuzzy Sets and Systems, 2012 (Ćirić, Ignjatović, Damljanović, Bašić) Fuzzy Sets and Systems, 2012 (Ćirić, Ignjatović, Jančić, Damljanović) simulations and bisimulations for fuzzy automata algorithms for testing the existence of a simulation/bisimulation between two fuzzy automata, and computing the greatest one bisimulation equivalence of fuzzy automata Information Sciences, 2014 (Ćirić, Ignjatović, Bašić, Jančić) Theoretical Computer Science, 2014 (Damljanović, Ćirić, Ignjatović) simulations and bisimulations for weighted automata over an additively idempotent semiring relative residuation Boolean residuals 29 Jelena Ignjatović General Problems of Fuzzy Automata Theory

35 Our main results THANK YOU FOR YOUR ATTENTION 30 Jelena Ignjatović General Problems of Fuzzy Automata Theory