Fundamental Problems of Fuzzy Automata Theory Jelena Ignjatovi´ c Department of Computer Science Faculty of Sciences University of Niˇ s, Serbia jelena.ignjatovic@pmf.edu.rs ARISTOTLE UNIVERSITY OF THESSALONIKI April 2016, Thessaloniki, Greece 1 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory
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] G¨ odel , 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 Ignjatovi´ c General Problems of Fuzzy Automata Theory
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 Ignjatovi´ c General Problems of Fuzzy Automata Theory
Complete residuated lattices Complete residuated lattice ⋆ a tuple L = ( 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 ) , � ⋄ G¨ odel structure : x → y = 1 if x � y and = y otherwise x ⊗ y = x · y ) , � x → y = 1 if x � y and = y ⋄ product structure : x otherwise 4 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory
Fuzzy sets and fuzzy relations Fuzzy sets ⋆ fuzzy subset of a set A is a function α : 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 ) , α i ( a ) = α i ( a ) i ∈ I i ∈ I i ∈ I i ∈ I ⋆ if A is finite, with | A | = n , then α is 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 , i.e., 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 5 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory
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 Ignjatovi´ c General Problems of Fuzzy Automata Theory
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 ) b ∈ B Composition of a fuzzy set and a fuzzy relation (vector-matrix products) ⋆ for R ∈ L A × B , α ∈ L A and β ∈ L B , the compositions α ◦ 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 b ∈ B Composition of fuzzy sets (scalar product, dot product) ⋆ for α, β ∈ L A , the composition α ◦ β ∈ L is defined by � α ◦ β = α ( a ) ⊗ β ( a ) a ∈ A 7 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory
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 ) c ∈ 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 8 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory
Historical overview Concept of fuzzy automata ⋆ natural generalization of the concept of non-deterministic automata Moˇ ckoˇ r, Bˇ elohl´ avek, Li and Pedrycz ⋆ Moˇ ckoˇ r – fuzzy automata represented as nested systems of non-deterministic automata ⋆ Bˇ elohl´ avek – 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 ⋆ quantuple A = ( A , X , δ, σ, τ ) A � ∅ – set of states , X � ∅ – input alphabet δ ⊆ A × X × A ( 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 Ignjatovi´ c General Problems of Fuzzy Automata Theory
Nondeterministic automata Transition relations, sets of initial and terminal states ⋆ represented by Boolean matrices and vectors: 1 1 0 1 1 0 0 � � 0 1 1 0 0 1 1 0 0 0 δ x = , δ y = , σ = , τ = . 1 0 0 0 0 1 1 x , y x x a 0 a 1 y y x x a 2 y ⋆ Extended transition relations – the family { δ u } u ∈ X ∗ ⊆ A × A defined by u , v ∈ X ∗ , x ∈ X δ ε = ∆ A , δ ux = δ u ◦ δ x ( δ uv = δ u ◦ δ v ) , X ∗ – monoid of words over X , ε – empty word 10 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory
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 by A ] ⊆ X ∗ – set of all words accepted by A ⋆ [ [ A ] ⋆ u ∈ [ [ A ] ] ⇔ ( ∃ a 1 , a 2 ∈ A ) a 1 ∈ σ ∧ ( a 1 , a 2 ) ∈ δ u ∧ a 2 ∈ τ ] = { u ∈ X ∗ | ( σ ◦ δ u ) ∩ τ � ∅} = { u ∈ X ∗ | σ ◦ δ u ◦ τ = 1 } ⋆ [ [ A ] ⋆ automata A and B are equivalent if [ [ A ] ] = [ [ B ] ] 11 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory
Fuzzy automata Fuzzy automaton ⋆ quantuple A = ( 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 by δ x ( a , b ) = δ ( a , x , b ) ⋆ the family { δ u } u ∈ X ∗ ⊆ L A × A defined by u , v ∈ X ∗ , x ∈ X δ ε = ∆ A , δ ux = δ u ◦ δ x ( δ uv = δ u ◦ δ v ) , Fuzzy language recognized by A ] ∈ L X ∗ is defined by [ for u ∈ X ∗ ⋆ [ [ A ] [ A ] ]( u ) = σ ◦ δ u ◦ τ , ⋆ fuzzy automata A and B are equivalent if [ [ A ] ] = [ [ B ] ] 12 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory
Determinization Crisp-deterministic fuzzy automaton (CDFA) ⋆ δ is a crisp function of A × X into A , i.e., any δ 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 )) c, ´ Fuzzy Seets and Systems, 2010 (Ignjatovi´ Ciri´ c, Bogdanovi´ c, Petkovi´ c) – 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 Ignjatovi´ c General Problems of Fuzzy Automata Theory
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