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Fundamental Problems of Fuzzy Automata Theory Jelena Ignjatovi c Department of Computer Science Faculty of Sciences University of Ni s, Serbia jelena.ignjatovic@pmf.edu.rs ARISTOTLE UNIVERSITY OF THESSALONIKI April 2016, Thessaloniki,

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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

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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¨

  • del, 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

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Fuzzy vs. classical logic

Classical Fuzzy logics with Fuzzy logics with Boolean logic linearly ordered more general structures structures

  • f truth values
  • f truth values

(not necessarily linearly

  • rdered)

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

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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): ⋄ Łukasiewicz structure: x ⊗ y = max(x + y − 1, 0), x → y = min(1 − x + y, 1) ⋄ G¨

  • del structure:
  • x ⊗ y = min(x, y),

x → y = 1 if x y and = y otherwise ⋄ product structure:

  • x ⊗ y = x · y),

x → y = 1 if x y and = y x otherwise

4 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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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

  • i∈I

αi

  • (a) =
  • i∈I

αi(a),

  • i∈I

αi

  • (a) =
  • i∈I

αi(a) ⋆ if A is finite, with |A| = n, then α is an n-dimensional fuzzy vector ⋆ LA – 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 ⋆ LA×B – the set of all fuzzy relations between A and B

5 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Fuzzy equivalences and fuzzy quasi-orders

a fuzzy relation R ∈ LA×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 Ea ∈ LA defined by Ea(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 Qa ∈ LA defined by Qa(b) = Q(a, b) is an afterset of E determined by a

6 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Compositions Composition of fuzzy relations (matrix product)

⋆ for R ∈ LA×B and S ∈ LB×C, the composition R ◦ S ∈ LA×C is defined by (R ◦ S)(a, c) =

  • b∈B

R(a, b) ⊗ S(b, c)

Composition of a fuzzy set and a fuzzy relation (vector-matrix products)

⋆ for R ∈ LA×B, α ∈ LA and β ∈ LB, the compositions α ◦ R ∈ LB and R ◦ β ∈ LA are defined by (α ◦ R)(b) =

  • a∈A

α(a) ⊗ R(a, b), (R ◦ β)(a) =

  • b∈B

R(a, b) ⊗ β(b)

Composition of fuzzy sets (scalar product, dot product)

⋆ for α, β ∈ LA, the composition α ◦ β ∈ L is defined by α ◦ β =

  • a∈A

α(a) ⊗ β(a)

7 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Residuals

⋆ right residual of T by S (for S ∈ LA×B, T ∈ LA×C): S\T ∈ LB×C is given by (S\T)(b, c) =

  • a∈A

S(a, b) → T(a, c) ⋆ left residual of T by S (for S ∈ LB×C, T ∈ LA×C): T/S ∈ LA×B is given by (T/S)(a, b) =

  • c∈C

S(b, c) → T(a, c) ⋆ residuation property: S ◦ T U ⇔ T S\U ⇔ S U/T ⋆ S\U = max {X ∈ LB×C | S ◦ X U}, U/T = max {X ∈ LA×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

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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 (δx ⊆ A × A) – transition relation (a, x, b) ∈ δ ⇔ (a, b) ∈ δx, 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

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Nondeterministic automata Transition relations, sets of initial and terminal states

⋆ represented by Boolean matrices and vectors:

δx =         1 1 1 1 1         , δy =         1 1 1 1         , σ =

  • 1
  • ,

τ =         1         .

a1 a0 a2 x y x y x x, y x y

⋆ Extended transition relations – the family {δu}u∈X∗ ⊆ A × A defined by δε = ∆A, δux = δu ◦ δx (δuv = δu ◦ δv), u, v ∈ X∗, x ∈ X X∗ – monoid of words over X, ε – empty word

10 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Nondeterministic automata Successful path

⋆ start in an initial, terminate in a final state ⋆ u ∈ X∗ is accepted (recognized) word (label of successful path); (σ ◦ δu) ∩ τ ∅

Language recognized by A

⋆ [ [A] ] ⊆ X∗ – set of all words accepted by A ⋆ u ∈ [ [A] ] ⇔ (∃a1, a2 ∈ A) a1 ∈ σ ∧ (a1, a2) ∈ δu ∧ a2 ∈ τ ⋆ [ [A] ] = {u ∈ X∗| (σ ◦ δu) ∩ τ ∅} = {u ∈ X∗| σ ◦ δu ◦ τ = 1} ⋆ automata A and B are equivalent if [ [A] ] = [ [B] ]

11 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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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 {δx}x∈X ⊆ LA×A defined by δx(a, b) = δ(a, x, b) ⋆ the family {δu}u∈X∗ ⊆ LA×A defined by δε = ∆A, δux = δu ◦ δx (δuv = δu ◦ δv), u, v ∈ X∗, x ∈ X

Fuzzy language recognized by A

⋆ [ [A] ] ∈ LX∗ is defined by [ [A] ](u) = σ ◦ δu ◦ τ, for u ∈ X∗ ⋆ fuzzy automata A and B are equivalent if [ [A] ] = [ [B] ]

12 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Determinization Crisp-deterministic fuzzy automaton (CDFA)

⋆ δ is a crisp function of A × X into A, i.e., any δu is a crisp function on A σ is a crisp singleton – σ = {a0} τ is a fuzzy set and [ [A] ](u) = τ(δu(a0)) Fuzzy Seets and Systems, 2010 (Ignjatovi´ c, ´ 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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Our determinization methods

Fuzzy Seets and Systems, 2008 ⋆ Nerode automaton of A – Accessible Subset Construction (ASC) Aσ = (Aσ, σε, X, δσ, τσ) Aσ = {σu | u ∈ X∗} σu = σ ◦ δu, for u ∈ X∗ (equivalently σux = σu ◦ δx) initial state – σε = σ, transition function – δσ(σu, x) = σux fuzzy set of final states – τσ(σu) = σu ◦ τ ⋆ we build the transition tree of the new CDFA ⋆ smaller automata than Bˇ elohl´ avek (INS, 2002), Li and Pedrycz (FSS, 2005) a1 a0

1 x/0.5, y/1 x/0, y/0 x/1, y/0.5 x/0, y/0

σe σx σy

1 0.5 1

x y y x, y x

14 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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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 ◦ δx δx ◦ R (x ∈ X)

Left invariant fuzzy quasi-orders/equivalences

⋆ fuzzy quasi-orders/equivalences which are solutions to system σ ◦ R σ, δx ◦ R R ◦ δx (x ∈ X) Information Sciences, 2011 (Janˇ ci´ c, Ignjatovi´ c, ´ Ciri´ c) Fuzzy Sets and Systems, 2016 (Janˇ ci´ c, Mici´ c, Ignjatovi´ c, ´ Ciri´ c) ⋆ Simultaneous determinization and state reduction Q – right invariant fuzzy quasi-order AQ = (AQ, Qε, X, δQ, τQ) AQ = {Qu | u ∈ X∗} states – Qε = σ ◦ Q, Qux = Qu ◦ δx ◦ Q transitions – δQ(Qu, x) = Qux terminal states – τQ(Qu) = Qu ◦ τ

15 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Our determinization methods

⋆ we build the transition tree of the new CDFA in polinomial time ⋆ Q – left invariant fuzzy quasi-order AQ = (AQ, Qε, X, δQ, τQ) AQ = {Qu | u ∈ X∗} states – Qε = Q ◦ τ, Qxu = Q ◦ δx ◦ Qu transitions – δQ(Qu, x) = Qxu terminal states – τQ(Qu) = σ ◦ Qu

a1 a2 a3 x, y x x y y x y

16 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Our determinization methods

σε σx σy σ

x2

σ

xy

σ

x2y

σ

xyx

x y x y x x y y x x, y y x, y Qε Qx Qy Q

x2

Q

xy

x y y x y y x x x, 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 Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Determinisation and state reduction

⋆ Further improvement: children automaton related to deterministion using transition sets ⋆ Q – the greatest right invariant fuzzy quasi-order Ac

Q = (Ac Q, Qc ε, X, δc Q, τc Q)

X = {x1, . . ., xm} – finite alphabet Ac

Q = {Qc u | u ∈ X∗}

states – Qc

u = (Qux1, . . ., Quxm, Qu ◦ τ)

transitions – δc

Q(Qc u, x) → Qc ux

terminal states – τc

Q(Qc u) = Qu ◦ τ

18 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Our canonization methods Canonization problem

⋆ Construction a minimal crisp-deterministic fuzzy automaton equivalent to A Fuzzy Sets and Systems, 2014 (Janˇ ci´ c, ´ Ciri´ c) ⋆ 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 Ignjatovi´ c General Problems of Fuzzy Automata Theory

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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 of A ⋆ the state minimization of fuzzy automata is computationally hard ⋆ Practical state reduction problem: replace A by an equivalent automaton with as small as possible number of states, which need not be minimal but must be effectively computable

20 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Factor automata. Afterset automata Factor automaton A/E of A w.r.t. E

⋆ E – fuzzy equivalence on the set of states A ⋆ A/E = (A/E, X, TE, σE, τE) A/E = {Ea | a ∈ A} – factor set or quotient set states – σE(Ea) = (σ ◦ E)(a) transitions – δE

x(Ea, Eb) = (E ◦ δx ◦ E)(a, b)

terminal states – τE(Ea) = (E ◦ τ)(a)

Afterset automaton A/Q

⋆ a fuzzy equivalence is replaced by a fuzzy quasi-order, and classes by aftersets

Fuzzy language recognized by A/Q

⋆ [ [A/R] ](e) = σ ◦ R ◦ τ ⋆ [ [A/R] ](u) = σ ◦ R ◦ δx1 ◦ R ◦ δx2 ◦ R ◦ · · ·◦ R ◦ δxn ◦ R ◦ τ, u = x1x2 . . . xn ∈ X∗

21 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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The general system The equivalence of A/Q and A – The general system

⋆ A/Q is equivalent to A if and only if Q is a solution to the system σ ◦ τ = σ ◦ R ◦ τ, σ ◦ δx1 ◦ δx2 ◦ · · · ◦ δxn ◦ τ = σ ◦ R ◦ δx1 ◦ R ◦ δx2 ◦ R ◦ · · · ◦ R ◦ δxn ◦ R ◦ τ x1, x2, . . ., xn ∈ X (R is an unknown taking values in LA×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 Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Our main results

Journal of Computer and System Sciences, 2010 ( ´ Ciri´ c, Stamenkovi´ c, Ignjatovi´ c, Petkovi´ c) ⋆ 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 (Stamenkovi´ c, ´ Ciri´ c, Ignjatovi´ c) ⋆ 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 Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Reduction of fuzzy automata

1 2 3 x / , y / x / , y / 1 x/0, y/0 x/0, y/1 x / , y / x / , y / x/0, y/1 x/0, y/0 x/1, y/1

R2 R1

x/1, y/1 x/0, y/1 x/1, y/1 x/0, y/1

24 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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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

  • ptimization,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 Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Equivalent fuzzy automata

a0 a1 a2 1 0.5 1 1 x / . 3 , y / . 6 x/0.5, y/0.6 x/0.4, y/0.2 x/0.4, y/0.7 x / . 3 , y / . 4 x / . 6 , y / . 7 x/1, y/0.5 x/1, y/0.3 x/0.7, y/1

b1 b0

1 1 1 0.5 x/0.6, y/0.6 x/0.6, y/0.7 x/0.7, y/1 x/1, y/0.6

26 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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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, δA, σA, τA), B = (B, δB, σB, τB) ⋆ ϕ : A × B → L

Forward simulations

σA σB ◦ ϕ−1 ϕ−1 ◦ δA

x δB x ◦ ϕ−1

ϕ−1 ◦ τA τB

Backward simulations

σA ◦ ϕ σB δA

x ◦ ϕ ϕ ◦ δB x

τA ϕ ◦ τB

27 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Forward and backward simulations

A B

σA σB τA τB

a0 . . . ak ak+1 . . . an b0 . . . bk bk+1 . . . bn x1 xk xk+1 xk+2 xn x1 xk xk+1 xk+2 xn ϕ

28 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Forward and backward simulations

A B

σA σB τA τB

a0 . . . ak ak+1 . . . an b0 . . . bk bk+1 . . . bn x1 xk xk+1 xk+2 xn x1 xk xk+1 xk+2 xn ϕ forward simulation

28 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Forward and backward simulations

A B

σA σB τA τB

a0 . . . ak ak+1 . . . an b0 . . . bk bk+1 . . . bn x1 xk xk+1 xk+2 xn x1 xk xk+1 xk+2 xn ϕ arbitrary successful run forward simulation

28 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Forward and backward simulations

A B

σA σB τA τB

a0 . . . ak ak+1 . . . an b0 . . . bk bk+1 . . . bn x1 xk xk+1 xk+2 xn x1 xk xk+1 xk+2 xn ϕ arbitrary successful run forward simulation δA

xk+1 28 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Forward and backward simulations

A B

σA σB τA τB

a0 . . . ak ak+1 . . . an b0 . . . bk bk+1 . . . bn x1 xk xk+1 xk+2 xn x1 xk xk+1 xk+2 xn ϕ arbitrary successful run forward simulation δA

xk+1

building a successful run which simulates the original run

28 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Forward and backward simulations

A B

σA σB τA τB

a0 . . . ak ak+1 . . . an b0 . . . bk bk+1 . . . bn x1 xk xk+1 xk+2 xn x1 xk xk+1 xk+2 xn ϕ arbitrary successful run forward simulation δA

xk+1

δB

xk+1

building a successful run which simulates the original run

28 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Our main results

Fuzzy Sets and Systems, 2012 ( ´ Ciri´ c, Ignjatovi´ c, Damljanovi´ c, Baˇ si´ c) Fuzzy Sets and Systems, 2012 ( ´ Ciri´ c, Ignjatovi´ c, Janˇ ci´ c, Damljanovi´ c) ⋆ 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 ( ´ Ciri´ c, Ignjatovi´ c, Baˇ si´ c, Janˇ ci´ c) Theoretical Computer Science, 2014 (Damljanovi´ c, ´ Ciri´ c, Ignjatovi´ c) ⋆ simulations and bisimulations for weighted automata over an additively idempotent semiring ⋆ relative residuation – Boolean residuals

29 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory

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Our main results

THANK YOU FOR YOUR ATTENTION

30 Jelena Ignjatovi´ c General Problems of Fuzzy Automata Theory