Formal Calculi of Fuzzy Relations

Transcription

1 Formal Calculi of Fuzzy Relations Libor B hounek Institute for Research and Applications of Fuzzy Modeling University of Ostrava libor.behounek@osu.cz SFLA, ƒeladná, 16 August 2016 Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 1 / 63

2 Outline 1 Fuzzy Relations: Why Formalize? 2 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations 3 Tarski-style Fuzzy Relational Calculi 4 References Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 2 / 63

3 Fuzzy Relations: Why Formalize? Outline 1 Fuzzy Relations: Why Formalize? 2 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations 3 Tarski-style Fuzzy Relational Calculi 4 References Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 3 / 63

4 Fuzzy Relations: Why Formalize? Fuzzy sets Recall: Fuzzy set = a set with weighted membership, A: V L Conventions: L usually a (residuated) lattice, often [0, 1] R Crisp subsets of V identied with fuzzy sets C : V {0, 1} L Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 4 / 63

5 Fuzzy Relations: Why Formalize? Fuzzy sets Notation: Membership degrees: A(x) or just Ax Write A = {x f(x)} if Ax = f(x) for all x V Examples: A B = {x Ax Bx} (A B)x min(ax, Bx) V = {x 1} analogously = {x 0} ker A = {x Ax = 1} analogously supp A Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 5 / 63

6 Fuzzy Relations: Why Formalize? Fuzzy relations Recall: (Binary) fuzzy relation = a fuzzy set of pairs, R: V 2 L n-ary fuzzy relation = a fuzzy set of n-tuples, S : V n L Fuzzy relation from X to Y (for X, Y V )... T : X Y L (treat as V 2 L: add 0's elsewhere, ignore V 2 (X Y )) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 6 / 63

7 Fuzzy Relations: Why Formalize? Fuzzy relations Notation: Membership degrees: Rxy, Sxyz,... Write R = {xy f(x, y)} if Rxy = f(x, y) for all x, y V Examples: supp R = {xy Rxy > 0} Id = {xy x = y} Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 7 / 63

8 Fuzzy Relations: Why Formalize? Representation of fuzzy relations Graphs (3D or contours = a system of horizontal cuts) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 8 / 63

9 Fuzzy Relations: Why Formalize? Representation of fuzzy relations Matrices (esp. for nite fuzzy relations): Rx 1 x 1 Rx 1 x 2 Rx 1 x n Rx 2 x 1 Rx 2 x 2 Rx 2 x n R = , e.g., R = Rx n x 1 Rx n x 2 Rx n x n Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 9 / 63

10 Fuzzy Relations: Why Formalize? Representation of fuzzy relations Co-graphs (weighted co-graphs of classical relations): x 1 x 2 x 3 Rx 1x 1 x 1 Rx 1x 2 x 2 Rx 3x 2 x 3 Rx 3x 3 Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 10 / 63

11 Fuzzy Relations: Why Formalize? Representation of fuzzy relations For a binary fuzzy relation R dene: Ry = {x Rxy} foreset (of x V ) xr = {y Rxy} afterset (of x V ) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 11 / 63

12 Fuzzy Relations: Why Formalize? Formalization of the theory of fuzzy relations Formalization = specication of: A formal language (formulae for statements about fuzzy relations) An interpretation of the formal language (translation of formulae into statements about fuzzy relations) Derivation rules (to derive true statements about fuzzy relations) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 12 / 63

13 Fuzzy Relations: Why Formalize? Formalization of the theory of fuzzy relations Why formalize? We restrict the language in order to enable formal (symbolic) manipulation with statements about fuzzy relations Derivation of schematic theorems Automatic (machine) deduction Metamathematical results (on expressivity, complexity,... ) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 13 / 63

14 Fuzzy Relations: Why Formalize? Predicate fuzzy logic: language Example of formalization: predicate fuzzy logic We restrict the mathematical language of fuzzy set theory to: Algebraic operations in L:,,,,,... (logical connectives) Inma and suprema, (written as the quantiers, ) Optionally: Crisp equality = of elements and fuzzy sets (=, on L denable) (Eliminable) set-terms { x ϕ} Variables for fuzzy sets of fuzzy relations (A, B,... ) etc. (then add the axioms of equality, comprehension, and extensionality) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 14 / 63

15 Fuzzy Relations: Why Formalize? Predicate fuzzy logic: interpretation The meaning of symbols in predicate fuzzy logic: Atomic formulae: Unary predicate symbols P : fuzzy sets Unary atomic formulae P x: membership degree of x in P n-ary predicate symbols R: fuzzy relations n-ary atomic formulae Rxy, Sxyz,... : membership degree of the n-tuple in the fuzzy relation Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 15 / 63

16 Fuzzy Relations: Why Formalize? Predicate fuzzy logic: interpretation Connectives: Min-conjunction : minimum (in the lattice L of degrees) Max-disjunction : maximum Strong conjunction &: a (left-)continuous t-norm T-norm = : [0, 1] 2 [0, 1] commutative associative monotone, unit 1 E.g.: product, min, the Šukasiewicz t-norm max(α + β 1, 0) More generally, the monoidal operation of the residuated lattice L Implication : the residuum of (α β) = sup γ α β γ (α β) = 1 i α β Negation : the function α 0 (e.g., 1 α for Šukasiewicz) Equivalence : the biresiduum min(α β, β α) Delta : indicator of α = 1 (in linear L) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 16 / 63

17 Fuzzy Relations: Why Formalize? Predicate fuzzy logic: interpretation Quantiers: Universal quantier : inmum (in the complete lattice L) Existential quantier : supremum Example: ( x)( y)(p x Qy Rxy) x y (min(p x, Qy) Rxy) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 17 / 63

18 Fuzzy Relations: Why Formalize? Predicate fuzzy logic: axiomatic system Aim: generate only (and all?) formulae ϕ s.t. always ϕ = 1 = soundness (and, sometimes, completeness) (a) For a given (left-)continuous t-norm : For the Šukasiewicz : Šukasiewicz logic For the minimum: Gödel logic For the product: product logic, etc. (b) For a given class of continuous t-norms: All continuous t-norms: Hájek's logic BL All left-continuous t-norms: MTL, etc. (c) Variations: adding connectives, relaxing conditions,... Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 18 / 63

19 Fuzzy Relations: Why Formalize? Predicate fuzzy logic: axiomatic system Example: Šukasiewicz fuzzy logic Axioms: ϕ (ψ ϕ) α (β α) (ϕ ψ) ((ψ χ) (ϕ χ)) α β (β γ) (α γ) ((ϕ ψ) ψ) ((ψ ϕ) ϕ) max(α, β) max(β, α) ( ϕ ψ) (ψ ϕ) (α 0) (β 0) (β α) ( x)p x P y x P x P y ( x)(ϕ P x) (ϕ ( x)p x) x (α P x) α x P x Rules: Uniform substitution and ϕ, ϕ ψ ψ if α = 1 and α β then β = 1 P x ( x)p x if P x = 1 for all x, then x P x = 1 Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 19 / 63

20 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Outline 1 Fuzzy Relations: Why Formalize? 2 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations 3 Tarski-style Fuzzy Relational Calculi 4 References Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 20 / 63

21 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Fuzzy relational operations (formalized) Recall: In formal fuzzy logic, predicates are interpreted by fuzzy relations (and unary predicates by fuzzy sets) Predicate fuzzy logic is in fact a formal calculus for fuzzy relations! Recall: Restricting the language enables formal manipulation, automatic deduction, schematic theorems, metamathematical results Large expressive power of predicate fuzzy logic can be used as the language of the theory of fuzzy relations Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 21 / 63

22 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Fuzzy relational operations (formalized) Notice: Fuzzy relations are fuzzy sets Fuzzy set-theoretical operations apply to fuzzy relations, too R S = df {xy (Rxy Sxy)} min-intersection (R S)xy = Rxy Sxy analogously:, R = df {xy Rxy)} complement ( R)xy = (Rxy 0) also written R, R c, \R,... ker R = df {xy Rxy)} xy ker R i Rxy = 1 kernel analogously: supp ( ) A =df {xy ( R)(AR Rxy)} fuzzy setintersection ( ) A xy = R (A(R) Rxy) analogously: Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 22 / 63

23 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Properties of fuzzy relations Fuzzy settheoretical properties of fuzzy relations: Hgt R df ( xy)rxy height Hgt R = x,y Rxy analogously: Plt ( / ) R S df ( xy)(rxy Sxy) (R S) = x,y (Rxy Sxy) graded inclusion R S df ( xy)(rxy & Sxy) graded compatibility (R S) = x,y (Rxy Sxy) = Hgt(R S) Notice: is graded (R S) = 1 i Rxy Sxy for all xy... (R S) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 23 / 63

24 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Properties of fuzzy relations Example (for Šukasiewicz): = Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 24 / 63

25 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Properties of fuzzy relations Fuzzy settheoretical laws apply to fuzzy relations, too. Examples: (R S) T = (R T ) (S T ) R S R S Hgt(R S) Hgt R Hgt S, etc. Derivation in fuzzy logic (MTL, using Lemma: ( x)(ϕ ψ) ( x)ϕ ( x)ψ): Hgt(R S) ( xy)((r S)xy) ( xy)(rxy Sxy) ( xy)rxy ( xy)sxy Hgt R Hgt S Hgt(R S) = xy ((R S)xy) = xy(rxy Sxy) xy Rxy xysxy = Hgt R Hgt S Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 25 / 63

26 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Basic fuzzy relational operations 2 = df {xy 0} empty fuzzy relation 2 xy = 0 for all xy analogously: V 2 = {xy 1} Id = df {xy x = y} Id xy = 1 if x = y Id xy = 0 otherwise R 1 = df {xy Ryx} R 1 xy = Ryx crisp identity relation converse relation Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 26 / 63

27 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Basic fuzzy relational operations dom R = df {x ( y)rxy} (dom R)x = y Rxy domain analogously: rng R A = df {y ( x)(ax & Rxy)} image (R A)y = x (Ax Rxy) analogously: preimage A B = df {xy Ax & By} Cartesian product (A B)xy = Ax By analogously: n-ary, powers Example: dom = Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 27 / 63

28 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Dened properties of fuzzy relations Refl R df ( x)rxx Refl R = x Rxx Sym R df ( xy)(rxy Ryx) Sym R = x,y (Rxy Ryx) Trans R df ( xyz)(rxy & Ryz Rxz) Trans R = x,y,z (Rxy Ryz Rxz) reexivity symmetry transitivity Notice graded properties again: Trans R = 1 i Rxy Ryz Rxz for all x, y, z (crisp ` -transitivity') Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 28 / 63

29 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Dened properties of fuzzy relations Compound properties: Sim R df (Refl R Sym R Trans R)... similarity relation Sim R = 1 i Refl R = Sym R = Trans R = 1 (a.k.a. indistinguishability, fuzzy equivalence relation) Similarly: fuzzy preorders, fuzzy orders,... Modications (Höhle, Bodenhofer): e.g., Refl E R df ( xy)(exy Rxy) for an indistinguishability E Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 29 / 63

30 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations (Sup-T) composition of fuzzy relations Fuzzy relational composition R S = df {xy ( z)(rxz & Szy)} (R S)xy = z (Rxz Szy) sup-t composition Fore/afterset characterization: (R S)xy xr Sy xr Sy (crisp) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 30 / 63

31 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations (Sup-T) composition of fuzzy relations Fuzzy relational composition R S = df {xy ( z)(rxz & Szy)} (R S)xy = z (Rxz Szy) sup-t composition Fore/afterset characterization: (R S)xy = xr Sy = Hgt(xR Sy) (fuzzy) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 31 / 63

32 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations (Sup-T) composition of fuzzy relations Fuzzy relational composition R S = df {xy ( z)(rxz & Szy)} (R S)xy = z (Rxz Szy) sup-t composition Co-graph visualization: x ( R S ) xy Rxz 1 Sz 1 y Rxz 2 Sz 2 y Rxz 3 y Sz 3y Rxz 4 Sz 4 y Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 32 / 63

33 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations (Sup-T) composition of fuzzy relations Fuzzy relational composition R S = df {xy ( z)(rxz & Szy)} (R S)xy = z (Rxz Szy) sup-t composition Matrix `multiplication': Rx 1x 1 Rx 1x n Rx nx 1 Rx nx n Sx 1x 1 Sx 1x n Sx nx 1 Sx nx n (R S)x 1x 1 (R S)x 1x n (R S)x nx 1 (R S)x nx n Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 33 / 63

34 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Properties of sup-t composition Theorem (R S) T = R (S T ) R Id = Id R = R (R S) 1 = S 1 R 1 R 1 R 2 R 1 S R 2 S ( R A R) S = R A (R S) ( R A R) S R A (R S) (associativity) (identity) (transposition) (monotony) (union distributivity) (intersection subdistributivity) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 34 / 63

35 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Derivation in predicate fuzzy logic (R S) 1 = S 1 R 1 (transposition) Proof: (R S) 1 xy (R S)yx ( z)(ryz & Szx) ( z)(s 1 xz & R 1 zy) (S 1 R 1 )xy (R S) 1 xy = (R S)yx = z (Ryz Szx) = z (S 1 xz R 1 zy) = (S 1 R 1 )xy Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 35 / 63

36 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Derivation in predicate fuzzy logic ( R A R) S R A (R S) Proof: (intersection subdistributivity) Q Lxy df (( R A R) S ) xy Q Lxy = (( R A R) S ) xy = ( z)(( R)(A(R) Rxy) & Szy) ( z R (A(R) Rxy) Szy) ( z)( R)(A(R) Rxy & Szy) (A(R) (Rxy Szy)) ( R)( z)(a(r) Rxy & Szy) z R R z (A(R) (Rxy Szy)) ( R)(A(R) ( z)(rxy & Szy)) R (A(R) z (Rxy Szy)) = ( R)(A(R) (R S)xy) R (A(R) (R S)xy) = ( R A (R S)) ( xy df Q Rxy R A (R S)) xy = Q Rxy Thus provably, Q Lxy Q Rxy Q Lxy Q Rxy (Q Lxy Q Rxy) = 1 ( xy)(q Lxy Q Rxy) x,y (QLxy QRxy) = 1 Q L Q R (Q L Q R) = 1 Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 36 / 63

37 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Inf-R compositions of fuzzy relations Compositions of fuzzy relations (a.k.a. fuzzy relational products): R S = df {xy ( z)(rxz & Szy)} (R S)xy = z (Rxz Szy) sup-t composition R S = df {xy ( z)(rxz Szy)} inf-r composition (R S)xy = z (Rxz Szy) a.k.a. BK-(sub)product R S = df {xy ( z)(szy Rxz)} (R S)xy = z (Szy Rxz) R S = df {xy ( z)(rxz Szy)} (R S)xy = z (Rxz Szy) BK-superproduct BK-squareproduct Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 37 / 63

38 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Inf-R compositions of fuzzy relations Characterization in terms of foresets and aftersets: (R S)xy = xr Sy (R S)xy = xr Sy (R S)xy = Sy xr Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 38 / 63

39 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Inf-R compositions of fuzzy relations Example (crisp/graded): Rps = patient p shows symptom s Ssd = s is a symptom of disease d (R S)pd = some symptoms shown by patient p are symptoms of disease d (R S)pd = all symptoms shown by patient p are symptoms of disease d (R S)pd = all symptoms of disease d are shown by patient p (R S)pd = patient p shows exactly the symptoms of disease d Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 39 / 63

40 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Expressivity of fuzzy relational products Many relational properties can be characterized by means of fuzzy relational products Examples: Trans R R R R Refl R R R R Trans R R R R 1 Refl R R R 1 R Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 40 / 63

41 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Properties of BK-products Theorem R S = (S 1 R 1 ) 1 R S = (R S) (R S) (interdenability) (R S) 1 = S 1 R 1 (R S) 1 = S 1 R 1 (transposition) R (S T ) = (R S) T (R S) T = R (S T ) (residuation) (R S) T = R (S T ) (quasi-associativity) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 41 / 63

42 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Properties of BK-products Theorem R 1 R 2 R 2 S R 1 S S 1 S 2 R S 1 R S 2 R A (R S) = ( R A R) S S A (R S) = R S A S R A (R S) ( R A R) S S A (R S) R S A S (monotony) (union subdistributivity) (intersection distributivity) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 42 / 63

43 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations De BaetsKerre modication of BK-products De BaetsKerre modication of BK-products R S = df {xy ( z)(rxz Szy) ( z)rxz} (analogously for, ) (R S)xy = z (Rxz Szy) z Rxz Motivation: to avoid the `useless pairs' (good for some applications, many BK-properties preserved) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 43 / 63

44 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations Modications of BK-products Example: Rps = patient p shows symptom s Ssd = s is a symptom of disease d (R S)pd = all symptoms shown by patient p are symptoms of disease d (trivially true if p shows no symptoms) (R S)pd = patient p shows some symptoms, all of which are symptoms of d (much more relevant for diagnosis) Other side-condition modications of fuzzy relational products possible e.g., to exclude incompatible pairs ( t pni ka, Hol apek, Cao) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 44 / 63

45 Tarski-style Fuzzy Relational Calculi Outline 1 Fuzzy Relations: Why Formalize? 2 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations 3 Tarski-style Fuzzy Relational Calculi 4 References Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 45 / 63

46 Tarski-style Fuzzy Relational Calculi Tarski's relational calculus Tarski's relational calculus for crisp relations = an equational calculus using,,, 1, Id Strong expressive powercan express: Classical rst-order logic with max 3 variables Peano arithmetic incomplete (Gödel's theorems apply) ZFC set theory math can be done w/o quantiers Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 46 / 63

47 Tarski-style Fuzzy Relational Calculi Tarski's relational calculus Axioms of crisp relational calculus R S = S R R (S T ) = (R S) T ( R S) ( R S)) = R R (S T ) = (R S) T R Id = R (R 1 ) 1 = R (R S) 1 = S 1 R 1 (R S) 1 = R 1 S 1 (R S) T = (R T ) (S T ) (R 1 (R S)) S = S = axioms of BA Rules: uniform replacement, substitution of equals for equals Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 47 / 63

48 Tarski-style Fuzzy Relational Calculi Fuzzy relational calculus Fuzzy relational calculus = Tarski's calculus adapted for fuzzy relations A simple way: replace the axioms of BA by the axioms of fuzzy logic Example for Šukasiewicz logic: add, 2, replace BA-axioms with MV-axioms: R S = S R R (S T ) = (R S) T R 2 = R R 2 = 2 ( R) = R R S = ( R S) S R S = S R Theorem: The `non-logical' relational axioms carry over to fuzzy relations based on any (left-)continuous t-norm (Kohout 2002) Only sketched by Kohout, many open problems (expressive power,... ) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 48 / 63

49 Tarski-style Fuzzy Relational Calculi Composition-based calculus for fuzzy sets and relations Extension to fuzzy set-theoretical notions (B hounekda ková 2009) by including BK-products and identication of fuzzy sets and truth degrees with certain fuzzy relations Compare: R S = df {xy ( z)(rxz & Szy)} (R S)xy = z (Rxz Szy) R A = df {x ( z)(rxz & Az)} (R A)x = z (Rxz Az) Idea: Add a dummy constant o and identify Ax with S A xo This reduces to : R S A = df {xo ( z)(rxz & S A zo)} (R S A )xo = z (Rxz S Azo) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 49 / 63

50 Tarski-style Fuzzy Relational Calculi Composition-based calculus for fuzzy sets and relations Consequently, properties of transfer to, e.g.: ( (R 1 R 2 ) (R 1 S R 2 S) R A R) S R A (R S) (R 1 R 2 ) (R 1 A R ( 2 A) R A R) A R A (R A) Similarly for many other fuzzy set- and relation-theoretical notions Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 50 / 63

51 Tarski-style Fuzzy Relational Calculi Fuzzy relational representation of fuzzy sets In general, identify a fuzzy set A with the fuzzy relation A = A {o}: (i.e., a 1 N fuzzy relation if nite) Ax 1 A =. Ax n A: x 1 Ax 1 x 2 Ax 2 Ax o x 3 3 Ax 4 x 4 Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 51 / 63

52 Tarski-style Fuzzy Relational Calculi Fuzzy relational representation of fuzzy sets Rx 1 x 1 Rx 1 x n Ax 1 (R A)x 1 Then R A = R A = =. Rx n x 1 Rx n x n Ax n (R A)x n Similarly: R A = R 1 A, dom R = R V, A B = A B 1, etc. ( ) Bx 1 Bx n Ax 1. Ax n (A B)x 1 x 1 (A B)x 1 x n (A B)x n x 1 (A B)x n x n Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 52 / 63

53 Tarski-style Fuzzy Relational Calculi α = ( α ) α: o α Fuzzy relational representation of truth degrees Furthermore, identify a truth degree α with the fuzzy set α = {o α} (i.e., a 1 1 fuzzy relation) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 53 / 63

54 Tarski-style Fuzzy Relational Calculi Fuzzy relational representation of truth degrees Then, e.g., A B = ( x)(ax Bx) = ( x)(axo Bxo) = ( x)(a 1 ox Bxo) = A 1 B Bx 1. Bx n ) ( ) (Ax 1 Ax n A B Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 54 / 63

55 Tarski-style Fuzzy Relational Calculi Fuzzy relational representation of truth degrees Similarly: Hgt A = V 1 A, Plt A = V 1 A, (A B) = A 1 B, etc. Moreover, propositional connectives reduce to relational products, too: (α & β) = α β (α β) = α β (α β) = α β ( α ) ( β ) ( α & β ) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 55 / 63

56 Tarski-style Fuzzy Relational Calculi Transfer of properties of compositions Properties of compositions apply to the notions reduced to compositions: Examples: ( ( R A R) S = R A (R S) ( R A R) A = R A (R A) ( R A R) A = R A (R A) A A A) B = A A (A B) Dom( R A R) = R A Dom R Rng( R A R) = R A Rng R Hgt( A A A) A A (Hgt A) R S A S = S A (R S) R A A A = A A (R A) R A A A = A A (R A) A B A B = B A (A B) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 56 / 63

57 Tarski-style Fuzzy Relational Calculi Equational derivations of fuzzy set-theoretic laws Example: R (rng S) = rng(s R) Recall: R A = R 1 A rng R = R V = R 1 V Derivation: R (rng S) = R 1 (S 1 V ) = (R 1 S 1 ) V = (S R) 1 V = rng(s R) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 57 / 63

58 Tarski-style Fuzzy Relational Calculi Equational derivations of fuzzy set-theoretic laws Example: Hgt(dom R) = Hgt(rng R) Recall: Hgt A = A 1 V = V 1 A dom R = R V = R V rng R = R V = R 1 V Derivation: Hgt(dom R) = V 1 (R V ) = (V 1 R) V = (R 1 V ) 1 V = Hgt(rng R) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 58 / 63

59 References Outline 1 Fuzzy Relations: Why Formalize? 2 Predicate Fuzzy Logic as a Calculus for Fuzzy Relations 3 Tarski-style Fuzzy Relational Calculi 4 References Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 59 / 63

60 References References: Predicate fuzzy logic P. Hájek: Metamathematics of Fuzzy Logic (Ch. 15, 7) Kluwer, L. B hounek, P. Cintula, & P. Hájek: Introduction to mathematical fuzzy logic (Ÿ12, Ÿ5) In P. Cintula, P. Hájek, & C. Noguera (eds.): Handbook of Mathematical Fuzzy Logic, Vol. I, pp College Publications, (Available at Mathematical_Fuzzy_Logic.) S. Gottwald: A Treatise on Many-Valued Logics (esp. Part III) Research Studies Press, Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 60 / 63

61 References References: Fuzzy relations S. Gottwald: A Treatise on Many-Valued Logics (Part IV) Research Studies Press, L. B hounek, U. Bodenhofer, & P. Cintula: Relations in Fuzzy Class Theory: Initial steps. Fuzzy Sets and Systems 159 (2008) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 61 / 63

62 References References: Fuzzy relational compositions W. Bandler & L.J. Kohout: Semantics of implication operators and fuzzy relational products. International Journal of ManMachine Studies 12 (1980) B. De Baets & E. Kerre: Fuzzy relational compositions. Fuzzy Sets and Systems 60 (1993) W. Bandler & L.J. Kohout: On the universality of the triangle superproduct and the square product of relations. International Journal of General Systems 25 (1997) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 62 / 63

63 References References: Tarski-style fuzzy relational calculi A. Tarski: On the calculus of relations. Journal of Symbolic Logic 6 (1941) L.J. Kohout: Fuzzy logic-based extension of Tarski's relational systems using non-associate BK-products of relations. In O. Majer (ed.): The Logica Yearbook 2001, pp Filosoa, Prague, L. B hounek & M. Da ková: Relational compositions in Fuzzy Class Theory. Fuzzy Sets and Systems 160 (2009) Libor B hounek (IRAFM) Formal Calculi of Fuzzy Relations 63 / 63