Some Remarks on Minimal Clones
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1 ALGORITHMIC COMPLEXITY and UNIVERSAL ALGEBRA Szeged, Hungary Some Remarks on Minimal Clones Hajime MACHIDA (Tokyo) Joint Work with Michael PINSKER (Wien)
2 ALGORITHMIC COMPLEXITY and UNIVERSAL ALGEBRA Szeged, Hungary Some Remarks on Minimal Clones A Minimal Adventure to the World of Béla Csákány Hajime MACHIDA (Tokyo) Joint Work with Michael PINSKER (Wien)
3 1 Where to Start 3
4 1 Where to Start Béla Csákány 4
5 1 Where to Start Béla Csákány In 1983, B. Csákány published the following paper : B. Csákány, All minimal clones on the three element set, Acta Cybernet., Vol. 6, , 1983 where he determined all minimal clones on a three-element set. Number of minimal clones = 84 5
6 0 Just in Case... Definitions clone C ( O k ) : clone on E k (i) (ii) C : closed under composition C J k L k : lattice of all clones on E k minimal clone C ( L k ) : minimal clone (i) C J k (ii) for C L k, J k C C = C = C 6
7 1 Where to Start... Continued Generators of Minimal Clones on a Three-Element Set ( B. Csákány, 1983 ) Each of the following operations generates mutually distinct minimal clone. 7
8 (II) Binary idempotent operations b 0 = b 364 = b 728 = b 8 = b 368 = b 80 = b 36 = b 40 = b 692 = b 10 = b 280 = b 458 = b 20 = b 448 = b 188 =
9 b 11 = b 286 = b 215 = b 16 = b 281 = b 296 = b 47 = b 205 = b 179 = b 17 = b 287 = b 53 = b 38 = b 43 = b 206 = b 26 = b 449 = b 37 =
10 b 33 = b 122 = b 557 = b 35 = b 125 = b 71 = b 42 = b 41 = b 530 = b 68 = b 528 = b 116 = b 178 = b 290 =
11 0 2 1 b 624 =
12 (III) Ternary majority operations m 0 = m 364 = m 728 = m 109 = m 473 =
13 m 510 = m 624 =
14 (IV) Ternary semiprojections s 0 = s 364 = s 728 = s 8 = s 368 =
15 s 80 = s 36 = s 40 = s 692 = s 26 = s 449 = s 37 =
16 s 76 = s 684 = s 332 = s 424 =
17 2 Tools of Our Research Polynomials over a Finite Field 17
18 2 Tools of Our Research Polynomials over a Finite Field Let k (> 1) be a power of a prime. We consider the base set E k = {0, 1,..., k 1} as a finite field (Galois field) GF(k), and express a function defined on E k as a polynomial over GF(k). 18
19 QUIZ ( Easy ) 19
20 QUIZ ( Easy ) Given polynomials f, g over GF(2) ( i.e., f, g : Boolean function ) f(x, y) = x y + 1 g(x, y) = x y + x + y 20
21 QUIZ ( Easy ) Given polynomials f, g over GF(2) ( i.e., f, g : Boolean function ) f(x, y) = x y + 1 g(x, y) = x y + x + y QUESTION : Which is weaker, f or g? ( with respect to the productive power of functions ) 21
22 ANSWER : g is much weaker than f 22
23 ANSWER : g is much weaker than f Because : f(x, y) = NAND (x, y) g(x, y) = OR (x, y) 23
24 QUIZ ( Hard ) 24
25 QUIZ ( Hard ) Given polynomials u v w over GF(3) u(x, y) = x 2 y 2 + xy 2 + x 2 y + 2xy + x + y v(x, y) = x 2 y 2 + xy 2 + x 2 y + xy + x + y w(x, y) = x 2 y 2 + xy 2 + x 2 y + 2xy + x + y
26 QUIZ ( Hard ) Given polynomials u v w over GF(3) u(x, y) = x 2 y 2 + xy 2 + x 2 y + 2xy + x + y v(x, y) = x 2 y 2 + xy 2 + x 2 y + xy + x + y w(x, y) = x 2 y 2 + xy 2 + x 2 y + 2xy + x + y + 1 QUESTION : Which is the weakest among u v and w? ( with respect to the productive power of functions ) 26
27 ANSWER : u is the weakest. 27
28 ANSWER : u is the weakest. Because : (1) u(x, y) generates a minimal clone. (3) w(x, y) is Webb function which generates all functons. (2) v(x, y) is inbetween. 28
29 Our Hope!! To characterize polynomials generating minimal clones in terms of the form of polynomials or To discover principles or rules that are common to polynomials generating minimal clones
30 3 Basic Facts on Minimal Clones Lemma A minimal clone is generated by a single function, i.e., for any minimal clone C L k there exists f O k such that C = f Proposition For any f O k, suppose f satisfies the following : (i) (ii) f J k For h f, if h J k then f h Then f generates a minimal clone. 30
31 Type Theorem for Minimal Clones An function f on E k is minimal if (i) it generates a minimal clone, and (ii) every function from f whose arity is smaller than the arity of f is a projection 31
32 Type Theorem for Minimal Clones An function f on E k is minimal if (i) it generates a minimal clone, and (ii) every function from f whose arity is smaller than the arity of f is a projection Theorem ( I. G. Rosenberg, 1986 ; Inspired by Csákány s work,... ) Every minimal function is classified in one of the following five types: (1) Unary function (2) Idempotent binary function (3) Majority function (4) Semiprojection (5) If k = 2 m, the ternary function f(x, y, z) := x + y + z where E k ; + is an elementary 2-group 32
33 Note f(x 1,..., x n ) : idempotent f(x,..., x) = x for x E k 33
34 Generators of all minimal clones of type (2) over GF(3) (Originally from B. Csákány) b 11 = xy 2 b 624 = 2x + 2y b 68 = 2x + 2xy 2 b 0 = 2x 2 y + 2xy 2 b 449 = x + y + 2x 2 y b 368 = x + y 2 + 2x 2 y 2 b 692 = x + 2y 2 + x 2 y 2 b 33 = x + 2x 2 y + xy 2 b 41 = x 2 + xy 2 + 2x 2 y 2 b 71 = 2x 2 + xy 2 + x 2 y 2 b 26 = 2x + x 2 + 2xy + 2x 2 y b 37 = 2x + 2x 2 + xy + 2x 2 y b 17 = 2x + x 2 + 2xy 2 + 2x 2 y 2 b 38 = 2x + 2x 2 + 2xy 2 + x 2 y 2 b 10 = xy + 2x 2 y + 2xy 2 + 2x 2 y 2 34
35 b 20 = 2xy + 2x 2 y + 2xy 2 + x 2 y 2 b 43 = x + xy + 2x 2 y + xy 2 + 2x 2 y 2 b 53 = x + 2xy + 2x 2 y + xy 2 + x 2 y 2 b 35 = x + xy + x 2 y + 2xy 2 + 2x 2 y 2 b 42 = x + 2xy + x 2 y + 2xy 2 + x 2 y 2 b 530 = x + y + y 2 + 2x 2 y + 2x 2 y 2 b 125 = x + y + 2y 2 + 2x 2 y + x 2 y 2 b 116 = x + y + xy + 2y 2 + 2xy 2 b 528 = x + y + 2xy + y 2 + 2xy 2 b 206 = x + 2y + y 2 + x 2 y + 2x 2 y 2 b 287 = x + 2y + 2y 2 + x 2 y + x 2 y 2 b 215 = x + 2y + 2xy + y 2 + xy 2 b 286 = x + 2y + xy + 2y 2 + xy 2 b 122 = y + x 2 + 2y 2 + 2x 2 y + xy 2 b 557 = y + 2x 2 + y 2 + 2x 2 y + xy 2 b 16 = 2x + x 2 + xy + 2x 2 y + x 2 y 2 b 47 = 2x + 2x 2 + 2xy + 2x 2 y + 2x 2 y 2 b 178 = 2x + 2y + x 2 + xy + y 2 b 290 = 2x + 2y + 2x 2 + 2xy + 2y 2 b 40 = x 2 + xy + 2x 2 y + 2xy 2 + x 2 y 2 35
36 b 80 = 2x 2 + 2xy + 2x 2 y + 2xy 2 + 2x 2 y 2 b 364 = x 2 + xy + y 2 + 2x 2 y + 2xy 2 b 728 = 2x 2 + 2xy + 2y 2 + 2x 2 y + 2xy 2 b 448 = x + y + xy + x 2 y + xy 2 + 2x 2 y 2 b 458 = x + y + 2xy + x 2 y + xy 2 + x 2 y 2 b 205 = x + 2y + xy + y 2 + xy 2 + x 2 y 2 b 296 = x + 2y + 2xy + 2y 2 + xy 2 + 2x 2 y 2 b 188 = 2x + 2y + x 2 + 2xy + y 2 + 2x 2 y 2 b 280 = 2x + 2y + 2x 2 + xy + 2y 2 + x 2 y 2 b 8 = 2x + x 2 + xy + x 2 y + xy 2 + x 2 y 2 b 36 = 2x + 2x 2 + 2xy + x 2 y + xy 2 + 2x 2 y 2 b 179 = 2x + 2y + x 2 + y 2 + x 2 y + 2xy 2 + x 2 y 2 b 281 = 2x + 2y + 2x 2 + 2y 2 + x 2 y + 2xy 2 + 2x 2 y 2 36
37 4 More Properties of Minimal Clones For f, g O k we write f g if g f ( Note : Binary relation on O k is a quasi-order ) Lemma Let f O (2) k be an essentially binary function. If f satisfies (1) and (2) then f is minimal. (1) f is idempotent (2) For any g O (m) k ( m 2 ) if f g then g is a projection or g f 37
38 Let m 3 and g O (m) k. We say g is a quasi-projection if g becomes a projection whenever two arguments of g are identified, i.e., if g(x 1,..., x i,..., x i,..., x m ) is always a projection. Lemma Let f O (2) k be a binary idempotent function. Then f is minimal (1) and (2) hold (1) For g O (2) k \ J k if f g then g f (2) For m 3 and g O (m) k \ J k if f g then g is not a quasiprojection. 38
39 For f O (2) k let Γ (x,y) f be : { f(f(x, y), x), f(f(x, y), y), f(x, f(x, y)), f(y, f(x, y)), f(f(x, y), f(y, x)) } Then let Γ f = Γ (x,y) f Γ (y,x) f Lemma Let f O (2) k be a binary idempotent function which is not a projection. Suppose that, for any γ Γ f, one of the following holds: γ(x, y) f(x, y) or γ(x, y) f(y, x) Then f is minimal ( Here, by h 1 (x, y) h 2 (x, y) we mean h 1 (x, y) = h 2 (x, y) for all (x, y) E 2 k ) 39
40 5 Polynomials generating Minimal Clones Here we consider only Binary Idempotent Minimal Functions 40
41 5 Polynomials generating Minimal Clones Here we consider only Binary Idempotent Minimal Functions Our Strategy Step 1 : Take arbitrary f(x, y) O (2) 3 from Csákány s list. Step 2 : Search for a polynomial g(x, y) O (2) k for k 3 whose counterpart for k = 3 is f(x, y) Step 3 : Examine if g is minimal. 41
42 (1) Linear Polynomials For k = 3, the binary linear polynomial is minimal. f(x, y) = 2x + 2y For k = 5, the binary linear polynomial f(x, y) = 2x + 4y is minimal, whose Cayley table is : f(x, y) = 2x + 4y x\y
43 Theorem (Á. Szendrei) Let k be a prime. Let f(x, y) be a binary linear polynomial on E k Then f is minimal if and only if f(x, y) = ax + (k + 1 a)y for some 1 < a < k Moreover, all such linear polynomials generate the same minimal clone. 43
44 (2) Monomials Consider x s y t for 1 s t < k For k = 3, f(x, y) = x y 2 is minimal. For k = 5, f(x, y) = x y 4 is minimal whose Cayley table is : f(x, y) = x y 4 x\y
45 Theorem Let k be a prime. (1) x y k 1 is a minimal function. (2) For any 1 < s < k 1, x s y k s is not a minimal function. 45
46 (3) More Examples 46
47 (3) More Examples Recall!! Our Strategy Step 1 : Take arbitrary f(x, y) O (2) 3 from Csákány s list. Step 2 : Search for a polynomial g(x, y) O (2) k for k 3 whose counterpart for k = 3 is f(x, y) Step 3 : Examine if g is minimal. 47
48 (3) More Examples (3-1) Example A Step 1 : k = 3 : x + y + 2 x y 2 48
49 (3) More Examples (3-1) Example A Step 1 : k = 3 : x + y + 2 x y 2 Step 2 : k 3 :??? 49
50 (3) More Examples (3-1) Example A Step 1 : k = 3 : x + y + 2 x y 2 Step 2 : k 3 : x + y + (k 1) x y k 1 50
51 (3) More Examples (3-1) Example A Step 1 : k = 3 : x + y + 2 x y 2 Step 2 : k 3 : x + y + (k 1) x y k 1 Example. k = 5 : f(x, y) = x + y + 4 x y 4 x\y
52 (3-2) Example B Step 1 : k = 3 : x + 2y 2 + x 2 y 2 52
53 (3-2) Example B Step 1 : k = 3 : x + 2y 2 + x 2 y 2 Step 2 : k 3 :??? 53
54 (3-2) Example B Step 1 : k = 3 : x + 2y 2 + x 2 y 2 Step 2 : k 3 : x + (k 1)y k 1 + x k 1 y k 1 54
55 (3-2) Example B Step 1 : k = 3 : x y x 2 + x 2 y 2 Step 2 : k 3 : x + (k 1)y k 1 + x k 1 y k 1 Example. k = 5 : f(x, y) = x + 4 y 4 + x 4 y 4 x\y
56 (3-3) Example C Step 1 : k = 3 : x y x 2 + x 2 y 2 56
57 (3-3) Example C Step 1 : k = 3 : x y x 2 + x 2 y 2 Step 2 : k 3 :??? 57
58 (3-3) Example C Step 1 : k = 3 : x y x 2 + x 2 y 2 Step 2 : k 3 : x y k 1 +(k 1)x k 1 +x k 1 y k 1 58
59 (3-3) Example C Step 1 : k = 3 : x y x 2 + x 2 y 2 Step 2 : k 3 : x y k 1 +(k 1)x k 1 +x k 1 y k 1 Example. k = 5 : f(x, y) = x y x 4 + x 4 y 4 x\y
60 (3-4) Example D A bit more difficult example! 60
61 (3-4) Example D Step 1 : k = 3 : 2 x + 2 x y 2 61
62 (3-4) Example D Step 1 : k = 3 : 2 x + 2 x y 2 Step 2 : k 3 :??? 62
63 (3-4) Example D Step 1 : k = 3 : 2 x + 2 x y 2 Step 2 : k 3 : (k 1) x + (k 1) x y k 1 is not good! 63
64 (3-4) Example D Step 1 : k = 3 : 2 x + 2 x y 2 Step 2 : k 3 : (k 1) x + 2 x y k 1 64
65 (3-4) Example D Step 1 : k = 3 : 2 x + 2 x y 2 Step 2 : k 3 : (k 1) x + 2 x y k 1 Example. k = 5 : f(x, y) = 4 x + 2 x y 4 x\y
66 (3-5) Example E This example requires better skill even to find a candidate of generalization!! 66
67 (3-5) Example E Step 1 : k = 3 : 2 x 2 y + 2 x y 2 67
68 (3-5) Example E Step 1 : k = 3 : 2 x 2 y + 2 x y 2 Step 2 : k 3 :??? 68
69 (3-5) Example E Step 1 : k = 3 : 2 x 2 y + 2 x y 2 Step 2 : k 3 : (k 1) x k 1 y + (k 1) x y k 1 is not good! 69
70 (3-5) Example E Step 1 : k = 3 : 2 x 2 y + 2 x y 2 Step 2 : k 3 : (k 1) x k 1 y + (k 1) x y k 1 is not good! 2 x k 1 y + (k 1) x y k 1 is also not good!! 70
71 (3-5) Example E Step 1 : Step 2 : k = 3 : 2 x 2 y + 2 x y 2 k 3 : (k 1) k 1 i=1 xk i y i 71
72 (3-5) Example E Step 1 : Step 2 : k = 3 : 2 x 2 y + 2 x y 2 k 3 : (k 1) k 1 i=1 xk i y i Step 3 (Sketch of Proof): f(x, x) = x since (k 1) 2 = 1 For x y, let D = k 1 i=1 xk i y i Then xy 1 D = D Hence xd = yd which implies D = 0 Therefore f(x, y) = x if x = y and f(x, y) = 0 if x y It is easy to see that f is minimal 72
73 (3-5) Example E Step 1 : Step 2 : k = 3 : 2 x 2 y + 2 x y 2 k 3 : Example. k = 5 : (k 1) k 1 i=1 xk i y i f(x, y) = 4 x 4 y + 4 x 3 y x 2 y x y 4 x\y
74 More and More Examples 74
75 More and More Examples You will hear at the conference celebrating the 80th Birthday of Béla Csákány!! 75
76 More and More Examples You will hear at the conference celebrating the 80th Birthday of Béla Csákány!! Thank you 76
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