Local monotonicities and lattice derivatives of Boolean and pseudo-boolean functions

Transcription

1 Local monotonicities and lattice derivatives of Boolean and pseudo-boolean functions Tamás Waldhauser joint work with Miguel Couceiro and Jean-Luc Marichal University of Szeged AAA 83 Novi Sad, 16 March 2012

2 Partial derivatives Boolean function: f : {0, 1} n {0, 1}

3 Partial derivatives Boolean function: f : {0, 1} n {0, 1} pseudo-boolean function: f : {0, 1} n R

4 Partial derivatives Boolean function: f : {0, 1} n {0, 1} pseudo-boolean function: f : {0, 1} n R The partial derivative of f : {0, 1} n R w.r.t. x k is the function k f : {0, 1} n R defined by k f (x) = f (x 1 k) f (x 0 k) = f (x 1,..., 1,..., x n ) f (x 1,..., 0,..., x n ). Observe that k f does not depend on x k.

5 Partial derivatives Boolean function: f : {0, 1} n {0, 1} pseudo-boolean function: f : {0, 1} n R The partial derivative of f : {0, 1} n R w.r.t. x k is the function k f : {0, 1} n R defined by Example k f (x) = f (x 1 k) f (x 0 k) = f (x 1,..., 1,..., x n ) f (x 1,..., 0,..., x n ). Observe that k f does not depend on x k. The partial derivatives of the Boolean sum f (x 1, x 2 ) = x 1 x 2 = x 1 + x 2 2x 1 x 2 are 1 f (x 1, x 2 ) = f (1, x 2 ) f (0, x 2 ) = 1 2x 2, 2 f (x 1, x 2 ) = f (x 1, 1) f (x 1, 0) = 1 2x 1.

6 Monotonicity f is isotone (positive, order-preserving, nondecreasing) in x k if k f (x) 0 for all x {0, 1} n.

7 Monotonicity f is isotone (positive, order-preserving, nondecreasing) in x k if k f (x) 0 for all x {0, 1} n. f is antitone (negative, order-reversing, nonincreasing) in x k if k f (x) 0 for all x {0, 1} n.

8 Monotonicity f is isotone (positive, order-preserving, nondecreasing) in x k if k f (x) 0 for all x {0, 1} n. f is antitone (negative, order-reversing, nonincreasing) in x k if k f (x) 0 for all x {0, 1} n. f is monotone in x k if it is either isotone or antitone in x k, i.e., if k f (x) does not change sign.

9 Monotonicity f is isotone (positive, order-preserving, nondecreasing) in x k if k f (x) 0 for all x {0, 1} n. f is antitone (negative, order-reversing, nonincreasing) in x k if k f (x) 0 for all x {0, 1} n. f is monotone in x k if it is either isotone or antitone in x k, i.e., if k f (x) does not change sign. f is monotone (isotone, antitione) if it is monotone (isotone, antitone) in all of its variables.

10 Monotonicity f is isotone (positive, order-preserving, nondecreasing) in x k if k f (x) 0 for all x {0, 1} n. f is antitone (negative, order-reversing, nonincreasing) in x k if k f (x) 0 for all x {0, 1} n. f is monotone in x k if it is either isotone or antitone in x k, i.e., if k f (x) does not change sign. f is monotone (isotone, antitione) if it is monotone (isotone, antitone) in all of its variables. All unary functions are monotone.

11 Monotonicity f is isotone (positive, order-preserving, nondecreasing) in x k if k f (x) 0 for all x {0, 1} n. f is antitone (negative, order-reversing, nonincreasing) in x k if k f (x) 0 for all x {0, 1} n. f is monotone in x k if it is either isotone or antitone in x k, i.e., if k f (x) does not change sign. f is monotone (isotone, antitione) if it is monotone (isotone, antitone) in all of its variables. All unary functions are monotone. The only non-monotone binary Boolean functions are x 1 x 2 and x 1 x 2 1.

12 Local monotonicities Definition We say that f : {0, 1} n R is p-locally monotone if, for every k [n] and every x,y {0, 1} n, we have x i y i < p k f (x) k f (y) 0. i [n]\{k}

13 Local monotonicities Definition We say that f : {0, 1} n R is p-locally monotone if, for every k [n] and every x,y {0, 1} n, we have x i y i < p k f (x) k f (y) 0. i [n]\{k} p-local monotonicity implies (p 1)-local monotonicity.

14 Local monotonicities Definition We say that f : {0, 1} n R is p-locally monotone if, for every k [n] and every x,y {0, 1} n, we have x i y i < p k f (x) k f (y) 0. i [n]\{k} p-local monotonicity implies (p 1)-local monotonicity. An n-ary function is n-locally monotone iff it is monotone.

15 Local monotonicities Definition We say that f : {0, 1} n R is p-locally monotone if, for every k [n] and every x,y {0, 1} n, we have x i y i < p k f (x) k f (y) 0. i [n]\{k} p-local monotonicity implies (p 1)-local monotonicity. An n-ary function is n-locally monotone iff it is monotone. Every function is 1-locally monotone.

16 Local monotonicities Definition We say that f : {0, 1} n R is p-locally monotone if, for every k [n] and every x,y {0, 1} n, we have x i y i < p k f (x) k f (y) 0. i [n]\{k} Theorem p-local monotonicity implies (p 1)-local monotonicity. An n-ary function is n-locally monotone iff it is monotone. Every function is 1-locally monotone. A Boolean function f : {0, 1} n {0, 1} is 2-locally monotone iff k f (x) k f (y) i [n]\{k} x i y i.

17 Lattice derivatives We define the partial lattice derivatives of f : {0, 1} n R w.r.t. x k by k f : {0, 1} n R, k f (x) = f (x 0 k) f (x 1 k) = min ( f (x 0 k), f (x 1 k) ), k f : {0, 1} n R, k f (x) = f (x 0 k) f (x 1 k) = max ( f (x 0 k), f (x 1 k) ).

18 Lattice derivatives We define the partial lattice derivatives of f : {0, 1} n R w.r.t. x k by k f : {0, 1} n R, k f (x) = f (x 0 k) f (x 1 k) = min ( f (x 0 k), f (x 1 k) ), k f : {0, 1} n R, k f (x) = f (x 0 k) f (x 1 k) = max ( f (x 0 k), f (x 1 k) ). Example The lattice derivatives of the Boolean sum f (x 1, x 2 ) = x 1 x 2 are 1 f (x 1, x 2 ) = f (1, x 2 ) f (0, x 2 ) = (1 x 2 ) x 2 = 0, 1 f (x 1, x 2 ) = f (1, x 2 ) f (0, x 2 ) = (1 x 2 ) x 2 = 1. The second-order lattice derivatives are 2 1 f (x 1, x 2 ) = 2 0 = 0, 1 2 f (x 1, x 2 ) = 1 1 = 1.

19 Permutable lattice derivatives Theorem A Boolean function f : {0, 1} n {0, 1} is 2-locally monotone iff k j f = j k f for all j = k.

20 Permutable lattice derivatives Theorem A Boolean function f : {0, 1} n {0, 1} is 2-locally monotone iff Definition k j f = j k f for all j = k. We say that f : {0, 1} n R has p-permutable lattice derivatives, if O k1 O kp f = O kπ(1) O kπ(p) f holds for every p-element set {k 1,..., k p } {1,..., n}, for all operators O ki { ki, ki } and for every permutation π S p.

21 Permutable lattice derivatives Theorem A Boolean function f : {0, 1} n {0, 1} is 2-locally monotone iff Definition k j f = j k f for all j = k. We say that f : {0, 1} n R has p-permutable lattice derivatives, if O k1 O kp f = O kπ(1) O kπ(p) f holds for every p-element set {k 1,..., k p } {1,..., n}, for all operators O ki { ki, ki } and for every permutation π S p. Theorem If a function has p-permutable lattice derivatives, then it has (p 1)-permutable lattice derivatives.

22 Local monotonicities vs. permutable lattice derivatives Theorem If a function is p-locally monotone, then it has p-permutable lattice derivatives.

23 Local monotonicities vs. permutable lattice derivatives Theorem If a function is p-locally monotone, then it has p-permutable lattice derivatives. Example Let f : {0, 1} n {0, 1} be the function that takes the value 0 on all tuples of the form m {}}{ ( 1,..., 1, 0,..., 0) with 0 m n, and takes the value 1 everywhere else. Then f has n-permutable lattice derivatives, but it is only 2-locally monotone.

24 Local monotonicities vs. permutable lattice derivatives Theorem If a function is p-locally monotone, then it has p-permutable lattice derivatives. Example Let f : {0, 1} n {0, 1} be the function that takes the value 0 on all tuples of the form m {}}{ ( 1,..., 1, 0,..., 0) with 0 m n, and takes the value 1 everywhere else. Then f has n-permutable lattice derivatives, but it is only 2-locally monotone. Theorem For symmetric functions, p-local monotonicity is equivalent to p-permutability of lattice derivatives.

25 Sections A section of a function f is any function g that can be obtained from f by substituting constants to some of the variables of f. For example, if f : {0, 1} 3 R, then g : {0, 1} 2 R, g (x 1, x 2 ) := f (x 1, x 2, 0) is a section of f. f(0, 1, 1) f(1, 1, 1) f(0, 1, 0) f(1, 1, 0) f(0, 0, 0) f(1, 0, 0) f(1, 0, 1)

26 Sections A section of a function f is any function g that can be obtained from f by substituting constants to some of the variables of f. For example, if f : {0, 1} 3 R, then g : {0, 1} 2 R, g (x 1, x 2 ) := f (x 1, x 2, 0) is a section of f. f(0, 1, 0) f(1, 1, 0) f(0, 0, 0) f(1, 0, 0)

27 Sections A section of a function f is any function g that can be obtained from f by substituting constants to some of the variables of f. For example, if f : {0, 1} 3 R, then g : {0, 1} 2 R, g (x 1, x 2 ) := f (x 1, x 2, 0) is a section of f. g(0, 1) g(1, 1) g(0, 0) g(1, 0)

28 Forbidden sections Theorem If a function is nice, then all of its sections are also nice, where nice stands for any of the previously discussed properties.

29 Forbidden sections Theorem If a function is nice, then all of its sections are also nice, where nice stands for any of the previously discussed properties. ugly nice

30 Forbidden sections Theorem If a function is nice, then all of its sections are also nice, where nice stands for any of the previously discussed properties. ugly nice Corollary A function is nice iff none of the minimal ugly functions appear among its sections.

31 Forbidden sections Theorem A Boolean function is isotone iff x 1 1 does not appear among its sections.

32 Forbidden sections Theorem A Boolean function is isotone iff x 1 1 does not appear among its sections. Theorem A Boolean function is 2-locally monotone iff neither x 1 x 2 nor x 1 x 2 1 appears among its sections.

33 Forbidden sections Theorem A Boolean function is isotone iff x 1 1 does not appear among its sections. Theorem A Boolean function is 2-locally monotone iff neither x 1 x 2 nor x 1 x 2 1 appears among its sections. Conjecture A Boolean function has permutable lattice derivatives iff none of the following functions appear among its sections:

34 References M. Couceiro, J.-L. Marichal, T. Waldhauser, Locally monotone Boolean and pseudo-boolean functions, to appear in Discrete Applied Mathematics, arxiv:

35 Advertisement Conference on Universal Algebra and Lattice Theory Szeged, Hungary, June 21 25, Dedicated to the 80th birthday of Béla Csákány