Update digraphs and Boolean networks Julio B. Aracena Lucero (J. - PowerPoint PPT Presentation

Contents Update digraphs and Boolean networks Julio B. Aracena Lucero (J. Demongeot, E. Fanchon, E. Goles, L. G´ omez, M. Montalva, A. Moreira, M. Noual and L. Salinas) ıa Matem´ Departamento de Ingenier´ atica Facultad de Ciencias F´ ısicas y Matem´ aticas Universidad de Concepci´ on Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Contents Contents Boolean Networks 1 Definition Connection Digraph Deterministic Update Schedule Update Digraph 2 Necessary conditions Sufficient conditions Some combinatorics problems about update digraphs 3 Inverse Problem 4 Complexity Open Problems Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Contents Contents Boolean Networks 1 Definition Connection Digraph Deterministic Update Schedule Update Digraph 2 Necessary conditions Sufficient conditions Some combinatorics problems about update digraphs 3 Inverse Problem 4 Complexity Open Problems Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Contents Contents Boolean Networks 1 Definition Connection Digraph Deterministic Update Schedule Update Digraph 2 Necessary conditions Sufficient conditions Some combinatorics problems about update digraphs 3 Inverse Problem 4 Complexity Open Problems Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Contents Contents Boolean Networks 1 Definition Connection Digraph Deterministic Update Schedule Update Digraph 2 Necessary conditions Sufficient conditions Some combinatorics problems about update digraphs 3 Inverse Problem 4 Complexity Open Problems Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Boolean Networks Definition Update Digraph Connection Digraph Some combinatorics problems about update digraphs Deterministic Update Schedule Inverse Problem Boolean Network (BN) A BN N = ( F , s ) is defined by: 4 3 A global transition function F = ( f 1 , . . . , f n ) : { 0 , 1 } n → { 0 , 1 } n . 5 2 f i local activation function. 6 1 An update schedule s . . . n 7 x i ( t ) ∈ { 0 , 1 } is the node state i on time t . Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Boolean Networks Definition Update Digraph Connection Digraph Some combinatorics problems about update digraphs Deterministic Update Schedule Inverse Problem Boolean Network (BN) A BN N = ( F , s ) is defined by: 4 3 A global transition function F = ( f 1 , . . . , f n ) : { 0 , 1 } n → { 0 , 1 } n . 5 2 f i local activation function. 6 1 An update schedule s . . . n 7 x i ( t ) ∈ { 0 , 1 } is the node state i on time t . Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Boolean Networks Definition Update Digraph Connection Digraph Some combinatorics problems about update digraphs Deterministic Update Schedule Inverse Problem Boolean Network (BN) A BN N = ( F , s ) is defined by: 4 3 A global transition function F = ( f 1 , . . . , f n ) : { 0 , 1 } n → { 0 , 1 } n . 5 2 f i local activation function. 6 1 An update schedule s . . . n 7 x i ( t ) ∈ { 0 , 1 } is the node state i on time t . Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Boolean Networks Definition Update Digraph Connection Digraph Some combinatorics problems about update digraphs Deterministic Update Schedule Inverse Problem Connection Digraph Given N = ( F , s ) , the connection digraph G F = ( V , A ) is defined as: V = { 1 , . . . , n } , ( i , j ) ∈ A ⇐ ⇒ f j depends on x i , f j ( x 1 , . . . , x i − 1 , 0 , x i + 1 , . . . , x n ) � = f j ( x 1 , . . . , x i − 1 , 1 , x i + 1 , . . . , x n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . . ✲ . . . . . . . . . . . . . . . . . ♠ . ♠ . . . . . . . f 1 ( x ) = ( x 1 ∧ x 2 ) ∨ x 4 . . . 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . � ✻ f 2 ( x ) = 0 � f 3 ( x ) = x 3 ∧ ( x 4 ∨ ¯ x 4 ) � ✠ � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . ✛ . . . . . . . . . . . . . . . f 4 ( x ) = x 2 ∧ ¯ . . . ♠ ♠ x 3 . . . . . . . . . . 1 2 . . . . . . . . . . . . . . . . . . . . . . . V − ( j ) = { i : ( i , j ) ∈ A } Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Boolean Networks Definition Update Digraph Connection Digraph Some combinatorics problems about update digraphs Deterministic Update Schedule Inverse Problem Connection Digraph Given N = ( F , s ) , the connection digraph G F = ( V , A ) is defined as: V = { 1 , . . . , n } , ( i , j ) ∈ A ⇐ ⇒ f j depends on x i , f j ( x 1 , . . . , x i − 1 , 0 , x i + 1 , . . . , x n ) � = f j ( x 1 , . . . , x i − 1 , 1 , x i + 1 , . . . , x n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . . ✲ . . . . . . . . . . . . . . . . . ♠ . ♠ . . . . . . . f 1 ( x ) = ( x 1 ∧ x 2 ) ∨ x 4 . . . 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . � ✻ f 2 ( x ) = 0 � f 3 ( x ) = x 3 ∧ ( x 4 ∨ ¯ x 4 ) � ✠ � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . ✛ . . . . . . . . . . . . . . . f 4 ( x ) = x 2 ∧ ¯ . . . ♠ ♠ x 3 . . . . . . . . . . 1 2 . . . . . . . . . . . . . . . . . . . . . . . V − ( j ) = { i : ( i , j ) ∈ A } Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Boolean Networks Definition Update Digraph Connection Digraph Some combinatorics problems about update digraphs Deterministic Update Schedule Inverse Problem Connection Digraph Given N = ( F , s ) , the connection digraph G F = ( V , A ) is defined as: V = { 1 , . . . , n } , ( i , j ) ∈ A ⇐ ⇒ f j depends on x i , f j ( x 1 , . . . , x i − 1 , 0 , x i + 1 , . . . , x n ) � = f j ( x 1 , . . . , x i − 1 , 1 , x i + 1 , . . . , x n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . . ✲ . . . . . . . . . . . . . . . . . ♠ . ♠ . . . . . . . f 1 ( x ) = ( x 1 ∧ x 2 ) ∨ x 4 . . . 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . � ✻ f 2 ( x ) = 0 � f 3 ( x ) = x 3 ∧ ( x 4 ∨ ¯ x 4 ) � ✠ � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . ✛ . . . . . . . . . . . . . . . f 4 ( x ) = x 2 ∧ ¯ . . . ♠ ♠ x 3 . . . . . . . . . . 1 2 . . . . . . . . . . . . . . . . . . . . . . . V − ( j ) = { i : ( i , j ) ∈ A } Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Boolean Networks Definition Update Digraph Connection Digraph Some combinatorics problems about update digraphs Deterministic Update Schedule Inverse Problem Connection Digraph Given N = ( F , s ) , the connection digraph G F = ( V , A ) is defined as: V = { 1 , . . . , n } , ( i , j ) ∈ A ⇐ ⇒ f j depends on x i , f j ( x 1 , . . . , x i − 1 , 0 , x i + 1 , . . . , x n ) � = f j ( x 1 , . . . , x i − 1 , 1 , x i + 1 , . . . , x n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . . ✲ . . . . . . . . . . . . . . . . . ♠ . ♠ . . . . . . . f 1 ( x ) = ( x 1 ∧ x 2 ) ∨ x 4 . . . 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . � ✻ f 2 ( x ) = 0 � f 3 ( x ) = x 3 ∧ ( x 4 ∨ ¯ x 4 ) � ✠ � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✲ . . . . . . ✛ . . . . . . . . . . . . . . . f 4 ( x ) = x 2 ∧ ¯ . . . ♠ ♠ x 3 . . . . . . . . . . 1 2 . . . . . . . . . . . . . . . . . . . . . . . V − ( j ) = { i : ( i , j ) ∈ A } Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks

Boolean Networks Definition Update Digraph Connection Digraph Some combinatorics problems about update digraphs Deterministic Update Schedule Inverse Problem Deterministic Update Schedule A deterministic update schedule is a function s : { 1 , . . . , n } → { 1 , . . . , n } such that s ( { 1 , . . . , n } ) = { 1 , . . . , m } , m ≤ n . s (2) = 4 2 Sequential: s (3) = 3 s (1) = 5 3 1 s ( { 1 , . . . , n } ) = { 1 , . . . , n } . Parallel ( s p ): s ( { 1 , . . . , n } ) = { 1 } Block-Sequential: s (4) = 2 4 5 s (5) = 1 s ( { 1 , . . . , n } ) = { 1 , . . . , m } , m < n . Workshop DISCO, ISCV 2011 Update digraphs and Boolean networks