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# 711 Class Note for PHYS 597A with Professor Albert at PSU

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 24 views.

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Date Created: 02/06/15

Boolean logic and algebra Developed by George Boole 1850 Basic Boolean operators NOT AND OR Can be defined based on set intersection and union or inputoutput relations gates truth tables Assume you have a network where each node can have two states Denote these states by ON 1 or OFF 0 Assume that interactions between the nodes makes their state change Then you can use Boolean rules to describe the relation between the states of nodes that act on a node inputs and the new state of the node output Truth tables for Boolean operators NOT AND OR n1 n2 Out n1 n2 Out In Out 0 O O O O O 0 1 O 1 O 0 1 1 1 o 1 O O 1 O 1 1 1 1 1 1 1 Out NOT In Out n1 AND n2 Out n1 OR n2 Alternative notation aANDb gt aob aORb gt ab NOT a gt 1 Evaluating expressions invert first then perform AND then OR Ex Write NOT a AND b AND NOT a OR o in the alternative notation Evaluate it for several values of a b 0 Complete the truth table Random Boolean networks Construct a network where each node is connected randomly to K other nodes Assume that the state of each node can be 0 or 1 o The state of each node is updated each timestep The rule giving the new state of each node is determined by a Boolean function of the states of the nodes it is connected to o The Boolean function is selected randomly from all possible K input Boolean functions The state of the whole network changes in time Ex Construct a network of four nodes each with two edges Associate a Boolean rule to each node Start with an initial state and update the state of the nodes 10 times What is happening to the state of the network Start from a different initial state Will the final behavior be the same How many different states can the network have I I I o 1101 o o o I I I D39I 0 0391 0 o I o o mm o I o I 1 0 0 1 o o I I o I I o suomsueJi 91218 edwexe skexgw Concepts in RBN Attractor a set of states that repeats itself in afixed sequence can be periodic or a fixed point All states lead to or are part of an attractor Basin of attraction all states leading to a given attractor In a network of N nodes the maximum possible length of a periodic attractor is the total number of states 2N In practice the period length of the attractor is much shorter than this maximum Cause many nodes become frozen due partly to canalizing functions an attractor state shown in detail transient tree and sub trees Andy Wuenche wwwddabcom Canalizing forcing functions At least one of the inputs has the property that the output is fixed if this input has one particular value eg a AND b is canalizing because a0 implies a AND b 0 Ex 2 How many twoinput Boolean functions are there How many of them are canalizing

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