# Predecessor and permutation existence problems for sequential dynamical systems

### Description

A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was introduced in BMR99, BR991 as a formal model for analyzing simulation systems. An SDS S is a triple (G, F,n ),w here (i) G(V,E ) is an undirected graph with n nodes with each node having a state, (ii) F = (fi, fi, . . ., fn), with fi denoting a function associated with node ui E V and (iii) A is a permutation of (or total order on) the nodes in V, A configuration of an SDS is an n-vector ( b l, bz, . . ... continued below

25 p.

## Who

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## What

### Description

A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was introduced in BMR99, BR991 as a formal model for analyzing simulation systems. An SDS S is a triple (G, F,n ),w here (i) G(V,E ) is an undirected graph with n nodes with each node having a state, (ii) F = (fi, fi, . . ., fn), with fi denoting a function associated with node ui E V and (iii) A is a permutation of (or total order on) the nodes in V, A configuration of an SDS is an n-vector ( b l, bz, . . ., bn), where bi is the value of the state of node vi. A single SDS transition from one configuration to another is obtained by updating the states of the nodes by evaluating the function associated with each of them in the order given by n. Here, we address the complexity of two basic problems and their generalizations for SDSs. Given an SDS S and a configuration C, the PREDECESSOR EXISTENCE (or PRE) problem is to determine whether there is a configuration C' such that S has a transition from C' to C. (If C has no predecessor, C is known as a garden of Eden configuration.) Our results provide separations between efficiently solvable and computationally intractable instances of the PRE problem. For example, we show that the PRE problem can be solved efficiently for SDSs with Boolean state values when the node functions are symmetric and the underlying graph is of bounded treewidth. In contrast, we show that allowing just one non-symmetric node function renders the problem NP-complete even when the underlying graph is a tree (which has a treewidth of 1). We also show that the PRE problem is efficiently solvable for SDSs whose state values are from a field and whose node functions are linear. Some of the polynomial algorithms also extend to the case where we want to find an ancestor configuration that precedes a given configuration by a logarithmic number of steps. Our results extend some of the earlier results by Sutner [Su95] and Green [@87] on the complexity of the PREDECESSOR EXISTENCE problem for 1-dimensional cellular automata. Given the underlying graph G(V, E), and two configurations C and C' of an SDS S, the PERMUTATION EXISTENCE (or PME) problem is to determine whether there is a permutation of nodes such that 8 has a transition from C' to C in one step. We show that the PME problem is NP-complete even when the function associated with each node is a simple-threshold function. We also show that a generalized version of the PME( GEN-PMEp)r oblem is NP-complete for SDSs where each node function is NOR and the underlying graph has a maximum node degree of 3. When each node computes the OR function or when each node computes the AND function, we show that the GEN-PMEpr oblem is solvable in polynomial time.

25 p.

### Source

• Submitted to: 2nd IFIP International Conference on Theoretical Computer Science, August 25-30, 2002, Montreal, Canada

### Identifier

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• Report No.: LA-UR-02-0180
• Report No.: LA-UR-02-180
• Grant Number: none
• Office of Scientific & Technical Information Report Number: 975942

### Collections

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## When

### Creation Date

• January 1, 2002

### Added to The UNT Digital Library

• Nov. 13, 2016, 7:26 p.m.

### Description Last Updated

• Dec. 12, 2016, 6:54 p.m.

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