Asynchronous Cellular Automaton - Update Schemes

Update Schemes

Several studies have implemented asynchronous models and found that their behaviour differs from the synchronous ones. Bersini and Detours (1994) have shown how sensitive Conway's Game of Life is to the updating scheme. Any interesting behaviour disappears in the asynchronous case. Harvey and Bossomaier (1997) pointed out that stochastic updating in random boolean networks results in the expression of point attractors only: there is no repeatable cyclic behaviour, although they introduced the concept of loose cyclic attractors. Kanada (1994) has shown that some one-dimensional CA models that generate non-chaotic patterns when updated synchronously generate edge of chaos patterns when randomised. Orponen (1997) has demonstrated that any synchronously updated network of threshold logic units (see Artificial neuron) can be simulated by a network that has no constraints on the order of updates. Sipper et al. (1997) investigated the evolution of non-uniform CAs that perform specific computing tasks. These models relax the normal requirement of all nodes having the same update rule. In their models, nodes were organised into blocks. Nodes within a block were updated synchronously, but blocks were updated asynchronously. They experimented with three schemes: (1) at each time step, a block is chosen at random with replacement; (2) at each time step, a block is chosen at random without replacement; (3) at each time step, a block is chosen according to a fixed update order.

There are different types of asynchronous updating, and different authors have described these in different ways. The schemes shown in the images below are as follows (Cornforth et al. 2005):

• The synchronous scheme - all cells are updated in parallel at each time step. This is the conventional model, stated here for comparison.
• The random independent scheme - at each time step, a cell is chosen at random with replacement, and updated.
• The random order scheme - at each time step, all nodes are updated, but in random order.
• The cyclic scheme - at each time step a node is chosen according to a fixed update order, which was decided at random during initialisation of the model.
• The self-clocked scheme - each cell has an independent timer, initialised to a random period and phase. When the period has expired, the cell is updated and the timer reset. Updating is autonomous and proceeds at different rates for different cells.
• The self-sync scheme - the same as the clocked scheme, but the phase of the timers are affected by local coupling to neighbours, and so are able to achieve local synchrony.

The time-state diagrams below show the differences that are caused by changing the update scheme of the cellular automata model without changing any other parameters. The rule used, rule 30, is the same for each diagram.