Abstract And-Parallel Machines

And-Parallel Machines* Nachum Dershowitz and Naomi Lindenstrauss Department of Computer Science, The Hebrew University, Jerusalem 91904, Israel The deterministic Turing machine, though abstract, can still be seen as a model of a realistic computer. The same cannot be said for the nondeterministic Turing machine as a model of parallel computing. We introduce several abstract machines with fine-grained parallelism--the and-parallel Turing machine, the stronger parallel rewriting machine, and extensions of both with an interrupt capability. These machines are very powerful: the parallel rewriting machine can compute the permanent in polynomial time and the and-parallel machine with interrupt can simulate nondeterministic and alternating Turing machines of polynomial time complexity in polynomial time. All the same, they may be viewed as realistic models if time and space are suitably restricted. A n d P a r a l l e l T~arlng Mach ine s . One way of viewing the nondeterministic Turing machine is to say that at each stage, when confronted with k choices, it splits into k replicas of itself, each of which makes one of the choices. The input string is accepted if one of these machines enters an accepting state. This kind of parallelism can be termed "or-parallelism". However, the nondeterministic Turing machine does not model real parallel machines in which different processors communicate with each other. Also, when faced with an input of size n that must be read, it will need at least n units of time. But by then the machine may have arrived at an exponential number of possibilities, an unrealistic scenario. Our and-parallel machine works with an unbounded binary tree. (We take a binary tree for simplici ty--any bounded arity will do.) Instead of the square scanned by the "head" (representing the memory cell dealt with at the moment by a sequential computer), the tree has a "frontier" of nodes, representing the memory cells dealt with at the moment by the processors comprising the parallel computer. (By "frontier" we mean a subset of the binary tree such that none of its nodes is an ancestor of another, while every node in the binary tree is either an offspring or an ancestor of a node in the frontier.) In the beginning the frontier consists of the root node. The input is a finite binary tree. Each node in the frontier is in one of a finite number of states, and can be thought of as being taken care of by a different processor. Given a binary tree with frontier, the transition function ~ acts on its frontier nodes and transforms it into another binary tree with frontier. It may change the content of the nodes of the frontier in the following way: if N is a frontier node in state q with symbol t, (f can change the symbol at N. According to q and t, N may remain a frontier node, although its state may change. It can also be replaced in the frontier by its two children, who get states determined by g. If q and t are suitable, and N's sibling N ' also has suitable symbol and state, then both N * The full version of this extended abstract can be found on the World Wide Web at http ://www. cs. huj i . ac. i l /~naomil .