2.4 Complexity of Deciding Whether There Is a Solution
2.4.6 Networks of worm Relations Can Be Solved in Polynomial Time
about. It may be the most unnatural of all the relations in figure 2.2, but like anything else, one can get used to it. We can think of the worm relation as a negation on a wire, where the end of a third wire, the controller, controls the negation. If the controller is 0, then the negation operates normally. But if the controller is 1, then it allows the other two wires to also be 1s, if they don’t want to be the negation of each other.
We will consider the wires to be in loops and paths. We will think of a negation as being on a wire, so the wire is thought of as being the same wire on both sides of the negation, so for example we may speak of how many negations are on the wire. The two non-controller wires at a wormrelation will similarly be considered to be on the same loop or path, and the relation will count as a negation. Thus every wire is either a loop, or a path with a controller at each end.
If a loop has an even number of negations, then it may be assigned values that alternate at each negation, so that none of the impinging controllers are relevant. A wire that is irrelevant at one end can be said to “burn,” meaning that it can be oriented to point away from the source of burning, and if it is not a controller, it can devote itself to successfully satisfying the relation it points to (whetherwormor negation), thus freeing the other wires at that relation to burn, and so on.
A loop or path containing an even number of negations may burn. (The loop, by alternating values around the loop, and the path, by alternating values so that both ends are 1.) A negation with a burnt controller is a two-wayorrelation, which if on a loop allows the loop to burn, and if it is an odd-numbered negation along a path, allows the path to burn. An even-numbered negation along a path cannot make any use of a controller being 1, so any controllers on even-numbered negations along a path may burn.
At this point, all loops and paths have an odd number of negations, and paths only have controllers at odd positions along them. If anything burns now, the whole thing will burn, meaning the original network is satisfiable. If there is a cycle that passes through a controller, we can set the controller to 1 to start things burning, and when the burning comes around the cycle to that controller, that shows that it was ok to set it to 1.
If anything is left at this point, it is a tree (treating loops as points) whose loops and
paths all have an odd number of negations, and only odd-numbered negations on paths have controllers. Now each wormrelation has to choose at least one side: either the controller side (by enforcing the negation on the wire), or the wire side (by setting the controller to 1). Since the structure is a tree (except for loops), some loop or path must get left out, with no relation choosing it. (This loop or path may be found by following the “choice directions” backwards until we get stuck.) That loop or path cannot be assigned values in a satisfactory way, so the network is not satisfiable.
Chapter 3
Decidability
This chapter will discuss issues of decidability. There are two main areas in which decidabil- ity results are examined. The first is questions of implementability among relations without fan-out. (Some old and new results on the case when fan-out is available are given as well.) The second is questions of implementability among functions without fan-out. This is a topic that has not received as much attention in the literature as one might expect. We show that the question of implementability in this context is equivalent to the question of reachability for Petri nets, chemical reaction networks, and other systems.
For chemical reaction networks, we prove that using standard reaction rate kinetics, Turing machines can be reliably simulated using a probabilistic method. Without probabil- ities, the power disappears, and the systems become only as powerful as primitive recursive functions. This power of probability is the first example we are aware of where treating a system probabilistically dramatically increases the range of functions it is able to compute, in this case from primitive recursive or lower to general recursive. (Several examples are already known where treating a system probabilistically can dramatically improve the ef- ficiency of the task at hand, with respect to some precious resource, for example reducing communication in communication complexity or reducing congestion in network routing.)
For Petri nets, our results show that there is a big difference between two probabilistic models that might seem similar at first. If at each step, a transition is chosen at random, and then it fires if it is enabled, then the probability of reaching any given state can be calculated to any desired precision. However, if at each step, a token is chosen at random, and the token chooses an enabled transition that will absorb it upon firing, then even the approximate probability of reaching a given state can be undecidable.