2.2 Trivalent Boolean Networks
2.2.2 The Final Eighteen Proofs
implementable target relations are ones reducible to relations implementable by the original
monotone relation with some legs forced to 1.
We note that although this theorem applies to many relations, we only need it here for 31 (in which a 1 on the controller forces a 0 on the other two wires), and its dual, 234.
Theorem 43 (Singletons and pairs of 1s) The following property is preserved under implementation: For any tuplet accepted by the relation, the 1s of that tuple can be grouped into disjoint singletons and pairs such that the zeroing of any single group results in another acceptable tuple.
Proof: Juxtaposition: Easy. Jumper: For any tuple setting the jumper to 1, group the zero,
one, or two partners of the jumper legs.
Theorem 44 (Singletons and pairs of 0s) The following property is preserved under implementation: For any tuplet accepted by the relation, the 0s of that tuple can be grouped into disjoint singletons and pairs such that converting any single group to 1 results in another acceptable tuple.
Proof: Similar to the previous theorem.
• 45⇛{43,151,235}
• 61⇛{9,153,155,173}
• 111⇛{9,153}
• 126⇛{189}
• 151⇛{159,191}
• 155⇛{153}
• 159⇛{191}
Many of these proofs share a common technique. Assume the network is in a stable state X, and consider another stable stateY. Paint all wires differing in the two states. Flippers can be sent forth, with instructions to stay on those wires. When a flipper gets to a vertex, it chooses some minimal acceptable set of flipped wires which is a subset of the painted wires and a superset of the wires flipped so far (including the incoming one). The number of exiting flippers may be 0, 1, or 2. It is rarely 2. By giving the flippers instructions on how to make the choice, or by wisely choosing X and/or Y in the first place, we can often show something about the structure of the network, or show that flippers going in different directions on the same subset could avoid proper operation of the allegedly implemented relation. A “down flipper” will mean a flipper that only flips 1s to 0s, while an “up flipper”
only flips 0s to 1s.
Also, a relation that is at least 1-robust (every bad vector is isolated) is “flammable,”
meaning that if any leg or wire is “don’t-care” (DC), then it “burns,” meaning that given the two outgoing burnt wire values (at a ternary relation), one can always find a value for the originating burnt wire that satisfies the relation. Thus all the leaves of a burnt tree (counting places where it burned into itself in mid-edge as leaves) may be set arbitrarily.
256⇛{175}
25 has two legs equal and not all three may be 1. 175 is the≤relation on two legs. In a 25 net, started with all wires 0, an up flipper will only propagate along an equal-leg route. So if it pops out another leg, the implications all work in reverse too. This is unlike 175.
276⇛{9,43,153}
27 isA≤B ≤C. OnlyB can be forced to 1. 9 is equality on two legs with the third forced to 0. 43 is maj with two legs negated. 153 is equality on two legs. Equality done with a 27 net would need to useBs for the ends. Suppose the network is in a state corresponding to 1 = 1. A down flipper coming in a B can only continue out A. Entering anywhere else it always dies. So path cannot be bidirectional like = needs. If 43 has last leg negated, then 011 and 101 are acceptable, and in each case a down flipper sent in the last leg must come out the other 1 leg. But a non-dying down flipper’s path (enter B, leave A) is fixed by topology, and cannot depend on current wire values.
456⇛{43,151,235}
45 is an and-gate with one negated input. 43 is maj with two legs negated. 151 is Σ6= 2.
235 is two wire ends whose controller, if 0, forces =. We will refer to the output of 45 as A, the negated input as B, and the plain input as C. Starting from a 45 net with all wires 0, there are only up flippers. An up flipper in a 45 net never splits into two. Entering B, it dies. Entering A, it exits C. EnteringC, it can chooseA orB. Suppose a net has leg x which if it flips up forcesy to flip up too (if nothing else does). Then the path must contain only A→C,C→A, and C →B transitions at the vertices (but note that the connection type need not be the same at both ends of an edge). To do 235, there is a reversible forcing path from 0s, so it doesn’t useB. But there is also some path to the third leg, so it must use aB. This cannot implement 235, since routes to B cannot merge from other two legs.
For 43, the contradiction is because routes to negated leg cannot merge from other two legs.
For 151, we will need to consider a general state of the network. The 1-valued wires form chains that do not split, though they may end. Any chain may be fully converted from 1 to 0 and the network remains satisfied. Consider a network implementing 151, with every input a 1. At least one of the three legs must start a chain that ends somewhere inside the network. This may be converted to 0, so it’s not a 151.
616⇛{9,153,155,173}
61 is a negated wire whose controller, if 0, allows 00. 9 is equality on two wires and a forced 0. 153 is equality on two wires. 155 is a wire (B,C) whose controller (A), if 0, allowsB = 0
andC = 1. 173 is a wire (A,C) whose controller (B), if 1, allowsA= 0 andC= 1. In a 61 net, starting from all 0s, an up flipper dies unless it enters C, in which case it can choose where to leave. Since this is a unidirectional process, equality (9, 153) cannot be done. A 61 net doing 155, started with 0s, could take an up flipper at 155’s B. This must emerge at 155’s C. But in a 61 net, we can instruct the flipper to head straight towards 155’s A.
Similarly for 173, a flipper starting atA can emerge at B (or nowhere) instead of C.
1116⇛{9,153}
111 is two loose wires whose controller, if 1, connects them negatedly. It is flammable. 9 is equality and 0. 153 is equality and DC. Connecting the two loose wires forces the controller to 0. So equality is the only question. A cycle of wires burns if length is even. If odd, then at least one controller must be 0. A 111 net, started from 0s, kills up flippers unless they enter the controller, in which case they may leave where they like. Thus, (proof 1) they may avoid any particular destination (by heading for a different destination), and (proof 2) their route is not reversible (a flipper started from the other end could die immediately).
1266⇛{189}
126 is 6=3. Any cycle is equivalent to DC on the legs sticking out from the cycle, because given any legs, we can walk around the cycle setting its edges so as to satisfy all the nodes.
Any tree is6=k with legs negated according to the two-coloring of the tree. Any tree with v vertices has v+ 2 legs. 189 is 6=3 with one leg negated, which is prime and must be implementable (if at all) with a network of a single component. Since legs are not all negated the same, it must have at least two vertices. But this is impossible: Since it has only three legs, it must be implemented with just one vertex.
1516⇛{159,191}
151 is Σ6= 2. 159 is two loose ends whose controller, if 1, connects them (forces equality).
191 is or3 with two legs negated. Any DC on a 151 burns the whole thing. Any cycle can be set to 0 and is a giant DC. So assume the network is a tree. Any tree with v vertices has v+ 2 legs. 159 and 191, being prime, cannot be interpreted as disconnected trees. A tree with 3 legs has a single vertex, and can only be 151.
1556⇛{153}
153 is equality on two wires. 155 is a wire (B,C) whose controller (A), if 0, allows B = 0 and C = 1. Start with the network of 155s having all wires 1. A nondying flipper’s path through 155 must use one of the following transitions at each vertex: B → A, B → C, or C→B. If the path ever enters atB, it has a choice, but the two choices can never remerge, so no particular outcome can be forced. So the path forcing equality must always enter at C and leave at B. But it can’t do this both ways.
1596⇛{191}
159 has a controller leg, which, if 1, forces the other two legs to be equal. 191 is or with two legs negated.
Two legs on the 191 can be forced to 0 (by 01), and one can be forced to 1 (by 11). An up flipper coming in 159’s controller can die or exit as either an up or down flipper. A flipper coming in another 159 wire can die, continue on the other wire, or exit as a down flipper on the controller. If the 159 network has a cycle of the controlled wire, then it becomes a DC on all the controllers. A DC on any 159 leg makes both other legs DC, so the whole component burns up. So all wires end at a leg or a controller. If a wire has controllers at both ends, it can all be set to 0 and the component burns. So all wires have at least one end at a leg. So there are at most three wires in a network implementing a 191. Three or two.
Counting wire ends, the number of wires is 3 plus the number of vertices, divided by two.
So three wires means three vertices, and two wires means one vertex. These possibilities can be exhaustively verified not to implement 191.