2.2 Trivalent Boolean Networks
2.2.1 Preserved Properties
Here we prove that many specific properties are preserved under implementation, meaning that if a relation X has the property, and X can implement Y, then Y must have the property as well. For the proof, we will usually use the “juxtaposition / jumper” method of theorem 5. Occasionally, for properties that only apply to prime factors, we will use a similar method but instead of plain juxtaposition, we will consider juxtaposition with a wire connecting the two juxtaposed relations to maintain connectivity.
Theorem 30 (Convexity via the or) The following property is preserved under imple- mentation: If a relation accepts A and B, then it also accepts any vector C that (1) is directly between them in Hamming distance and (2) is bitwise≥ either A or B.
Proof: Juxtaposition: Walk both relations to peak, then walk them both down. Jumper:
If A and B are accepted with the same jumper value, then so is C. Otherwise, if A uses jumper=1 and B uses jumper=0, and C is a superset of A, then it is accepted with jumper=1. IfC is a superset of B, then it is accepted with either jumper.
Theorem 31 (Convexity via the and) The following property is preserved under im- plementation: If a relation accepts A and B, then it also accepts any vector C that (1) is directly between them in Hamming distance and (2) is bitwise≤ either A or B.
Proof: This can be proved just like the previous theorem.
Theorem 32 (Conjunctivity) The following property is preserved under implementa- tion: If a network accepts A and B, then it also accepts the bitwise conjunction A or B.
Proof: Juxtaposition: Easy. Jumper: Easy.
Theorem 33 (Disjunctivity) The following property is preserved under implementation:
If a network accepts A and B, then it also accepts the bitwise disjunction A and B.
Proof: Juxtaposition: Easy. Jumper: Easy.
Theorem 34 (Parity of 1s) The following property is preserved under implementation:
All accepted vectors have even parity.
Proof: Juxtaposition: Easy. Jumper: Taking away two identical bits always preserves the
parity.
Theorem 35 (Parity of 0s) The following property is preserved under implementation:
All accepted vectors have an even number of 0s.
Proof: Juxtaposition: Easy. Jumper: Taking away two identical bits preserves the parity
of the number of 0s.
Theorem 36 (Superset of parity of 1s) The following property is preserved under im- plementation: The relation’s prime factors accept all vectors having an even number of 1s.
Proof: Juxtaposition with wire: This construction still yields a superset of parity. Reducible:
If a relation is reducible and is a superset of parity then it accepts everything, and its factors will have the property too, so we can just check prime factors. Jumper: Still superset of
parity.
Theorem 37 (Superset of parity of 0s) The following property is preserved under im- plementation: The relation’s prime factors accept all vectors having an even number of 0s.
Proof: Similar to the previous theorem.
Theorem 38 (Low weight) The following property is preserved under implementation:
All vectors with weight ≤k are acceptable.
Proof: Juxtaposition: Any giant vector with weight ≤k has this property on both subparts too. Jumper: For any small vector, the large vector with jumper=0 is ok.
Theorem 39 (High weight) The following property is preserved under implementation:
All vectors with at most k zeros are acceptable.
Proof: Similar to the previous theorem.
Theorem 40 (Almost monotone) The following property is preserved under implemen- tation: From an acceptable tuple of values, any leg may change from 0 → 1, requiring at most one other to do so.
Proof: Juxtaposition: The other leg, if any, will be in the same subpart. Jumper: If the other leg is not in the jumper, we’re fine. If the other leg is in the jumper, then we cannot stop there, but must change the other jumper leg too, which may result in yet another leg changing. If that other leg is the first jumper leg, or the original changed leg, then we’re having bad luck, but actually those can’t happen, as we’re only changing from 0→1.
Theorem 41 (Almost negative-monotone) The following property is preserved under implementation: From an acceptable tuple of values, any leg may change from 1 → 0, requiring at most one other to do so.
Proof: Similar to the previous theorem.
Theorem 42 (Monotone with specific factors) A monotone (respectively negative-monotone) relation X can only implement relations that are factorable into relations that can be im-
plemented simply by forcing legs to 1 (respectively 0).
Proof: If a relation is monotone, then the set of vectors satisfying the network is the same as the set of vectors satisfying the netwok when all intenal wires are forced to 1. In the latter case it is clear that inputs connected to separate nodes are independent. So the only
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.