Variations on a Theorem Regarding Dogs and Cats

This is a transcription of an email I wrote to my research friends one late night last weekend after a bout of data analysis. One of them said I should post it on this blog, so I said I’d consider it. To be honest it’s a silly thing, but I figure it’s for an important cause. Always a good thing to contribute to the body of literature on this sort of subject.

Theorem: Dogs > Cats

Note: The comparator may be interpreted as stating ‘Dogs greater than Cats’ or ‘Dogs better than Cats’ – though it should be noted that,  in most major languages, the ‘>’ operator has not been redefined nor extended to handle this sort of comparison. Those desiring to attempt the illustrious ‘Proof By Evaluation of Statement Which I Have Just Written in an Interactive Shell’ must then first implement this functionality before proceeding with said proof.

Proof, by Google Poetry:

Cats are...

Dogs are...

Cats: 2 of 4 prompts indicate negative qualities, 2 of 4 prompts indicate popular lunacy

–> 0% positive, 50% lunar

Dogs: 3 of 4 prompts indicate positive qualities, 1 of 4 prompts indicate popular confusion.

–> 75% positive, 25% confused

=> Cats < Dogs (p < .05, Nil Hypothesis Rejected)

Proof by evaluation of function:

dogs_better_than_cats() returns True, and forms the basis of a non-deterministic Turing Machine.

Proof by observation of worst-case running time:

Every operation dogs perform is O(lifetime of dog), and therefore constant-time – as well as constant-cluelessness. Except for love, which is Theta(infinity). They can even sort in constant time, by immediately proving P=NP (Problem = No Problem), forgetting about the not-problem and staring at you until you forget about it too and go for a walk instead. That, and cats hate everything. <– That was not a non-sequitur. Whoosh, proven by proof of P=NP and not having a concept of hate.

Corollary: Observe that everything operation cats are expected to do is O(infinite), because there’s a nonzero probability that they’ll never do it. In most real-world cases, the probability tends toward 1. The only constant-time operation performed by a cat is disregard(), which ASDFASDF

Proof by Memory Management:

Dogs typically come with about three words of memory, all of which will be overwritten following each new call to malloc(). Dogs will only execute one process at a time, during which time memory is locked. Furthermore, if no process is executed after 5 seconds of downtime, memory will be immediately overwritten with love, love, and the system constant ‘treat?’. This is not actually a proof, but it is important technical documentation.

Proof by Well-Ordering Principle:

Okay, dogs aren’t always well-ordered, but they’ll always become so immediately before you give them a treat. This allows for the immediate ordering of an arbitrarily large system of dogs. Note that this subsequently forms the basis for proofs by induction. Furthermore, note that cats aren’t ordered at all, which is why all previous attempts to prove anything about cats inductively has failed. Therefore, dogs are on their way to implementing Zermelo set theory, while cats are driving us inevitably toward the entropic heat-death of the universe. Way to go, cats.

Proof by Induction:

Base Case: One dog loves you. One cat wants to know why you haven’t fed it yet.

Inductive Hypothesis: Assume k cats and dogs. From the above theorems, we can assume WLOG that k dogs all love you and are already best friends with each other, you, and a tennis ball they found. By the time you’ve read this, k cats have optimized their global spacing function; equidistant from each other, while simultaneously minimizing their distance to a can opener and something really practical to sit on. But we can’t do induction on cats, so who cares? Moving on.

Inductive Step: Assume k+1 dogs, and a new tennis ball. Pick up the new tennis ball. k+1 dogs now love you. Furthermore, at least k dogs are now well-ordered, k+1 assuming the new one didn’t mistakenly think you’d thrown the ball already. Note that the new tennis ball will be more than adequate distraction to pick up the old tennis ball. As a result of induction, you now have control over all dogs in the universe, and they all love you too. Who needs Machiavelli?

Alternate Inductive Step: Assume k+1 dogs. Pet the new dog. k+1 dogs now love you.

Proof by Similarity to Very Useful Data Structure:

Dogs are un-unembarrassingly parallel, barring their memory implementation. They are the definition of Union-Find, except the other way around. Actually, Union-Find-Union. Just throw a stick. Bam, proof by game of fetch. Cats just watch you, until you Find. Their food. For them. And then they glare at you until you die after which they go find someone else to pretend to love. Whamo, proven by non-lack-of-love-nor-lack-of-compassion.

Need I go on? Zing, proven by appeal to logic. Wait, what?

Dan

p.s. Begin with P = NP. Let N=1. ☐

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