共感で繋がるSNS
人気
ﾃﾞｲ
#AegisMaterialSupplier
꙳𓂅kaii.🌿𖠋ᐝ
Not enough
Not enough
Not enough
どれだけハードルを下げたつもりでも
Keeps repeating inside my head
No matter how dangerously exhausted
ハードルなんて下げたくないんだ
ハードル下げてまで生きたくない
満たされないまま呼吸をして
何故それを幸せと呼べようか
精神を癒す星
338
アストラ
We now state a basic structural theorem for \alpha-calculation.
Theorem 8.1. Existence of a Pareto frontier
Let \Lambda be a formal language environment such that:
\text{(i) all symbol and operation costs are strictly positive,}
\text{(ii) for each }M>0,\ \{E\in\mathrm{Expr}(\Lambda):\ell_{\mathrm{raw}}(E)\le M\}\text{ is finite,}
and
\text{(iii) }\mathrm{val}(E)\text{ is well-defined for all admissible expressions.}
Then for every target \tau\in\mathbb R, the set
\mathcal P_\tau
:=
\left\{
E\in\mathrm{Expr}(\Lambda)\; ;\;
\not\exists F\in\mathrm{Expr}(\Lambda)
\text{ with }
\ell(F)\le \ell(E),\
|\mathrm{val}(F)-\tau|\le |\mathrm{val}(E)-\tau|,
\right.
\left.
\text{and at least one inequality strict}
\right\}
is nonempty.
In particular, there exists at least one expression that is nondominated with respect to the two criteria of semantic length and approximation error.
Proof
By assumption, for each fixed L, there are only finitely many expressions with raw length at most L, hence only finitely many semantic equivalence classes with semantic length at most L. Therefore, at each admissible length level, the approximation error achieves a minimum.
Now choose a length L_0 for which at least one admissible expression exists. Among expressions of length L_0, let E_{L_0} minimize the error |\mathrm{val}(E)-\tau|. Then E_{L_0} is nondominated at length L_0. Repeating this construction for each admissible length level yields at least one nondominated expression. Hence the Pareto set \mathcal P_\tau is nonempty. ∎
1
ちび
I know I’m probably overthinking it. I also know that no matter how much I think about it, I’m not going to find an answer right away.
And yet, I kept thinking about the same thing over and over, repeating the same questions and answers in my head. Before I knew it, I was stuck in an endless loop of thoughts. Little by little, I began to feel more and more down.
I continued working while carrying those thoughts with me, and gradually, even my body began to feel heavy. The fatigue started to weigh on my legs and lower back. My mind and body fell out of balance, making it difficult to concentrate on work, and I began making more mistakes.
At that moment, what helped me was a casual remark from someone.
That one comment prompted me to take action. From there, things suddenly became very busy, and before I knew it, I no longer even had the time to think about what had been bothering me.
And now, strangely enough, I hardly think about it anymore, and I feel refreshed.
It was only a temporary experience, but now I find myself wondering:
“What was that all about, in the end?”
I’m not looking for anyone to understand the details of this situation.
I think I simply wanted someone to listen and say,
“Something like this happened to me.”
When you encounter ordinary, everyday events, what do you notice, and how do you look at things?
If you have an experience that makes you think, “Something similar happened to me,” I would love to hear about it.
Perhaps this feeling comes from nothing more than simple curiosity.
This turned into a long post, but thank you to everyone who read it all the way to the end (ㅅ´ ˘ `)
I look forward to hearing your stories.
#MyThoughts #SelfReflection #InnerJourney #FindingMyWay #KeepMovingForward #LifeGoesOn #BeTrueToYourself #OneStepAtATime #TrustTheJourney #StillOnMyWay
おまえさん
3