I am stuck on how to prove this question in NP-complete.

Ver = {(p, x, 1^t) : there exists a string y of length t such that program p on input (x, y) stops and outputs 1 within t time steps}

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# Tag: langauge

## Prove that a langauge is in NP-Complete

## Is it possible to bootstrap a transpiled langauge?

## Prove langauge is not context-free using pumping lemma

## Non-normal numbers definable without parameters in the langauge of differential rings with composition

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I am stuck on how to prove this question in NP-complete.

Ver = {(p, x, 1^t) : there exists a string y of length t such that program p on input (x, y) stops and outputs 1 within t time steps}

There’re multiple topics about bootstrapping a compiled language and the benefits of it, However, I can’t wrap my head around a bootstrapping a transpiled language. For example, Language X is transpiled into Javascript, and Language X traspiler/compiler is written in Language X but it always ends up being javascript before it runs… so is that still bootstrapping?

It’s so confusing specially thinking about what happens if the language syntax changes after it’s bootstrapped would the parser/traspiler need to be re-written?

I have the following alphabet: $ Σ = {0, 1, . . . , 9}$

and the Language $ L$ defined as: $ L = \{ abc | a + b = c\} $

where substrings $ a$ , $ b$ and $ c$ are interpreted as ordinary integers.

**My answer so far**:

Assume $ L$ is context-free. Then the pumping lemma for context-free languages applies to $ L$ .

Let $ n$ be the the constant given by the pumping lemma.

Let $ z=10^n20^n30^n$ clearly $ z \in L$ and $ |z| \geq n$

By the lemma we know that $ z = uvwxy$ with $ |vwx| \leq n$ and $ |vx| \geq 1$

There exists possibilities…

**My questions:**

I can see 8 possibilities where $ vwx$ can be within $ z$ . For example in the beginning including the 1 and overlapping with the initial $ 0^n$ . Another example the initial $ 0^n$ . Is this one way to think in this particular question? How can I pump and show that the result does not belong to $ L$ ?

**Thank you for your time.**

**Background:** It is currently unknown whether $ e$ is normal. A natural way to approach this question is to find a class to which $ e$ belongs, and prove all members of that class are normal. For example, if we want to know whether $ \sqrt{2}$ is normal, it makes sense to consider the class of irrational algebraic numbers, but it is still an open problem whether every irrational algebraic number is normal. Finding a counter-example to this conjecture, or a similar conjecture for an appropriate class containing $ e$ , would be quite useful to understanding the problem in general.

**Differential Rings with Composition:** One natural class containing $ e$ seems to be numbers definable without parameters in the language of differential rings with composition, say in the space of analytic functions on $ \mathbb{C}$ . The language of differential rings with composition is $ (0,1,+,*,\partial,\circ)$ , where $ 0$ , $ 1$ are the additive and multiplicative identities (constant functions), $ +$ and $ *$ are addition and multiplication, $ \partial$ is a derivation (in this case differentiation), and $ \circ$ is composition. We can define the constant function $ e$ in this language by the formula $ $ \psi(x) : (\partial f = f) \wedge (f \circ 0 = 1) \wedge (f \circ 1 = x)$ $

**Question:** Is there a non-normal irrational number definable without parameters in the ring of analytic functions on $ \mathbb{C}$ in the language of differential rings with composition?

This is a larger class of numbers than algebraic numbers, so in principle it should be an easier question to answer in the positive than the corresponding question of whether there is a non-normal irrational algbraic number, but I expect this is also quite a hard question, so answers slightly modifying the question would also be welcome. For example, perhaps it helps to generalize to algebraic elements in this differential ring with composition, rather than just definable elements, or to work in a different differential ring into which $ \mathbb{R}$ embeds.

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