Previous work has shown that perturbation analysis in algorithmic information dynamics can uncover generative causal processes of finite objects and quantify each of its element's information contribution to computably constructing the objects. One of the challenges for defining emergence is that the dependency on the observer's previous knowledge may cause a phenomenon to present itself as emergent for one observer at the same time that reducible for another observer. Thus, in order to quantify emergence of algorithmic information in computable generative processes, perturbation analyses may inherit such a problem of the dependency on the observer's previous formal knowledge. In this sense, by formalizing the act of observing as mutual perturbations, the emergence of algorithmic information becomes invariant, minimal, and robust to information costs and distortions, while it indeed depends on the observer. Then, we demonstrate that the unbounded increase of emergent algorithmic information implies asymptotically observer-independent emergence, which eventually overcomes any formal theory that any observer might devise. In addition, we discuss weak and strong emergence and analyze the concepts of observer-dependent emergence and asymptotically observer-independent emergence found in previous definitions and models in the literature of deterministic dynamical and computable systems.

Economic theory states that incineration and landfill taxation can effectively diminish the environmental impacts of pollution and resource use by reducing their associated pollutants while stimulating the reuse and recycling of materials, and therefore, fostering a circular economy. The aim of this research is to assess the economic and environmental effects of these taxes in Spain under different scenarios with a detailed dynamic computable general equilibrium (CGE) model, as there are no studies analyzing this in detail. We focus on the economic impact on GDP and sectorial production and the environmental impact on different categories: global warming potential, marine eutrophication potential, photochemical ozone formation potential, particulate matter, human toxicity (cancer and noncancer), ecotoxicity, and depletion of fossil resources. We find in all scenarios that these taxes have a limited economic impact while reducing all of the environmental impact categories analyzed. The study reinforces the theory that policy makers need to impose taxes on landfill and incineration to reinforce the circularity of the economy and reduce environmental burdens, but also demonstrates that they can improve their design without additional costs.

The COVID-19 pandemic has not only affected public health but has also significantly impacted the economy. Bandung, a bustling city in Indonesia serving as a satellite to the capital, has been hit hard due to its high population density, mobility, and reliance on the tourism, trade, and transportation sectors. Using a Computable General Equilibrium (CGE) model developed at the inter-regional level of Indonesia, this study investigates the microeconomic indicators of several economic activities in Bandung to assess the impact of the pandemic. Additionally, the study examines the role of private sector actors in contributing to the sustainable recovery efforts toward achieving Sustainable Development Goals (SDGs) amidst the pandemic. The findings reveal that Bandung's transportation, accommodation, food and beverage, water supply, and trade sectors experienced a significant decline in economic activity. However, there was a gradual recovery with increased economic activity between 2019 and 2021. Private sector actors and the health sector were the main drivers of economic recovery, with other sectors also contributing to the efforts.

Let  β= (((24!)!)!)!, and let {${\mathcal P}_{n^2+1}$} denote the set of all primes of the form {$n^2+1$}. Let ${\mathcal M}$ denote the set of all positive multiples of elements of the set {${\mathcal P}_{n^2+1} \cap (\beta,\infty)$}. The set {${\mathcal X}=\{0,\ldots,\beta\} \cup {\mathcal M}$} satisfies the following conditions: (1) ${\rm card}({\mathcal X})$ is greater than a huge positive integer and it is conjectured that ${\mathcal X}$ is infinite, (2)} we do not know any algorithm deciding the finiteness of ${\mathcal X}$, (3)~a~known and short algorithm for every {$n \in N$} decides whether or not {$n \in {\mathcal X}$}, (4) a known and short algorithm returns an integer~$n$ such that ${\mathcal X}$ is infinite if and only if ${\mathcal X}$ contains an element greater than $n$. The following problem is open: {\em simply define a set {${\mathcal X} \subseteq N$} such that ${\mathcal X}$ satisfies conditions (1)-(4), and \underline{we do not know any representation of~~${\mathcal X}$~~as a finite union of sets whose definitions are simpler than} \underline{the definition of ${\mathcal X}$}} {\tt (5)}. Let {$f(1)=2$}, {$f(2)=4$}, and let {$f(n+1)=f(n)!$} for every integer {$n \geqslant 2$}. For a positive integer $n$, let {$\Psi_n$} denote the following statement: {\em if a system of equations ${\mathcal S} \subseteq \Bigl\{x_i!=x_k: i,k \in \{1,\ldots,n\}\Bigr\} \cup \Bigl\{x_i \cdot x_j=x_k: i,j,k \in \{1,\ldots,n\}\Bigr\}$ has only finitely many solutions in positive integers \mbox{$x_1,\ldots,x_n$}, then each such solution \mbox{$(x_1,\ldots,x_n)$} satisfies {$x_1,\ldots,x_n \leqslant f(n)$}.} We prove that for every statement $\Psi_n$ the bound {$f(n)$} cannot be decreased. The author's guess is that the statements {$\Psi_1,\ldots,\Psi_9$} are true. We prove that the statement $\Psi_9$ implies that the set ${\mathcal X}$ of all {non-negative} integers $k$ whose number of digits belongs to {${\mathcal P}_{n^2+1}$} satisfies conditions {\tt (1)-(5)}.

Let $f\left(1\right)=1$ , and let $f(n+1)=2^{2^{f\left(n\right)}}$ for every positive integer n. We consider the following hypothesis: if a system S ⊆ {x_i · x_j = x_k : i, j, k ∈ {1, . . . , n}} ∪ {x_i + 1 = x_k : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x₁, . . . , x_n, then each such solution (x₁, . . . , x_n) satisfies x₁, . . . , x_n ≤ f (2n). We prove: (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set $\mathfrak{M}$ ⊆ has a finite-fold Diophantine representation, then $\mathfrak{M}$ is computable.