Narrow Results

This paper is concerned with the study of possibility of performing changes to existing running programs with the use of the RAM and RASP models of computation. A new model of computation is defined with the capability of performing runtime changes. Theoretical properties, including time and space complexities, of the defined models are presented and proven. A number of simple empirical tests are conducted in order to prove the ability to perform runtime changes as well as support obtained theoretical results. The paper concludes that the defined model has virtually no affect on performance when there are no changes and the performance with changes is easily manageable. Moreover, the results can be used to develop runtime change capabilities for a wide range of programming languages and paradigms.

Consider orbits ${𝒪}(z,\kappa )$ of the fractal iterator $f_{\kappa }(z):=z^2+\kappa$, $\kappa \in ℂ$, that start at initial points $z\in {\widehat{K}}_{\kappa }^{(m)}\subset \widehat{ℂ}$, where $\widehat{ℂ}$ is the set of all rational complex numbers (their real and imaginary parts are rational) and ${\widehat{K}}_{\kappa }^{(m)}$ consists of all such $z$ whose complexity does not exceed some complexity parameter value $m$ (the complexity of $z$ is defined as the number of bits that suffice to describe the real and imaginary parts of $z$ in lowest form). The set ${\widehat{K}}_{\kappa }^{(m)}$ is a bounded-complexity approximation of the filled Julia set $K_{\kappa }$. We present a new perspective on fractals based on an analogy with Chaitin’s algorithmic information theory, where a rational complex number $z$ is the analog of a program $p$, an iterator $f_{\kappa }$ is analogous to a universal Turing machine $U$ which executes program $p$, and an unbounded orbit ${𝒪}(z,\kappa )$ is analogous to an execution of a program $p$ on $U$ that halts. We define a real number ${\mathrm{{\rm Y}}}_{\kappa }$ which resembles Chaitin’s $\mathrm{\Omega }$ number, where, instead of being based on all programs $p$ whose execution on $U$ halts, it is based on all rational complex numbers $z$ whose orbits under $f_{\kappa }$ are unbounded. Hence, similar to Chaitin’s $\mathrm{\Omega }$ number, ${\mathrm{{\rm Y}}}_{\kappa }$ acts as a theoretical limit or a “fractal oracle number” that provides an arbitrarily accurate complexity-based approximation of the filled Julia set $K_{\kappa }$. We present a procedure that, when given $m$ and $\kappa$, it uses ${\mathrm{{\rm Y}}}_{\kappa }$ to generate ${\widehat{K}}_{\kappa }^{(m)}$. Several numerical examples of sets that estimate ${\widehat{K}}_{\kappa }^{(m)}$ are presented.