Motivated by algorithmic information theory, the problem of program discovery can help find candidates of underlying generative mechanisms of natural and artificial phenomena. The uncomputability of such inverse problem, however, significantly restricts a wider application of exhaustive methods. Here we present a proof of concept of an approach based on IMP, a high-level imperative programming language. Its main advantage is that conceptually complex computational routines are more succinctly expressed, unlike lower-level models such as Turing machines or cellular automata. We investigate if a more expressive higher-level programming language can be more efficient at generating approximations to algorithmic complexity of recursive functions.

Is undecidability a requirement for open-ended evolution (OEE)? Using methods derived from algorithmic complexity theory, we propose robust computational definitions of open-ended evolution and the adaptability of computable dynamical systems. Within this framework, we show that decidability imposes absolute limits on the stable growth of complexity in computable dynamical systems. Conversely, systems that exhibit (strong) open-ended evolution must be undecidable, establishing undecidability as a requirement for such systems. Complexity is assessed in terms of three measures: sophistication, coarse sophistication, and busy beaver logical depth. These three complexity measures assign low complexity values to random (incompressible) objects. As time grows, the stated complexity measures allow for the existence of complex states during the evolution of a computable dynamical system. We show, however, that finding these states involves undecidable computations. We conjecture that for similar complexity measures that assign low complexity values, decidability imposes comparable limits on the stable growth of complexity, and that such behavior is necessary for nontrivial evolutionary systems. We show that the undecidability of adapted states imposes novel and unpredictable behavior on the individuals or populations being modeled. Such behavior is irreducible. Finally, we offer an example of a system, first proposed by Chaitin, that exhibits strong OEE.

Is undecidability a requirement for open-ended evolution (OEE)? Using algorithmic complexity theory methods, we propose robust computational definitions for open-ended evolution and adaptability of computable dynamical systems. Within this framework, we show that decidability imposes absolute limits to the growth of complexity on computable dynamical systems up to a logarithm of a logarithmic term. Conversely, systems that exhibit open-ended evolution must be undecidable, establishing undecidability as a requirement for such systems. Complexity is assessed in terms of three measures: sophistication, coarse sophistication and busy beaver logical depth. These three complexity measures assign low complexity values to random (incompressible) objects. We conjecture that, for similar complexity measures that assign low complexity values, decidability imposes comparable limits to the stable growth of complexity and such behaviour is necessary for non-trivial evolutionary systems. Finally, we show that undecidability of adapted states imposes novel and unpredictable behaviour on the individuals or population being modelled. Such behaviour is irreducible.