You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
The desire to use reinforcement learning in safety-critical settings has inspired a recent interest in formal methods for learning algorithms. Existing formal methods for learning and optimization primarily consider the problem of constrained learning or constrained optimization. Given a single correct model and associated safety constraint, these approaches guarantee efficient learning while provably avoiding behaviors outside the safety constraint. Acting well given an accurate environmental model is an important pre-requisite for safe learning, but is ultimately insufficient for systems that operate in complex heterogeneous environments. This paper introduces verification-preserving model updates, the first approach toward obtaining formal safety guarantees for reinforcement learning in settings where multiple possible environmental models must be taken into account. Through a combination of inductive data and deductive proving with design-time model updates and runtime model falsification, we provide a first approach toward obtaining formal safety proofs for autonomous systems acting in heterogeneous environments.

Cited Abstract(s):
[1]: This paper describes GenProg, an automated method for repairing defects in off-the-shelf, legacy programs without formal specifications, program annotations, or special coding practices. GenProg uses an extended form of genetic programming to evolve a program variant that retains required functionality but is not susceptible to a given defect, using existing test suites to encode both the defect and required functionality. Structural differencing algorithms and delta debugging reduce the difference between this variant and the original program to a minimal repair. We describe the algorithm and report experimental results of its success on 16 programs totaling 1.25 M lines of C code and 120K lines of module code, spanning eight classes of defects, in 357 seconds, on average. We analyze the generated repairs qualitatively and quantitatively to demonstrate that the process efficiently produces evolved programs that repair the defect, are not fragile input memorizations, and do not lead to serious degradation in functionality.

[2]: Inductive programming (IP)—the use of inductive reasoning methods for programming, algorithm design, and software development—is a currently emerging research field. A major subfield is inductive program synthesis, the (semi-)automatic construction of programs from exemplary behavior. Inductive program synthesis is not a unified research field until today but scattered over several different established research fields such as machine learning, inductive logic programming, genetic programming, and functional programming. This impedes an exchange of theory and techniques and, as a consequence, a progress of inductive programming. In this paper we survey theoretical results and methods of inductive program synthesis that have been developed in different research fields until today.

[3]: This work presents a novel approach for automatically repairing an erroneous program with respect to a given set of assertions. Programs are repaired using a predefined set of mutations. We refer to a bounded notion of correctness, even though, for a large enough bound all returned programs are fully correct. To ensure no changes are made to the original program unless necessary, if a program can be repaired by applying a set of mutations Mut, then no superset of Mut is later considered. Programs are checked in increasing number of mutations, and every minimal repaired program is returned as soon as found.
Our approach includes a model synthesis phase that is closely related to program synthesis and program repair algorithms [2] [1] [3] . Relative to work on program synthesis and repair, VPMUs are unique in several ways. We are the first to explore program repair. Our approach combines program verification with mutation. We treat programs as in which one part of the model is varied according to interactions with the environment and another part of the model is systematically derived (together with a correctness proof) from these changes. This separation of the dynamics into inductively synthesized models and deductively synthesized controllers enables our approach toward using programs as representations of dynamic safety constraints during reinforcement learning.