Low (computability)

Low, in computability theory, refers to a class of recursively enumerable (r.e.) sets. A set A is said to be low if its jump, A', is Turing reducible to the halting problem (∅'). In other words, A' ≤_T ∅'. This means that the information about the halting problem is sufficient to determine the halting problem for all Turing machines with an oracle for A. The class of low sets is denoted as Low.

The concept of low sets is significant because it identifies r.e. sets that are computationally "weak" in a specific sense. While they are still r.e., meaning their members can be enumerated, they do not provide much additional computational power beyond what's already available with the halting problem. Their jump, representing the complexity of the halting problem relative to A, doesn't surpass the inherent difficulty of the halting problem itself.

Low sets possess several important properties, often explored in the context of relative computability and degree theory. Their study provides insights into the structure of the r.e. sets and helps to categorize the different levels of unsolvability within the r.e. degrees. The relationships between low sets and other classes of r.e. sets, such as high sets (whose jumps are computationally "strong"), are crucial aspects of computability theory research. Furthermore, investigations into low sets contribute to understanding the limitations of computation and the relative complexity of decision problems. Research continues to refine our understanding of the properties and implications of low sets within the broader landscape of computability theory.