The earlier paper " Introduction to clarithmetic I " constructed an axiomatic system of arithmetic based on computability logic, and proved its soundness and extensional completeness with respect to polynomial time computability. The present paper elaborates three additional sound and complete systems in the same style and sense: one for polynomial space computability, one for elementary recurs...

In this survey, we discuss computability spectra of countable structures that provide a natural measure non-computability structure. This notion is main tool for investigating algorithmic properties structures. Along with review known results in field, present proofs some new to illustrate the method interpretation, which basic field. We also open questions.