These notes present various priority arguments in classical computability theory, effective model theory, and complexity theory in a uniform style. Our notation usually follows Soare (1986) with some exceptions. We view Turing functionals as c.e. sets Φ of triples ⟨x, y, σ⟩, denoting that Φ(σ;x) ↓= y. (Of course, we have to impose the obvious compatibility condition, namely, that if ⟨x, y, σ⟩, ...

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We present a new complexity theoretic approach to real-time computations. We define timed ω-languages as a new formal model for such computations, that we believe to allow a unified treatment of all variants of real-time computations that are meaningful in practice. To our knowledge, such a practically meaningful formal definition does not exist at this time. In order to support our claim that ...

“The big secret of quantum mechanics is how simple it is once you take the physics out of it.” The course website is http://www.scottaaronson.com/qct2016/, and the syllabus is at http://www. scottaaronson.com/qct2016/syllabus-qct2016.pdf. We’ll be mostly following lecture notes found at http://www.scottaaronson.com/barbados-2016.pdf. This lecture’s goal is to acquaint the listener with the basi...

We give a short introduction to some questions in complexity theory and proceed to give some recent developments. In particular, we discuss probabilistically checkable proofs and their applications in establishing inapproximability results. In a traditional proof the proofchecker reads the entire proof and decides deterministically whether the proof is correct. In a probabilistically checkable ...