Priority Arguments in Computability Theory, Model Theory, and Complexity Theory

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⟩, ⟨x, z, τ⟩ ∈ Φ where σ and τ are comparable then y = z. It is not hard to check that any given Turing program can be effectively transformed into a computable enumeration of such triples coding the corresponding partial computable Turing functional, and that conversely every such c.e. set of triples can be effectively transformed into a Turing program for the corresponding partial computable Turing functional.) The use of a computation Φ(X;x) is the largest number actually used in that computation, i.e., in the notation of the previous paragraph, the use equals |σ| − 1. Allowing a minor abuse of notation in the case when the oracle is given as the join of two or more sets, we then let the use be the largest number used on any set involved in the join. (Thus Φ((X1 ⊕ X2) (u + 1);x) is defined iff Φ((X1 (u + 1) ⊕ (X2 (u + 1));x) is.) We denote the use of a computation Φ(X;x) by the corresponding lower-case letter (i.e., φ(X;x), or simply φ(x) when the oracle is clear from the context). We adopt the following