Levels of Undecidability in Infinitary Rewriting: Normalization and Reachability

In [2] it has been shown that infinitary strong normalization (SN∞) is Π11-complete. Suprisingly, it turns out that infinitary weak normalization (WN∞) is a harder problem, being Π12-complete, and thereby strictly higher in the analytical hierarchy. We assume familiarity with infinitary term rewriting; we further reading we refer to [5,4]. 1 Infinitary Strong Normalization and Reachability Definition 1. A Turing machine M is a quadruple 〈Q,Γ , q0, δ〉 consisting of: – finite set of states Q, – an initial state q0 ∈ Q, – a finite alphabet Γ containing a designated symbol 2, called blank, and – a partial transition function δ : Q× Γ → Q× Γ × {L,R}. A configuration of a Turing machine is a pair 〈q, tape〉 consisting of a state q ∈ Q and the tape content tape : Z → Γ such that the carrier {n ∈ Z | tape(n) 6= 2} is finite. The set of all configurations is denoted Conf M. We define the relation →M on the set of configurations Conf M as follows: 〈q, tape〉 →M 〈q , tape 〉 whenever: – δ(q, tape(0)) = 〈q, f , L〉, tape (1) = f and ∀n 6= 0. tape (n+1) = tape(n), or – δ(q, tape(0)) = 〈q, f ,R〉, tape (−1) = f and ∀n 6= 0. tape (n− 1) = tape(n). Without loss of generality we assume that Q ∩ Γ = ∅, that is, the set of states and the alphabet are disjoint. This enables us to denote configurations as 〈w1, q, w2〉, denoted w −1 1 qw2 for short, with w1, w2 ∈ Γ ∗ and q ∈ Q, which is shorthand for 〈q, tape〉 where tape(n) = w2(n + 1) for 0 ≤ n < |w2|, and tape(−n) = w1(n) for 1 ≤ n ≤ |w1| and tape(n) = 2 for all other positions n ∈ Z. The Turing machines we consider are deterministic. As a consequence, final states are unique (if they exist), which justifies the following definition. Definition 2. Let M be a Turing machine and 〈q, tape〉 ∈ Conf M. We denote by finalM(〈q, tape〉) the →M-normal form of 〈q, tape〉 if it exists and undefined, otherwise. Whenever finalM(〈q, tape〉) exists then we say thatM halts on 〈q, tape〉 with final configuration finalM(〈q, tape〉). Furthermore we say M halts on tape as shorthand for M halts on 〈q0, tape〉. Turing machines can compute n-ary functions f : N → N or relations S ⊆ N . We need only unary functions fM and binary >M ⊆ N× N relations. Definition 3. Let M = 〈Q,Γ , q0, δ〉 be a Turing machine with S, 0 ∈ Γ . We define a partial function fM : N ⇀ N for all n ∈ N by: