§3 Advanced Computability

1. WHILE programs UTM Theorem Normalform Theorem SMN Theorem / Currying (Schönfinkeling) Fixedpoint Theorem, Quines Oracle WHILE programs Post‘s Problem / Friedberg&Muchnik Arithmetic Hierarchy §3 Advanced Computability

2. WHILE Programs Syntax in Backus—Naur Form: P := ( xj := 0 | xj := xi+ 1 | P ; P | LOOP xj DO P END | WHILE xj DO P END ) body bettermodify xj Semantics: loop executed as long asxj≠0 Observation: a) To every LOOP program P there is an equivalent WHILE program P'without LOOPs. b) As opposed to LOOP programs, WHILE programs have undecidable Halting Problem. Rado’s Corollary: WHILE programs do not admit a bound t(P,n) such that P on input xNk either at most t(P,||x||1) steps or runs indefinitely.

3. First UTM Theorem UTM-Theorem: There exists a LOOP program U' that, given PN and x1,…,xkNand NN, simulates P on input (x1,…,xk) for N steps. Proof (Sketch): Use one variable y for x1,…,xk, and z to store the current program counter of P: Switch/case P[z] of: „xj:=0“ : x1,…xj,…,xk := x1,…0,…,xk ; z:=z+1 „xi:=xj+1“ : x1,…xi,…xk := x1,…xj+1…xk ; z:=z+1 „WHILE xj DO“ : if xj=0 then z:=1+#of corresponding END „END“ : z := line# of corresponding WHILE Definition: Let PN denote the encoding of WHILE program P (e.g. as ascii sequence).

4. Normalform Theorem UTM-Theorem: There exists a LOOP program U' that, given PN and x1,…,xkNand NN, simulates P on input (x1,…,xk) for N steps. Normalform-Thm: To every WHILE program P there exists an equivalent one P' containing only one WHILE command (and several LOOPs). Corollary: A WHILE program U can semi-decide the Halting problem for WHILE programs. H = { (P,x1,…xk) : P terminates on input (x1…xk) }

5. SMN Theorem: Currying Definition: Let C = PN denote the encoding of WHILE program P , P = C its inverse/decoding. Type conversion examplef(x,y) = sin(x)·ey SMN-Theorem: There exists a WHILE program that, given PN and xN, returns P(x, · ),where P(x, · )(y) :P(x,y) UTM-Theorem: There is a WHILE program that, given PN, returns QN with Q(x,y) = P(x) (y) WHILE program that, given P,Q, returns Q◦P

6. Fixedpoint Theorem and Quines Def: For partial functions f,g:N→N write fgto mean dom(f)=dom(g) and xdom: f(x)=g(x). x y :x  y Theorem: Every total computable function φ:N→Nhas a „semantic fixedpoint“, i.e. xN s.t. φ(x) x. Proof: Let x := ψ(φ◦ψ), where ψ(y) := y (y) ψ'(y) := z →(y) (z) semantic extensionof (y) partial! defined? ' Application (Quines): Let A=A(p,y) be a program. Consider "fixedpoint" Pof (p) :=A(p, · ). NP = code of program P, C = program with code CN

7. Oracle WHILE programs xj := (xi) | P := ( xj := 0 | xj := xi+ 1 | P ; P | LOOP xj DO P END | WHILE xj DO P END ) Fix some arbitrary total :N→N Examples:▪  := χP characteristic function of Primality Probl. ▪  := χH characteristic function of Halting Problem ▪  := χT characteristic function of Totality Problem cmp. Cardinality…(Cantor–Schröder–Bernstein) χP≼ χH ≡χH ≼ χT For , N write ≼ if there is a WHILE program with oracle  computing . For L,L' N write L≼L' if there is a computable f: N N such that x: xL  f(x)L'. a) L' semi-/decidable  so L. b) L≼L'≼L''  L≼L'' b) L≼L'≼L''  L≼L'' a)  computable  so  b) ≼≼χ  ≼ χ

8. Arithmetic Hierarchy : P := ( xj := 0 | xj := xi+ 1 | P ; P | LOOP xj DO P END | WHILE xj DO P END ) Def:Δ1 =Σ0 =Π0= decidable Δk+1 = decidableΣk = decidableΠk Σk+1 = semi-decidableΣk Πk+1 = co-semi-decidableΣk Δ3 Σ2Π2 Fix some arbitrary total :N→N Σ2 Π2 Δ2 Σ1 Π1 Lemma: a) Δk= co-Δk b) Δk = Σk  Πk c) Σk  Πk Δk+1 Π1 Σ1 Δ1