1. I. Recap on the discrete Theory of Computation
un-/computability, Halting Problem
oracle computation
Asymptotic runtime and memory
Machine models: unit vs. bit cost
L, P, NP1, NP, #P, PSPACE, EXP
Reduction and completeness
Parameterized complexity

2. simulator/interpreter B ?
Alan M. Turing 1936
1941
first scientific calculations on digital computers
What are its fundamental limitations?
Uncountably many PN
but countably many 'algorithms'
Undecidable Halting Problem H: No algorithm B can always correctly answer the following question:
Given A,x, does algorithm A terminate on input x?
Proof by contradiction: Consider algorithm B' that,
on input A, executes B on A,A  B' A
decision problem B' A B +  B' A x B'
Halting Problem H
simulator/interpreter B ?
Proof by contradiction:
Logicians Tarski, Alonzo Church (PhD advisor)
Kurt Gödel (1931): There exist arithmetical statements which are true but cannot be proven so.
and, upon a positive
answer, loops infinitely.
How does B' behave on B' ?

3. Un-/Semi-/Decidability I
Definition: a) An 'algorithm' A computes a partial function f:{0,1}*  {0,1}* if it
on inputs xdom(f) prints f(x) and terminates,
on inputs xdom(f) does not terminate.
Injective string pairing function ("Hilbert Hotel")
x1,…,xn ; y1,…,ym := 0 x1 0 x2 0 … 0 xn 1 y1 … ym
b) A decides set L{0,1}* if it computes its total char. function: cfL(x):=1 for xL, cfL(x):=0 for xL.
c) A semi-decides L if terminates precisely on xL
d) A enumerates L if it computes some total bijection f:{0,1}*=N  L.

4. Un-/Semi-/Decidability II
Example: The Halting problem H, considered as subset of {0,1}*, is semi-decidable, not decidable.
Theorem: a) Every finite L is decidable.
b) L is decidable iff its complement L is.
c) L is decidable iff both L, L are semi-decidable.
d) L is enumerable iff infinite and semi-decidable.
b) A decides set L{0,1}* if it computes its total char. function: cfL(x):=1 for xL, cfL(x):=0 for xL.
c) A semi-decides L if terminates precisely on xL
d) A enumerates L if it computes some total bijection f:{0,1}*=N  L.

5. Comparing Decision Problems
Halting problem H = { A,x : A(x) terminates }
Nontriviality N = { A : y A(y) terminates }
Totality problem T = { A : z A(z) terminates}
H ≼ N
H ≼ T
N ≼ H ≼ H
unde-
cidable
{0,1}*
{0,1}* f
unde-
cidable L L' L L'
H ≼ T  T ≼ H
For L,L'{0,1}* write L≼L' if there is a computable
f:{0,1}*{0,1}* such that x: xL  f(x)L'.
a) L' semi-/decidable  so L.
b) L≼L'≼L''  L≼L''

6. Examples of Undecidability
Universes U other than {0,1}* (e.g. N): encode.
Halting problem: H = { A,x : A terminates on x }
Hilbert's 10th: The following set is undecidable:
{ p | pN[X1,…Xn], nN, x1…xnN p(x1,…xn)=0 }
Word Problem for finitely presented groups
Mortality Problem for two 2121 matrices
Homeomorphy of two finite simplicial complexes
For L,L'{0,1}* write L≼L' if there is a computable
f:{0,1}*{0,1}* such that x: xL  f(x)L'.
a) L' semi-/decidable  so L.
b) L≼L'≼L''  L≼L''

7. Exercise Questions is it a syntactically correct expression?
Which of the following are un-/semi-/decidable?
a) Given an integer, is it a prime number?
b) Given a finite string over +,,(,),0,1,X1,…Xn
is it a syntactically correct expression?
c) Given a Boolean expression (X1,…Xn),
does it have a satisfying assignment?
d) Given MZnn and bZn,
does there exists a vector x s.t. M·xb?
e) Given an algorithm A, input x, and integer N,
does A terminate on input x within N steps ?
f) Does a given algorithm terminate on all inputs?
g) Does given algorithm terminate on some input?
real
integer

8. Bit Model: Turing Machine
Alan M. Turing [1936/37]
Mathematical idealization and abstraction of his assistents (profess. „computers“)
unbounded tape
one bit (0/1) in each cell
initially: input + delimit.
finite program
one read/write/move operation per step
until stop: output.
MO computes f:{0,1}*{0,1}* in #steps using #cells O

9. Bit Model: Turing Machine
M = (Q,q0,q+,q-,), where Q is a finite set of states,
:{0,1}Q{L,R}{0,1}Q transition table,
q0Q initial, q+Q accepting, q-Q rejecting state.
A configuration of M is a triple (v,q,w), with qQ
and v,w{0,1}*. The initial configuration on input w is ( ,q0,w). A successor configuration to (v, q, b w) is • (v b', p, w) for (b,q)=(R,b',p), • (b,q)=(L,b',p)
Shoenfield's Limit Lemma: A partial f:N→N
is computable relative to H iff f(n)=limj g(n,j)
for some total (oracle-free) computable g:NN→N.