Universität Koblenz-Landau FB 4 Informatik
Prof. Dr. Viorica Sofronie-Stokkermans∗1 08.11.2017
M.Ed. Dennis Peuter∗2
Exercises for Advances in Theoretical Computer Science Exercise Sheet 3
Due at 13.11.2017, 10:00 s.t.
Exercise 3.1
LetL1,L2,L3 be languages, where L2 is recursively enumerable andL3 is decidable.
Prove or refute the following statements:
I) IfL1⊆L3, thenL1 is decidable.
II) IfL3⊆L1, thenL1 is decidable.
III) IfL1⊆L2, thenL1 is recursively enumerable.
IV) IfL2⊆L1, thenL1 is recursively enumerable.
Exercise 3.2
Are the following problems decidable or undecidable? Justify your answer.
I) P1:={n∈N| Mn does not hold on empty input} II) P2:={n∈N|L(Mn) =∅}
III) P3:={(m, n)∈N×N|L(Mm)∩L(Mn) =∅}
IV) P4:={(m, n)∈N×N|L(Mm)⊆L(Mn)}
V) P5:={(n, w)∈N×Σ∗|For input w,Mn does not reach another conguration after s,#w#where the head is on a blank (#)}
VI) P6:={(n, w, s)∈N×Σ∗×N| Mn halts on input wafter at most ssteps } Remark:
• Mn denotes the Turing machine with Gödel numbern.
• L(M) is the language accepted by the Turing machine M (i.e. the set of all words accepted by M).
Hint: To prove undecidability you can for instance use properties of decidable languages (e.g. the fact that the complement of a decidable language is decidable) or a reduction to a problem which was already proved to be undecidable: (1) You are allowed to use the undecidability of the halting problem HALT, of H0 or ofK (notation as in the lecture); (2) if you have proven the (un-)decidability of Pi, you may use this result for any of the next tasks.
∗1 B 225 sofronie@uni-koblenz.de https://userpages.uni-koblenz.de/~sofronie/
∗2 B 223 dpeuter@uni-koblenz.de https://userpages.uni-koblenz.de/~dpeuter/
If you want to submit solutions, please do so until 13.11.2017, 10:00 s.t. via the cardboard box in the shelf in room B 222 or via e-mail (with Homework ACTCS in the subject) to dpeuter@uni-koblenz.de.
