Universität Koblenz-Landau FB 4 Informatik
Prof. Dr. Viorica Sofronie-Stokkermans∗1 09.01.2020
M.Ed. Dennis Peuter∗2
Exercises for Advances in Theoretical Computer Science Exercise Sheet 10
Due at 13.01.2020, 10:00 s.t.
Exercise 10.1
LetS be a set of states (including the halting state). Are the following problems decidable or undecidable? Justify your answer.
I) P1:={(n, q)∈N×S|there exists a start conguration (s,#w#) ofMn from which state q is reachable}
II) P2:={(n, w)∈N×Σ∗ |there existsq ∈K which does not occur in the computation of Mn= (K,Σ, δ, s) with start conguration (s,#w#)}
Remark:
• Mn denotes the Turing machine with Gödel numbern.
Hint: To prove undecidability you can for instance use a reduction to a problem which was already proved to be undecidable in the lecture or a previous exercise.
Exercise 10.2
Prove that it is undecidable whether a WHILE program which computes a partial function f :N→Nterminates on input n.
Hint: One can give e.g. a proof by contradiction using the fact that the class of WHILE- computable functions coincides with the class of T M-computable functions.
Exercise 10.3
Prove that the following problems are undecidable using the theorem of Rice.
I) L1 ={n|Mn accepts an innite language} II) L2 ={n|Mn accepts a nite language} III) L3 ={n|Mn accepts a decidable language}
IV) Letk∈N andL4 ={n|Mn accepts only words which have length greater thank}
V) L5 ={n|L(Mn)is context sensitive }
VI) L6 ={n|the language accepted byMn is regular} VII) L7 ={n|Mn halts on all inputsw∈Σ∗}
∗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 13.01.2020, 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.
