WS 2016/2015 23.11.2016 Exercises to the lecture
Complexity Theory Sheet 5 Prof. Dr. Roland Meyer
Dr. Prakash Saivasan Delivery until 29.11.2016 at 10h
Exercise 5.1 (Completeness in L)
Let Σ be a finite alphabet. Prove the following two statements:
a) A language Aover Σ is in L if and only ifA≤logm {0,1}.
b) Any languageA inL that satisfiesA6=∅ andA6= Σ∗ is alreadyL-complete with respect to logspace-many-one reductions.
Exercise 5.2 (Acyclic reachability)
Show that we can reducePATH to ACYCLICPATH with respect to logspace-many-one reductions. Conclude thatACYCLICPATH is NL-complete.
Exercise 5.3 (Reducing ACYCLICPATH to 2SAT)
LetG be an acyclic graph ands andt vertices ofG. We construct a formula F in CNF as follows: for any edgex→y, we add a clause (¬x∨y). Moreover, we add the clauses (s) and (¬t). Show the following:
F is satisfiable ⇔ there is no path from stotinG.
Exercise 5.4 (Counter automata)
Let Σ be a finite alphabet and A ank-counter two-way automaton over Σ.
a) The counters ofAmay take values inZ. Construct ank0-counter two-way automaton A0 such that:
• A0 simulatesA, and
• the counters ofA0 only take values inN.
b) Assume thatA has linearly bounded semantics and that the counters can only take values in N. Construct ak0-head two-way finite automaton B that simulatesA.
Delivery until 29.11.2016 at 10h into the box next to room 343 in the Institute for Theoretical Computer Science, Muehlenpfordstrasse 22-23
